So, how did that go for you at the end of the game there, Mark? Yeah, it was an unbelievable feeling to finish with a penalty shoot-out.  You know, you're always going to make a few saves and thankfully you came in the penalty shoot-out. Do you practice them much? You've got all three of them right? Well, as we're saying, we do a bit after training sometimes. We have a bit of fun with it. Now Mossy, my goalkeeper and goalkeeper as well, he helped me do it.  And a lot of us amateur goalkeepers will know that it's one thing to get there. It's nothing to save there. A strong risk could save each one of them. Yeah, you just got to trust your instincts as well.  Thankfully kept them out. And then at the end of the game, we could hear Travis on fire, I believe, in the dressing room, what was all that about and how did you enjoy that? Yeah, it was a bit of fun, to be fair. Yeah, we did it last week against Aston Villa as well. It was a bit of fun after the game. I think it's always important to enjoy these moments. And I'm sure any top-keeper will tell you that it's about having that consistency. It's about being concentrated right to the very end. And again, that didn't necessarily have a lot to do. You need to be ready and ready at those right times. Definitely, that's the thing about these cool competitions. You've got to stay focused throughout the game. It was great to get back out on the pitch today. Get more minutes into me. I know, as you say, stay focused right to the end. I was able to do that, thankfully. And you played for the under-21s recently. How important was that in terms of building you up and getting you ready for today? Yeah, of course. It was good to get that game in last week. It gave me that confidence coming into today. Me and Mossy spoke about it actually before. It gave me that boost coming in. So, yeah, it's always important to get minutes. And we've seen, you know, we've got a really strong stable of goalkeepers here. What's it like being a part of those and trying to make sure you're the one that rises to the top? Yeah, definitely. It's a great competition. Obviously, Rammer's in goal at the minute. He's doing very well. So, yeah, just keep pushing each other on every day. Big man and Asmire as well. We work well together. You know, we enjoy these moments together. Even in the prime games when Rammer pulls off great saves, we enjoy it as well. So, yeah, we just got to keep pushing each other on. It's great to work with them. Thanks to your penalty saves. We've got another game in the Carmel Cup this season. I'm sure we'll be hoping that you're selected and you get the opportunity to show your best in that as well. Yeah, definitely. You know, it's always good to be in the competitions, especially the cup ones. So, something to aim for, something to keep working for. And look forward to that when it comes up.


Hello, everyone, and thank you for watching this video. I'm Jia Mincui, and today I will briefly introduce our work named on the field-based division property, applications to Mimise, Festum Mimise, and Gymimise, and it is a joint work with Kaihu, Meiqinwang, and Puwenwei. The main topic is symmetric key primitives that are efficient in several applications like multi-party computation, fully homomorphic encryption, and zero knowledge proof. In this case, the designers need to reconsider which operations are costly and which cost almost free, and it is quite different from usual symmetric primitives which are defined on binary fields. New cost metrics like low number of multiplications and low multiplicative depths are designed. These primitives usually work over large binary extension fields, F2 to the N, or large point fields. Since the size of the field is large, the S-box cannot be pre-compute or pre-stored, as in the traditional one. So they have simple algebraic representations like parmaps, x maps to x to the N, or the inverse of x. As a result, these new primitives are often algebraically simple. Since these primitives can be represented by simple algebraic representations, they are vulnerable to algebraic attacks naturally. And attacks like Gropner-Basey's attack and high-order differential attacks have shown powerful against these primitives. We have Gropner-Basey's attack on Javas and Friday, and high-order differential attacks on 4-round Mimici and Mimici. So a natural problem is how can we study the algebraic representation of the cipher? In this paper, we focus on the polynomial representation of ciphers over F2 to the N. First, we propose general monomial prediction, and this technique is used to detect the algebraic representation of field-based ciphers by tracing the so-called general monomial trials. It is an extension of division property over binary field. By using these two, we propose a new algorithm for degree evaluation based on the link between binary field and binary extension field, and we get tighter algebraic degree bounds. And we can no longer only rely on the theoretic proof to estimate the algebraic degree over finite fields. We analyze the security of Mimici, faster Mimici, and G-Mimici, and present the best high-order differential distinguisher and zero-sum distinguisher for different instances to guarantee the security level. Here is the result of our algorithm. For Mimici, we present the exact algebraic degree when D is equal to 3, the same as the theoretical proof in the recent paper. For faster Mimici, the previos only attack for 83 rounds, but we can attack for 124 rounds, and we can also give a 4-round knowing key zero-sum distinguisher. For Mimici, we search for a 50-round high-order distinguisher for the instance with block size 33 and 8 branches. This is the short summarization of our paper, and thank you for listening.


Thank you so much Flavia for the welcome. Today I wanted to talk about how we can use critical pedagogy to help us in our grammar teaching. I really didn't want to start this way when I thought about creating a presentation a few weeks ago but in light of recent events in the country I think that it's so important now more than ever to think about any kind of bias that we have towards our students of color within the curriculum and think about how we can start noticing those biases and correcting them so that we can teach all of our students better. So that's that's how I think I'll frame this today. The examples that I'll give you about using critical pedagogy are things that I use in a grammar class at St. Edwards. Most of the students in that class are heritage learners but I do have a few second language learners and every now and then even a few native speakers that are exchange students or who have recently arrived. So I think that these are strategies that work really well for a variety of different kinds of learners regardless of the kinds of experiences that they come from. So to go ahead and get started. If you haven't opened your chat yet please do. I would like for you to jot down just a few words about what comes to mind when you think about teaching grammar. These can be just little phrases or they can be longer beefs. If you will. Rules. Yes. Definitely right and wrong. That's great. And they really don't like the topic I do. Okay well I definitely think that rules and in structure yes logic patterns that are sometimes overwhelming all of those are things that happen when we teach grammar. I would also then like you to think about what your heritage learners would say about their own knowledge of grammar. I do think your students would say that they know about grammar. I like that they don't know not much. Definitely when I start a grammar class. I usually have students on the first day or sometimes even before the first day come up and say, Oh I don't think I can take this class because I don't know that kind of Spanish, right. I don't know grammar in reality that's that's not actually true our heritage learners know so much about grammar, but they might know it implicitly. A few examples, since I noticed that there were a couple of grad students in here would be. I noticed that heritage learners are really good at subject verb agreement, even when it's implicit knowledge. One example of grammar that they know really well is that they're super accurate on now an adjective agreement, much more than than second language learners, especially when it's not just an or an ending. And another thing that this is not research based but I noticed that heritage learners are really really good at interpreting a sentence when it departs from the typical subject verb object order. So if I give a sentence, it has happened recently something like la llamada la hizo Maria. Second language learners often go wait a minute so so somebody made a call to Maria, and they're not quite sure and then the heritage learners go no no no Maria hizo la llamada. And so they're really good about implicitly understanding in and out so grammar that second language learners don't yet know about. So when our students say that they don't know grammar. I think what they mean is that they don't have the meta linguistic knowledge to talk about all of the grammar that they already know. Yes, like complex subordination that our second language students get super overwhelmed with. I think there's this sort of ideological tension with teaching grammar, you guys definitely mentioned it in the chat of the kinds of competing desires that that we have when we teach grammar to heritage learners. So I put these two different ideas and tension in different colors. So on the one hand or the green side. We, we definitely want to teach students about structures in standard Spanish. If our goal is to help them communicate confidently and accurately in more and more domains in Spanish outside the home. Obviously we have to teach them about structures that they're not currently using in standard Spanish.  We need to teach them about the things that they're not yet doing. But at the same time, in the purple side or the on the other hand, we definitely don't want to privilege standard Spanish at the expense of our students home language practices. I think it's not only an example of linguistic violence, but I think it's also not great pedagogy to make our students think that everything that they already know and do in Spanish is not useful. So, today I have I brought some quotes from an article Flavia if you want to put this article in the chat this is the language Randolph article. You might not have access to it but you could at least read the abstract. But with this article is about high school heritage Spanish teachers. And we see that even when heritage teachers have learned about heritage students and worked with them for a while. When they report enjoying working with heritage students, they continue to teach from this perspective of bias against their home language practices. Thank you Flavia. So I'll read you some of the quotes here. If you have already closed the chat, I'd like you to open it again. And as I'm reading these quotes, I'll have to move the chat around so I can see everything I put as I'm reading these quotes, I would like you to jot down some of the words in them. So here's why these heritage Spanish teachers that appear problematic to you, either in a sense that this is an outright instance of linguistic violence, or maybe something that appears to be not a good pedagogical practice to you. So here's one of the teachers she said, Oh, there are many many Spanish speakers around the world that would agree with them that don't last this is probably perfectly fine. And at the same time, just pointing out that that really is substandard Spanish, and then she goes on to explain why she justifies harping on the two of last this four semesters at a time. Good, I agree. Substandard is, is a terrible way to talk about our students lived experiences. Another one, and you can do the same thing jot down some problematic ideas. There was another teacher she said, and you can write bad habits. I mean, if you point that out enough and you point out how many times do we spell that with an H things can be unlearned, but it does take a lot of repetition and almost, I made examples out of them and say, This is not going to be accepted. So yes, definitely bad habits. Yes, unaccepted, making samples of them. Perfect. You guys picked all the things that I said. So, I think before we talk about this idea of standard and non standards. I'd like to mention that in general we, we like to point out differences because we're language teachers and we're noticing details, but by and large, our students that are heritage generally tend to do things very similarly to the way they're done in other dialects of Spanish. If we think about a language as having multiple levels, like a phonological level and morphological lexical and syntactic level. And just of the levels, they're truly doing what monolingual Spanish speakers do, but sometimes slightly different. So, the idea of calling their language practices substandard as, as you pointed out, is saying that they're less than standard rather than different from standard. And it's also the idea of suggesting that their language practices are just bad habits that need to be unlearned. It is a really violent point of view. In my linguistics class recently we were talking about how from the period of infants and infancy all the way to adolescence, people increasingly start to assign linguistic characteristics to their identity, and they start having more and more linguistic like thoughts, right? At the beginning, our thoughts are more like feelings or reactions, and little by little they start to become more and more framed in linguistics and in language and telling somebody that the language in which they lived all of their experiences is just a bunch of bad habits that need to be reduced is not a great practice. So, I put our, I think our big question for today in this green box and that's how can we teach grammar in a way that validates our students' home language practices, so this is like that idea of tension, without ignoring standard Spanish, how do we do both at the same time? And I think critical pedagogy is a really helpful approach to addressing that question. So critical pedagogy is just the idea of doing a lot of things at once. It's teaching typical standard content and at the same time looking at how society's notions of privilege and discrimination have shaped our understanding of that content. I like to use the example of a history class when I explain what critical pedagogy is because I think that a lot of times we're used to thinking about history in that way. So critical pedagogy and history would be something like where we learn a couple of historical events and we read about them in the textbook, and then we think about how history was written in a certain way with certain definitions and certain inclusions in the text and how that benefits some people at the expense of other people. We can do the same kind of critical approach in grammar teaching. So I've listed four broad questions that I usually try to address with my students over the course of a whole semester. I decided to number them so that, again, in the chat, if one of these sticks out to you as something that either is impactful or something that you would like to share with your students, you could put that number in the chat and we could come back to that later. So here's some general questions. One would be how has the notion of standard Spanish been shaped by racism and classism. And then, number two is who gets to decide what's included in standard Spanish. And number three is what are the consequences of being excluded from participation and creating these norms. And finally, how have students been personally affected by these prejudices. So if one of those is particularly motivating or something you would like to share, you can include that number in the chat. All of them. Okay. Thank you. Thank you. Yes. So there's yes a lot of these are a lot of these are things that we can we can share with our students. You have this recording so you can come back to these questions and also revisit them again. I think I don't need to talk about this slide and I will go straight to an actual pedagogical example of how I apply critical pedagogy when we teach grammar. And most importantly, I want to emphasize before I do this and before you see all the beast this and con lucy second and feel a little bit nervous about that. There's a misconception that critical pedagogy means that I don't have to teach focus on form. And in reality, we teach a lot of forms, applying critical pedagogy. Another misconception is that I will accept anything wild on an exam. And in fact, when we use critical pedagogy students have to really construct an argument to defend the way that they are speaking using metal linguistic and social linguistic knowledge. So, this is not a throwaway kind of, I don't really have to teach that much now. So, okay, these would be three different steps that I use when applying critical pedagogy. I'm going to use the example here of predatory because that is usually a topic addressed in most grammar classes and it's a great, a great field for for looking at different kinds of variation. And this is probably what you would do in any typical grammar lesson, we look at all of the forms. So if we're doing predatory, we will learn as they almost add on each day you must get on and maybe the most other forms and then we learn what the two stem changes are and then we learn all of the irregular verbs and we learn when I becomes accentuated when you becomes accentuated. As you can see the other day we did when you I becomes you why and so we learn all of the different forms first, and then we start to notice them in context. One great resource for doing this Flavia if you'd like to put this in the chat is looking at the spend tax or the Spanish and Texas videos for real examples, and then starting to have students notice all the forms. One of the things that I've given to you in these green boxes is actually just an interview taken from, thank you Flavia it's from, from a textbook. And I start to have students notice okay what are all of the preterite looking forms in here, and they will say okay we've got foiste and con lucy and estube. So we have to take out all of our focus on form knowledge and notice which ones are regular and, and which ones follow a non standard pattern and they start to notice Oh well, we'll see. We learned that all of the seed verbs have a J in the preterite so so this is different from what we've looked at. And then we have to say okay well hold that thought let's think about why that happens. And then when we look at Oh well beast this is is definitely a non standard form because then you stay you almost get on. We don't have anything with an S on the end so that's different. And once we started to notice all of the different forms that are unfaithful to speakers, then we have to learn. This is a really important part we have to learn why these different variations occur. So one thing that I always teach them in looking at preterite is the concept of regularization. Sometimes I teach them the difference between regularization and analysis if they seem that they really like these terms, but otherwise we just call it all regularization. Basically regularization is this tendency that all language brains have that to create fewer and fewer different patterns to express the same thing. You can see how if a brain is is bilingual, having two separate systems in their brain, of course they're going to want to regularize because they have to manage twice the cognitive load twice the information. So regularization is this sort of quick trick that that brains use and that society is used to reduce the amount of different forms that we have in a language. And I like to show them how the Spanish that we have today is a product of many different cases of regularization so I like to show them older examples of regularization that are already incorporated into the way. The Spanish does things and then show them a couple of modern examples of regularization that are processes underway right now. So one example I give them is the verb in Spanish. I don't know why this is this but raise a nobody is mad that when somebody says it's in the past tense but really this is a product of a process of regularization that when Spanish was was just becoming Spanish and was not quite Latin anymore but not totally Spanish yet. What they say was was really quite irregular and the infinitive and the present tense and the preterite they all looked so different that that over time people started regularizing it to the form that we have today. A modern example that I give them is the verb on that. Most of you, me to teach underway as the past tense, but we know that we are now witnessing a process of regularization with that verb. In fact, about two years ago, a student told me that she really liked this group it was kind of like indie pop from Mexico. I wish I remembered the name. It was kind of a celestial sounding name of the group. I don't remember it sorry. In the first verse, the guy says and they instead of a new way. And so I started listening to how younger people talk and asking my little speakers and it seems that all of us who teach and the way as the preterite tense. The speakers even a monolingual communities are more and more starting to prefer and they, they say that and Louis sounds a bit pedantic, or something that their grandparents would say that they, they don't want to sound that way anymore. I also show them examples of regularization in English, because English has the same processes operating in order to create this more regularized system. Here's another example, although I've got to hear. So we do have these conversations in Spanish, even though we're talking also about English and I think this helps our second language learners to make some connections to the curriculum if they're not as familiar exposed to non standard Spanish each time. So here's an example that I show them, you know what you guys are so mad about regularization with going to see, but look, even in English we have increasingly regularized our preterite forums over time. So I'll show them an example of however like burn has become more and more regular taking burned instead of burnt over time and how this is happening with many other verbs. I don't know if you can can actually access this link, but really it's just an article about scientists predicting which verbs are going to regularize in the future in English, and the one that they came up with that's going to be the new regular one is wedded instead of wed. So feel free to check that out. So, once I teach them all of this focus on form knowledge and meddling with technology so we can talk about it, and then we learn the social linguistic knowledge for why these things change, then it's time to have critical reflection. And this is definitely not just a throwaway or enrichment step. This is like if grammar grammar would be the means to this end. So in the step of the lesson, we discuss why certain changes have been accepted in standard Spanish, and others have not been incorporated into standard Spanish and how that affects the way grammar is taught and how that affects the way real people think about their own language practices. And I take all of this and try to have students do something productive with all of this new knowledge so a good thing to do at the end of the lesson is teach them how to respond to a discriminatory attitude or comment about their own speech community. If they're using one of these non standard forms. So, those are basically the steps we look at the knowledge we find different examples, then we learn how things change and why they change, and then we learn to respond to discriminatory attitudes against some of these changes. So I've listed a couple of examples of other areas that I generally tend to address in a grammar class. You're welcome to ask me about any of these during the Q&A time. But I think now Flavia would be a good time to or Sarah to create some breakout rooms, some small groups. Let's see I think first one thing that you could do in your small groups is think about how critical pedagogy of using an approach like this might motivate your students to learn more about Spanish. And maybe think about one of the grammar topics that that you teach, it could be one of these listed here or something else. And think about what your students already know about variation or what kind of variations they're already using, and how you could validate those in a class. So, let me share again. I am fantastic. Okay, well, so I hope that you were able to discuss a little bit about what you think might be applicable from this idea of critical pedagogy in in your groups hopefully you met somebody that has a similar experience or challenge or thought, or maybe they gave you a new idea. Let me go to questions. I wanted to kind of this is being recorded. So I wanted to just go through some comments on some slides about how you might assess this kind of lesson. And I won't really get into the things that are on here but if you would like to assess critical pedagogy, you could always watch the video and then just pause it and read through the questions on these slides. So, first I give some examples of like question starters that you can use to assess all the different components in a lesson. I think if you're going to teach it all, you've got to assess it all to in this way. And then I screenshotted a little tiny chunk from one of my exams. This one was about different types of past constructions and so there were questions about regularization. And you'll see that they're focused on form and social linguistic knowledge and critical reflection questions. And then I screenshot it and answer. And I was very intentional about screenshotting and not fantastic answer, because I didn't want to blow you away with an amazing student who might feel intimidated. So this is like a really average B response. It's not perfectly accurate in terms of metalinguistic knowledge, but the idea of what language variation does in bilingual communities is definitely on. And finally, thinking about now that everything is online and we have to assess online, I think it's a good time to try incorporating critical pedagogy because the answers that you are assessing are not so easily found in Google. I really don't think that a focus on form type test makes very much sense now that we're online. You can do those kinds of questions and then add to them and I have many ideas of how to do that if you would like. Finally, I wanted to end with some quotes from my students. A good way to see if you think this is working is for your last question on the exam to be something like, What did you learn that you would like to share. You can see how they framed their own learning and then again I tried to give not the most fantastic blow you away answers but really solid one was a B and one was an A answer. And you can see that the students really understand the idea of languages change and people get hurt in the process when stigmatization occurs. So that's my presentation. I would love to answer any questions that you might have. I don't know Flavia just want to to read out the questions. I'll open the chat again. Okay, just go ahead and write any questions or comment or something you would like to ask so we can continue the conversation. While they are doing that I've got the chat open now so that I can see. I know that one of the questions was about resources and so a couple of ideas. One is using your students own speech as a great resource students, I thought they would think that this was weird but they like to be featured in my lessons and they like to be quoted in my lessons. I also brought I was just in a breakout room talking about this one. This is one that I really, really recommend it is gonna make a funny or that variation social by Kim Potoski. You can see it's a really skinny book. But it is a book that I use a lot. Because now that I work more with critical pedagogy I think it's really important to privilege like depths over breath. Somebody gave me this. Yes, thank you for the machine, and she is here. Wonderful. So thank you Kim Potoski for writing a wonderful book that I use in class a lot. I think somebody gave me the phrase throwing spaghetti at the wall, and I think, you know just seeing what sticks and I think teaching too many grammar topics is kind of like that. I would rather teach fewer grammar topics and incorporate them from different angles so that the students can decide how they want to connect with that material and, you know, if they want to incorporate it into their daily practices or not. I hope that you continue to revise it. It's already great though. Thank you. Do you introduce the concept of appropriateness as a replacement for grammaticality. I introduced the concept of appropriateness as different from grammaticality. I would say that usually in my classes we talk about grammaticality as a predictable pattern that speakers in a native speech community. I don't know if it's up by level right but in a speech community that uses Spanish natively would use, and then I definitely look at appropriateness as the idea of, well, you can decide how much of this you want to incorporate in your own Spanish practices. I don't know if you are speaking with somebody in your professional life or maybe in your creative life or your academic life, but I think Carl, like, in the end, what's appropriate for one student might not be appropriate for another student because not everybody wants to incorporate Spanish into their professional life. I wish they did, but they might not want to and that is also their personal choice. What else? Other questions? Yes, professional or personal. They can add that. Let's see what is here. How do you organize the syllabus for the class? I've taught heritage learners for a while. I kind of predict the topics that have lots of variation that I think they're going to need to know, and we just do one after the other. I usually start with a general idea of thinking about critical language awareness and what the right is and what different academies of the language would say about different varieties. And then I usually pick a couple of topics that I think are going to be most useful to students and I definitely teach fewer topics now. I like this question. Could you give an example of something students could say to someone who critiques their language usage? Well, so yeah, this is like a big part of what I teach is how to respond to that when students hear a comment about, let's say, the he's this. Well, you can't say the he's this because the he's this is wrong. I teach them to first start with the existence of knowledge that variations exist. And then I teach them to incorporate so silhouistic information so that they might say, well, the he's this occurs because of a process called regularization. And that process has occurred in lots of Spanish instances in the past, for example, and then they might come up with something like a reír, say how it's been regularized or how and that is becoming more and more regular. And then they can say that their own speech community has its own practices and its own patterns. And if it is a pattern preferred by the speakers of that speech community, then it is a grammatical pattern for them. I usually teach them to sort of acknowledge that variation exists and then explain why and then defend their own practice if they choose to. And so I was wondering why people focus that much on grammar variation and less on on lexical variation, which impacts communication, much more. I think it's easier to teach grammatical variation so we do it more because we could take one whole box and say that all of these instances of patterns fit in that one box and so we tend to do it more. I would say that when trying to select topics that are worth presenting in depth. So I do think about if there is some kind of variation that's non standard that my students will probably use which ones have the most informational value. So the other day Flavia and I were talking about the subjunctive and teaching the subjunctive and Spanish teachers are just so in love with teaching the subjunctive and I think it's because we break it up into nominal clauses and adjective clauses and there's many different ways to teach subjunctive and we just keep teaching it. But in reality, the amount of informational value in coding a verb for subjunctive versus indicative is so small, because the first part of the sentence tells you whether it's subjunctive or not it tells you the mood that the speaker has towards its action. So that is something that we teach in a very advanced class, but we don't teach that much in a lower level class because coding for subjunctive is going to improve their effectiveness of their communication very little. What about the swen? Do you have an example of critical pedagogy being used to learn about lexical variation? I don't know would pronoun usage be lexical variation? Those are grammar words. We definitely talk a lot about it in the linguistics class. We don't do as much of it in the grammar class, but we definitely like to approach lexical variation in class as something that's exciting. Students love to talk about it and they love to learn about new varieties, but I don't teach it that much in a grammar class. I wonder if it's necessary for heritage learners to learn all the rules for the uses of subjunctive. If we have this much time and this much information. It's fun to teach about in an advanced class because I love teaching about how different languages code for mood differently and how it's a really salient feature in some languages and not in other languages. But that's like really nerdy talk and only the students that get to take linguistics classes really find that information useful. I try to think like what could they choose to incorporate in their own professional or academic or creative life and that's what I would choose to teach first. So subjunctive, sometimes yes and sometimes no. I saw someone asked earlier about ideas for teaching this with high school students doesn't even have ideas about that. I teach this to college freshmen and college freshmen are a lot like high school students. So, yeah, it's almost like the same thing. I do think that again you can teach fewer topics, and maybe you're in your focus on form, you don't give so many forms, but you start with forms, and then you can still teach variations on those forms and I think you can still listen to real people using those variations and talk about why they occur, especially if you make the connection to how those examples occur in English. And I also think that this book, it was written for for college, but I do think that you could pick out plenty of lessons for high school. Definitely in terms of subject for agreement, things like that. What else did you see the questions go so fast I try to get one and see if there's any more. I got to scroll back up I lost someone asked about how do you organize the syllabus for the class. Yeah, so I definitely start with the idea of language variation in general, and then I just pick instances of variation that I know are going to occur in my students speech communities at home. That's what we talk about I would say that I'm like hesitant to answer this question with specifics now that we're online because I'm just not sure how many topics we're going to teach, but I would say that we change grammar topics, once a week in a college classroom because we can do a lot more and I can assign a lot more out of class and I assign a lot of the focus on form outside of class, but maybe if you were in high school. You might teach the same grammar topic for like two weeks so that you can look at it from many different angles. If you are not only teaching grammar that could be half of your class period could be something like that and the other half of the class could be a different kind of reading a more creative style reading and writing. But we do incorporate a lot of reading and writing in into these topics. What else. I'll let you address them since it looks like someone Donna asked any thoughts on immersion students with fossilized grammar. Yes, I don't have that many because unfortunately. Well, and I noticed that Paltowski said that she was here. It was the kinds of programs that she's talking about that we need gosh we just don't have them in Texas. So I don't have a lot of those students. But again, I would go back to picking your battles and thinking about because of informational value. What can we teach them that they could choose to incorporate in their own communication style that would actually really improve the value of the message. And I do think that something like preterite really improves the value of the message, because these verbs are marked for very saliently we clearly hear the different endings so I would pick that. I would pick pronoun usage as something that could really improve the quality of somebody's message because recently I noticed that students were. This is like wild grammar talk but I noticed that students were confusing the gender of the indirect object with the gender of the direct object when they were creating sentences with double object pronouns. And that could be confusing to somebody there's a lot of informational value in that because if it's only coded once in a sentence and a pronoun, I would definitely teach them that. So yeah with students that have fossilized grammar. Their language is their language the way they speak is the way they speak and I would not try to tackle everything at once just as I wouldn't do that with a heritage speaker because, you know, that's like that I that metaphor of throwing spaghetti at trying to create correct everything at once. Thank you. I think we will also want to ask what classes you teach and where can they order the book also what other classes do you teach. But I think after this, since we only have one minute left we can probably. Thank you for for all the questions and I teach a lot of different classes I teach Spanish for heritage learners, and I teach a grammar class I teach a linguistics class translation, a class on language and gender that only the really like nerdy Spanish language to take because it's a very top level, but a lot of the strategies that I mentioned today are strategies that I use in classes with mostly freshman students. And where can you order the book. You can just Google it, and there's a great website. She has a website. Thank you so much for this great presentation I think that I mean there are so many questions that we can continue discussing so I hope the conversation keeps going. Everybody knows that we attacks really try to connect people from different spaces, different educational context, different stages of the profession. So anytime that you have questions or comments just contact us and hopefully we will see you around in our next event.


Hello and welcome to this next lecture in the series on the topic of wavelets and multirate digital signal processing and the topic which deals with joint time frequency analysis in broader perspective. This particular lecture is titled as wavelets applications and we are going to run through few significant applications which makes wavelet transform extremely useful which makes the USP of wavelet transform. However, before starting that journey we are going to do a small recap of what we have done so far. In last three to four lectures we posed few significant questions and in a way we also try to answer those questions. Let us have a quick recap of what we have done so far and then we will start our journey towards solving few of the critical problems when it comes to practical applications of wavelet transform. So, quick recap we started our journey by saying that wavelet transform in a way decomposes signal into two separate series. A single series that would represent the approximations and that would actually emerge out of the underlying scaling functions phi of x and the double series to in a way represent the details the refined version of the signal and this is high pass filtering and this gets originated out of the underlying wavelet functions or psi of t or psi of x. However, the moment we started dealing with the two band Har filter bank structure we realized that it is very easy to move down the ladder because in a way we have to simply throw away the details and then we pose this challenge that can I build a framework using which I can think of moving up the ladder go on adding up the details and really go tantalizingly close to the actual signal or function under consideration for analysis. And we build that framework and that framework actually gave us power to actually move up the ladder and moving down the ladder is anyways pretty simple. So, now I can move in both the directions and in a way I can now say that now I indeed can zoom on or zoom out of any part of the underlying signal and this makes the entire analysis scalable and the scalability stems out of the underlying multi resolution framework. The whole framework in a way is based on this beautiful mathematical equation that we already seen this is a scaling equation of phi of t square root of 2 is the normalizing factor we are summing it over k where k is the translational parameter and h of k are the coefficients of scaling function. And then this function is at the heart of the entire framework of multi resolution analysis and that is because we realize that phi of t is indeed phi of 1 t and 1 essentially means 2 to the power 0 and then I can say this function belongs to a subspace v of 0. We also realized that on the right hand side we have phi of twice t minus k and twice t in a way indicates 2 to the power 1 and this function indeed belongs to a subspace v of 1 and that gave us this beautiful property and that tells us that v 0 is a subset of v 1 and so v 1 contains something that v 0 does not have and that additional part is indeed the details present in the signal and those details are in a way captured by the wavelet equation psi of t. And so we looked at psi of t and g of k are the coefficients of this psi equation and again we did the same analysis that phi of t has phi of 1 t and it indeed belongs to w of 0 and this function once again belongs to v of 1 because I have 2 to the power 1. And then this once again conveyed the same fact that indeed v 0 is a subspace of v of 1 and by virtue of adding these 2 factors by virtue of adding v of 0 and w of 0 by the way this is an orthogonal addition I can indeed move to v of 1 and this is something which is of great significance because this helps us move up the ladder. I can indeed find out the projections of the signal in w subspaces add those details into approximations and move into a higher subspace very important very significant. However, the same equations are recursive and the same equations can also be written down in this form for example, phi of t by 2 and then I have phi of t. So, I have t by 2 that is 2 to the power minus 1 and this would indeed belong to then v of minus 1 and phi of t minus 1 t minus k would then belong to v of 0 because I have 2 to the power 0. And correspondingly this would belong to w of minus 1 and this once again belongs to v of 0 and that conveys again the same underlying message the same current the same theme and I can once again go on adding up v of minus 1 and w of minus 1 orthogonally to generate projections in v of 0. What this essentially tells us is the strong framework composed of underlying nested subsets and these subsets can be represented like this. And now it does not really matter in which particular subspace I am starting my analysis for example, I can again think of starting in v of 0 by virtue of using this framework I can very easily move either down the ladder and that will make the whole presentation coarser and coarser and ultimately lead us to a null subset a tree well subset and I can also go on moving up the ladder and achieve the L 2 r norm as we have seen while studying the axioms of multi resolution analysis. This was the framework that we wrote down if j of x belongs to v of j and if my analysis scale that is analysis window is 1 upon 2 to the power j then in order to be able to span these subspaces I would require this particular basis function. And this factor 2 to the power j by 2 is used for normalization and now I can very well say that my orthogonal basis is now orthonormal. And I can very well find out f j of x which are the projections of underlying function f of x into that subspace v of j and it is a linear subspace. So, I can apply super position theorem and this addition over k which is my translational parameter would help me find out these projections. The main question is how do I find out alpha values which are the approximation values and I have this integration formula. So, integration between the function and the orthonormal basis and that will help me find out the alpha values which are the approximation values. Correspondingly we looked at the test signal because we wanted to indeed find out whether this actually works or not and we looked at this particular test signal f of x which exists only between 0 and 3 and between 0 and 3 f of x is equal to x and it is 0 elsewhere. And this signal looks like this a simple good looking linear function. And the moment we calculated the projections of this f into v of 0 we had the analysis window from 0 to 1 and we calculated the values of alpha 0 0 0 1 and 0 2 and correspondingly we were able to generate these projections. In the last to last lecture we also calculated the projections in W 0 which we named these projections as G 0 of x and we calculated corresponding beta parameters and once again the analysis window is of length 1. So, between 0 and 1 1 and 2 and then 2 2 3 and finally, we are able to do the orthogonal addition of projections in v 0 and W 0 to generate projections f 1 of x into v 1 of x v 1 and indeed these projections in v 1 are much much better much much finer compared to the projections in v of 0. And we know the reason the reason is we added these details in these projections in v of 0. So, this indeed worked and we also did MATLAB simulation we went on adding the details and then we realized that this is dyadic scale and I can go all the way till v of 10 or v of 20 and if I mean v of 20 the analysis window will be 1 upon 2 to the power 20. So, we are talking about a very very small window and to what value we can go we can actually move to any value of our choice. It boils down to how close we really want to go as far as the representation of the underlying signal is concerned and from the slides we can clearly sense that this is indeed moving very close. However, that led us to two important questions how do we go about selecting the mother wavelet and the scale of analysis and what is the procedure to calculate the scaling and wavelet coefficients. We in a way answered these questions we took the first question that how at all one should decide which mother wavelet should be used for what kind of an application for what kind of an analysis and to answer this question we invoked the very concept of vanishing moments and then we realized that vanishing moments they play very significant role in selecting the mother wavelet and we brought out this concept of vanishing moments through correlation and for that matter we revisited how exactly the Fourier transform works and then we realized in order to be able to find out the frequency content in any underlying signal a stationary signal like this we do correlation between the signal which is shown in red with the underlying basis function for example, the basis function shown is cos of 8 and it is all about finding out the correlation by the dot product. And the dot product gave us all positive values with very minimal negative values and when I integrate this green part green area then indeed I will generate a peak at a frequency of 8. So, we realize this concept of correlation is important and then we thought of bringing in this concept to define moments and we define moment of order m of given underlying function f of x on interval from a to b like this and this is once again a dot product between x to the power m and f of x a correlation dot product. And we realize that this indeed is of great importance and to bring out the significance of vanishing moments we looked at a particular example a particular application where we wanted to detect the hidden jump or hidden discontinuity. We started off with this function g of t and we can clearly sense that there is a discontinuity or a jump at t is equal to 0.5 and the function looks like this and at 0.5 there is a clear jump. However, such functions are easier to analyze the discontinuity can be sensed even with the naked eye. And so, we thought of complicating the matters little and we did the integration of this g of t and we generated h of t and then we realize that the clear jump has been converted into a cusp jump at time t is equal to 0.5 and now my h of t indeed looks like this and still there is this cusp jump. So, we thought of complicating the matters even further and we did the integration of h of t to actually find out f of t. And now this f of t is very smooth and at least with naked eye we cannot sense the presence of discontinuity or a jump. However, we realized that at second derivative of this function there exists a problem at t is equal to 0.5 and then we pose this question that can I select an appropriate mother wavelet to indeed find out that hidden jump or hidden discontinuity. And that is where the concept of vanishing moment really became very important because the moment you are doing integration or low pass filtering and the moment you say that my function now appears smooth. And if at all you want to find out the discontinuities it is a property that is associated with psi that is wavelet functions. Just a quick reminder that phi functions or the father functions are low pass filters and psi functions or wavelet functions are high pass filters because they do differentiation. And indeed if at all I want to find out this hidden discontinuity I will have to invoke the underlying psi function wavelet function, but which of them because once again we also saw in the last lecture that in case of Fourier transform the basis is known e to the power j of a variable it could be a frequency e to the power j omega. But when it comes to wavelet transform we have choice of selecting mother wavelet and so we pose this problem in the light of indeed finding out the hidden discontinuity. And then we realized that since the discontinuity lies at the second derivative I should at least select my mother wavelet with m 0 and m 1 getting vanished at least the zeroth moment and the first moment should get vanished. And that is because then I can think of doing the correlation for the second derivative and all the derivatives which are greater than order of 2. We invoked the wave menu the wavelet toolbox in matlab and then we realized in case of Haar wavelet which is dobash 1 this discontinuity is not sensed and that is because in case of Haar only the zeroth moment vanishes and so this discontinuity is not sensed. We did the same exercise with dobash 2 and since dobash 2 has m 0, m 1 these vanishing moments and which is badly needed to find out the discontinuity which is of the order of second derivative now we are able to sense and detect this particular discontinuity. We also did analysis with dobash 3 and then we realized that using dobash 3 the discontinuity was brought out even more prominently. From the slides we can once again see that we posed second important question. Once we realized that we have understood how to go about selecting the mother wavelet then the second important question is what should be the scale of analysis scale 1, scale 2, scale 3 and where exactly should I stop? Well honestly speaking there is no formal mathematical proof to this particular question. However, there is a thumb rule and the thumb rule says that the moment you lose more than 70 percent of the energy in your original signal one should typically stop doing the further analysis. That means you have lost enough information and probably you have brought out the approximations and details to a level where you can indeed find out the trends as well as the refinements. However, the third question was pending and from the slides we can once again sense the third question and the third important question was how to calculate the scaling coefficients and wavelet coefficients. This was one serious important question and we in a way answered this question. If you remember in the last lecture we said it is very easy to design your own wavelet but it is very difficult to design a useful wavelet. I can take a sine wave and I can cut it so that I am left with only the first period of that sine wave and that is a wavelet. I am letting that wave die out and so I have the restricted good compact support and it is a wavelet. It is hardly of any use. So, it is easy to design your own wavelet but it is very difficult to design a wavelet which is useful and so understanding how to find out the coefficients of scaling equation and wavelet equation is of great importance. We in a way rushed through this particular part in the last lecture. So, let us quickly also revisit that part and then we will move on to couple of important applications of wavelet transform. We have been using Haar mother wavelet and Haar scaling equation and we typically know how these two equations and functions they actually look and I can go about normalizing these values and this will be 1 upon square root of 2 minus 1 upon square root of 2. However, the important question is how Haar was able to find out these values. So, that is the important question and what properties these coefficients should obey so that they will be of some importance some significance. Now, referring back to the framework we can clearly sense that phi of twice t gives me phi of t as well as psi of t or in other words I can say once I calculate the coefficients of phi that is enough using these I can very well calculate coefficients of psi. So, this design of finding out the coefficients of phi and psi eventually boils down to just finding out the coefficients of phi because once I have these coefficients of phi I can very well calculate the coefficients of psi and it is quite evident from these two equations. So, the problem becomes the problem of searching for the scaling equation coefficients and then we can use this formula where once h of k are calculated g of k can be calculated like this minus 1 raise to k h of 1 minus k. It is very easy to actually show that this formula indeed holds true. So, now problem is very very clear we need to find out h of k values and then we pose this question that what different properties the coefficients of phi of t should have and then we also listed down those properties by virtue of looking at three important guiding theorems. The first guiding theorem essentially tells us that we are talking about the nested subsets and we are talking about the scaling equation which is indeed of finite duration. So, phi of x certainly has a compact support and outside this compact support it does not exist. So, theorem 2 in a way told us that for this phi of x which has a compact support I can very well find out finite value of this capital N that is the length of this phi of x and in a way this length is going to give me length of psi of x and length of psi of x is going to give me vanishing moments. So, this is of great importance and then we saw theorem number 3 which in a way talks about the discrete orthonormality of the sequences. So, this is of greater importance. Now, these were the properties that we derived last time 1, 2, 3 and then 4 to 6 and as first 4 properties 1, 2, 3 and the fourth one they predominantly are dependent on the normalizing factor. Property number 5 and 6 it emerges out of the fact that we are strictly talking about orthogonal and in fact orthonormal basis. And then we started our journey towards finding out these properties. For deriving property number 1 which is summation of all the scaling coefficients should be equal to square root of 2. We started once again with the scaling equation. We integrated both the sides and then we took out this summation part outside the integration and then we realize that this phi of twice t minus k can be written like this. One can substitute twice t minus k is equal to x for example and solve this. Very quickly we can show this and we actually showed this in the last lecture and then we use this normalization unit. We said when I integrate phi of t from minus infinity to plus infinity it should go to 1. Just a quick reminder if I am integrating psi of t that is my wavelet function it should go to 0 and that is the basic property of any wavelet. However, what phi of t should accumulate to this is called as normalizing factor and we can go about deciding this normalizing factor. And we have selected purposely this normalizing factor to be 1 because that gives us a constant of square root of 2 and square root of 2 traditionally has great significance. It is used in finding out the root mean square values. Square root of 2 also tells us roughly at what point the 70 percent energy would get consumed. So, this normalizing factor is also important in deciding the scale of analysis. Once we decided this normalization parameter then it was easy to actually plug in this into this formula. And then I already have the square root of 2 factor which comes out of the normalizing 2 to the power j by 2 parameter. And I can very well prove that summation of h of k should go to square root of 2. Now, deriving property number 5 was indeed challenging and we once again started of a scaling equation and by virtue of doing some mathematical jugglery that we did in the last lecture we will just skim through we will not go through each and every single step. And by virtue of doing this analysis we are able to finally prove that this would boil down to delta L 0. And this is indeed discretized version of orthogonality. And by the way not all the scaling functions are orthogonal in their nature. And we are going to see one example. But before that once we derived property number 5 it was easy to derive property number 4 because it is a specialized version of property number 5. And we are able to show this pretty quickly. Now, to find out property number 3 we did a small trick of dividing this equation into odd parts and even parts. So, twice k plus twice l as we saw in the last lecture is the even part and twice k plus 1 plus twice l is the odd part. And by virtue of having this going to be equal to 1 we are able to do the rest of the analysis. And because this is equal to 1 we could say let us say this even part goes to some constant let us say a. And this odd part goes to another constant let say b. And it was required to find out these two constants a and b. We have two constants and there is only one single common equation I can plug in a and b into this. And I would get 1 is equal to a square plus b square. So, this is one equation and there are two unknowns. And so we wanted one more equation. And we are able to generate that equation by looking at property number 1. And we once again split h of k into even and odd parts. And then we got the second equation a plus b to be equal to square root of 2. And this is a very special situation to be in because this is indeed equation of a circle. And this is equation of a straight line. However, they share a very special chemistry between each other. And then we saw last time that we are talking about a circle of radius 1. So, this is a unity circle.  And there is just one single point which is common. And that point exists at 1 upon square root of 2 and 1 upon square root of 2. So, both a and b values were calculated to be 1 upon square root of 2. And then once we plugged in the values of b and a, property number 3 and 2 were calculated. So, this is how we completed our journey. We did the verification of these properties in case of r. And first three properties were very easy to actually prove. Property number 4 was also comparatively easier to prove. However, property number 5 gave us great insight. And after solving through for property number 5, we realized that this is indeed the property that brings out discrete orthogonality. And like we said before, not all the functions are orthogonal. For example, if we see the roof scaling function whose formula goes like this. And if we apply the fifth property once again for the first translated version. So, we are taking a dot product between phi of x and phi of x minus 1, then this does not go to 0. And we can also see that graphically there is some amount of overlap. And because of this overlap, this dot product is not going to go to 0, because this overlap is in positive direction. And so, roof scaling function is not orthogonal. And it is very important to maintain the orthogonality, because only then we can claim that the two band MRA structures would be complements. And we can guarantee the power complementarity. And we can then also guarantee magnitude complementarity. So, in a way, we answered these three important questions. And now based on this, let us work out couple of applications of wavelet transform, which are of practical importance, practical significance. We have already seen one application, where we detected the hidden discontinuity. Now, that was the application where we relied heavily on the wavelet function. But if you remember, unirate DSP is all about designing filters and multirate DSP is all about designing filter banks. So, along with that high pass filter in psi function, we also have a low pass filter in phi functions. So, the second application is in a way dependent on the nature of phi function. So, from the slides, the second application is about suppressing the polynomials towards denoising the underlying signal. And in order to work out this particular application, we are going to invoke one MATLAB program. And it is a simple MATLAB program. So, we have specified the signal to noise ratio. We have also provided the initialization for the random seed. And then, we are generating a signal, which is the reference signal shown as x reference. And then, we are also adding noise, which is white Gaussian additive noise. And we are also generating x. So, x is the reference signal plus the added noise. And the noise is white Gaussian noise. And we are going to save this signal as test underscore signal underscore polynomial underscore noise. So, let us run this. So, this is how the original signal looks. And the moment you add noise on top of this, this is how the corrupted signal looks. And it is very difficult to actually see the underlying signal. The signal to noise ratio selected is 4. And so, it is good enough strength that in a way disturbs my underlying signal. And not all the details are visible. And there is a need to actually clean up this signal, so that I can bring out the underlying characteristics. How to do that? We have saved this signal with this name. So, let us once again invoke the wavelet toolbox. And once again, this is a one-dimensional application. We have signal, which is one-dimensional. And let us load the signal that we have generated. So, this is the signal. And this is how it looks. It is indeed noisy. We can see that. We can sense that. Now, typically a signal can be looked upon as a polynomial. And if the degree of the polynomial is 2, then it goes without saying that if I have a polynomial of degree 2, then db2 won't be able to do the neat analysis of that polynomial. And that is because we have already seen that the second derivative still exists. Or in other words, the moment corresponding to the second derivative does not vanish. I can very well do the analysis with db3. And then the second derivative is fine. I can also invoke the third derivative. And that is one serious important property that needs to be taken into account while doing the analysis. Now, let us start the analysis once again in db1. And db1 is indeed har as we know. And we know in case of har, only 0th moment vanishes. And as a result of that, if we focus our attention on d3, d4 and d5, well, hardly anything gets suppressed. From db1, we will move on to db2. And again, it is not very satisfactory. However, now if I invoke db3, now this is something which is of great importance. From at least d5 and d4, we can sense that things have started getting suppressed as far as the polynomial is concerned. And if this happens, then eventually what we will be left with is going to be suppressed. I am suppressing the whole of my polynomial. And I am left essentially with just the noise. And since this is an additive noise, I can also do the subtraction. And I can subtract this noise from the original signal to indeed retain back, generate back the original signal. So, let us do that exercise. And let us convince that indeed it will be possible to do denoising of the signal. We have once again selected db3. And the level selected is 5. We have selected the soft threshold. And now, if we denoise, then you can clearly sense that correspondingly, the residuals will be generated for the selected threshold values. And this is how we can get back the original signal. We started with this signal, which is indeed polluted. And we are able to generate back this signal. And many of the details are neatly restored, if you see here. To what level you want to restore your details, that is a matter of once again selecting the appropriate wavelet function and the corresponding father function or the scaling function. Now, once again from the slides, let us go back to another application that we want to study and understand. We have seen two one dimensional applications. And now, let us delve into adding up one more dimension. Let us look at a two dimensional application. And we will typically look at the compression and pattern recognition applications. As far as two dimensional or one dimensional signal or image analysis is concerned, typically the basis function that we try and generate should be of unitary nature. And only then we can guarantee two serious important qualities and properties. One is energy compaction. And second one is energy conservation. We should be able to preserve most of the energy. And whatever energy we have with us, we should be able to compactly represent. Only then we can achieve compression. Typically the formula of unitary matrix looks like this. And if I have a matrix which is A and if A inverse is equal to A transpose, let us keep it simple. Let us say I am talking about matrix which has all real entities and no imaginary part. Then A inverse is equal to A transpose in a way ensures that my matrix is unitary. And if my basis matrix is unitary, it guarantees decorrelation of the information. And once we decorrelate the information, we can guarantee energy preservation and also energy compaction. This is one example of 2D discrete cosine transform. This is a well known image of a cameraman. And by virtue of doing this kind of an operation where x is my input and c is my transformation matrix, then output y can be generated as y is equal to c into x into c transpose which is equal to c into x into c inverse, because this is now unitary. And there are only few coefficients which are non-zero. And essentially what I can do is, I can put a threshold value and I will retain only those coefficients which are significant and probably suppress all others. This will definitely bring out compaction, but this will also preserve most of my energy. Now, let us pose this important question. Can I preserve energy using wavelet transform W t? So, let us pose this question and let us try and answer this question quickly by using some sample signal x of n, let us say 3, 2, 5 and 1. So, there are only 4 elements and let us say this signal belongs to subspace V of 2. Now, I can very well calculate the approximations and let us call these approximations as a 1. And I can find out these approximations which would belong to V of 1 and I can also calculate the details and let us call them as d 1 and these d 1 details would belong to obviously W of 1 as we typically know. For doing the analysis, let us once again use Haar scaling and wavelet equation. And let us say we are using the normalized version. So, my h 0 is going to be equal to h of 1, it is going to be equal to 1 upon square root of 2. So, these are the coefficients of scaling equation. And as far as wavelet equation is concerned, let g 0 be equal to 1 upon square root of 2 and let g 1 be equal to minus 1 upon square root of 2. And then we know how to calculate a 1, there is a disseminator. So, I will have to do calculations between 3 and 2 and then 5 and 1. So, between 3 and 2, once again 3 into 1 upon square root of 2 plus 2 into 1 upon square root of 2. So, once I do that, I will end up with 5 square root of 2 divided by 2 and 3 square root of 2. And as far as details are concerned, I will have to make use of g 0 and g 1 and then I will be left with these two parameters. Now, if I calculate the energy in a 1, let us call it as E of a 1, then that is going to be 5 square root of 2 by 2 square plus 3 square root of 2 square. So, that will be equal to 61 by 2. And energy in details is going to be equal to, which is going to be equal to 1 by 2 plus 8. So, that is 17 by 2. And now, if we try and find out how much of energy is stored in details, if we pose this question, then that is going to be probably 17 upon 17 plus 61. And this is approximately 22 percent. So, almost 22 percent of the total energy in my underlying signal is stored in the details. And that is because this signal is fairly rough. There are transitions from 3 to 2 to 5 and then 5 to 1. So, fairly rough looking signal Now, as against this, if we consider a case of yet another signal, which is much more smoother, let us say x 1 of n, which looks like this 4, 6, 10 and 12. Then indeed, the details will be, let us say once again this belongs to V 2. And the details, these are the approximations. And the details will be, now these are same values. And that is because the difference between 4 and 6 is same as the difference between 10 and 12, as can be seen. So, this indeed gives us the detail values. And now, if I calculate E of A 1, it comes out to be 292. And if I calculate the energy stored in details, it comes out to be 4 plus 2 plus 2. So, basically 4. And now, if I want to calculate what is the percentage of energy stored in details with reference to the entire energy, then it boils down to 4 upon 292 plus 4. So, close to 1.4 percent. So, for smooth signals, most of the energy will be stored in approximations. And for fairly rough looking signals, where for example, you could be talking about a textured image, the energy distribution is in a way E 1. Although, most of the energy will always be retained in low pass quotient, which are the approximations, the details in case of fairly rough signals would also have fairly good amount of energy stored in them. Now, keeping this in our mind and realizing that indeed energy gets preserved in case of wavelet transform. Now, we can carry out two important tasks. The first task is going to be the task of denoising 2D toy image. And the sample image that we are going to play around with looks like this. It is some 4 by 4 part in the image. And let us say, I have her unitary matrix with me, which typically looks like this.  So, what we are going to do now is first of all find out. So, step number 1 will be to find out S in horizontal direction, which will be equal to S into A. And then 2, I am going to calculate S of H v horizontal and vertical direction, which will be equal to A into S of H. So, that is the plan that we have with us. And when we calculate S of H, which is equal to S into A, it turns out to be this fairly easy to calculate. For example, S into A, so we are talking about the row and the first column. So, 9 plus 7, because 9 into 1 plus 7 into 1, that is 16 divided by 2. So, that gave us the first element to be 8. So, that is how we calculated this. And then in order to find out S of H v, we have to do A into S of H. And this essentially gives us this matrix. And now, it is very interesting to understand how this typically looks. For example, if you sense these four coefficients 6, 1, 2 and 0. And if you see the original image and if you focus on these four elements 9, 7, 5 and 3, then indeed these four elements are responsible for generating this matrix. So, this matrix is a decomposed version of 9, 7, 5, 3. How? That is the question. It is very interesting to see what is really happening here. Keeping this in mind, we can clearly sense that indeed 6 is equal to 1 upon 4 into 9 plus 7 plus 5 plus 3. So, this is the low pass element. I can very well think of 2 to be like this. And what does this indicate? 9 minus 5 is 7 minus 3. That essentially indicates we are talking about the average of the differences in rows. Correspondingly, I can also say that 1 is indeed equal to 1 by 4 into 9 minus 7 plus 5 minus 3. So, this is an average of the difference in columns. And then 0 is equal to 1 by 4 into 9 plus 3 minus 5 plus 7. So, this is the average of difference between the diagonal elements. And so, these coefficients, they have great importance, great significance. And they are going to take us to how to build the final matrix and how to find out the final matrix when it comes to transforming the underlying image into a wavelet space. So, as far as finding out the transformed version is concerned, we can very well say from this particular matrix that we have already generated, I can very well sense that the analysis that we have done for this particular block is true for all the 4 blocks. And then 6, 4, 4 and 2, these are indeed the low pass coefficients. And I can bring them together. And probably I can say this is indeed going to look like a beautiful culmination of the thought process that we started with multi resolution analysis. I want to segregate the low pass and the high pass information. And then probably what I will say is if my S of i, so what we have done here is interesting. We have indeed calculated the transformed version of the original image that we wanted to analyze. And just like this particular matrix, we can also do a similar analytical understanding of the rest of the 2 by 2 matrices. And we will realize that 6, 4, 4 and 2, these are indeed the low pass values or the approximated values. Now if I want to take this analysis to next scale, then I will have to really focus on the approximation values 6, 4, 4 and 2. And let us do that. By the way, the reason as to why at all we want to do this is because if we have signal of x of n, then we have studied in multi resolution analysis that after doing low pass filtering and high pass filtering along with the decimation, we do not touch the high pass problem. Instead, we take this low pass branch because this contains maximum energy and then we split it up once again. So, we are going to segregate all the approximated values and we are going to call that as S of i. So, what I will have is 6, 4, 4 and 2. So, I am talking about only these values 6, 4, 4 and 2, only the approximated values. I am going to call this matrix a 2 by 2 matrix as S of i. And if I do multiplication of that with a and typically we know how 2 by 2 a would look, then it will be like this. Then what this product between S of i and a is going to give us? That is of great importance because that is indeed going to take me to next scale in my analysis. So, S of i into a is indeed going to give me and I will have to once again do because this is only in horizontal direction. So, I would call this as S i of h and then if I want to do it also in vertical direction, then a into this. So, a is like this into this matrix and this is going to produce 4, 1, 1, 0. And now using this information, if I want to finally write down how the matrix is going to look, then in the wavelet domain and let us call that matrix as w. This is the original signal that we wanted to analyze. And so I can very well write this down in an interesting manner. So, what I will do is the S i of h v that we have calculated. Now, I will have to distribute it back. So, I will distribute it back. So, that it will occupy only the approximation values and then I will have to fill in with the rest of the details. And where exactly do I get my details from? Well, I will still get my details from this. And so indeed I can fill in the rest of the details and it is going to look like this. So, this is the matrix that we have generated in wavelet domain. Now, let us say I want to do compression of this information. We have already realized from the energy point of view, my approximation values are of great importance. So, I cannot touch them.  And we will call this as W of D. We will also realize why at all we are doing this. And so this is the difference between W and W D. And this is called as W D because we are saying that this is the denoised version. And using this W D, now we will calculate the inverse transform. So, what we will do is, we will first of all calculate B which is indeed equal to 4, 1, 1 and 0 because I am going to redistribute back these coefficients. And for that I will have to first of all find out the inverse on them. So, using the philosophy of A inverse B A inverse, I am going to calculate the inverse from the second level of analysis. We have already pointed out that A, if A is equal to this, then A inverse looks like this. And that indeed makes my transformation matrix A to be unitary. Now, if we carry out these operations, what we will generate is this. I will distribute back these parameters in order to generate S D of H V. And I can very well say S D, this is the denoised version of H V is going to look like this now. Where again 6, 4, 4 and 2 is going to be equal to it is coming from here. And now if we apply the same philosophy and if we try and calculate A inverse and finally, in order to generate S of D which I can calculate as and if we do that, then what we are going to get is S of D. And this S of D I can compare with the S that we started of our analysis with. And you can clearly sense that in the original signal or the original image that we started our analysis with, the first two rows were relatively smoother. However, for row number 3 and row number 4, there was problem, especially you have variations like 8 to 2, 2 to 4. So, you have opposite side slopes and again going from 4 to 0, then 6 to 0, 0 to 2 opposite side slopes. So, row number 3 and 4 are indeed bumpy. And we really wanted to denoise particularly these two rows. And you will realize that in a way we have solved the problem of the third row. The first two rows are even better now, they are smoother. But row number 3 is significantly better because this 0 has become 1 and this 4 has now become 3. So, it is better. However, problem with the last row still remains. But, we know what is the solution. We can go on pursuing this analysis for next couple of scales and probably instead of using Haar wavelet, we can think of using D B 2 wavelet which will have 3 vanishing moments because it has 4 coefficients. So, 3 vanishing moments and that will give us results much quicker. So, this is one application that clearly indicates the intricate calculations and how the two dimensional analysis actually takes place. The next important thing that we are going to quickly cover is an application that deals with pattern recognition. Well, pattern recognition is one interesting branch and there is a very popular saying by Bob Dogwin's and he says that any and every system can be mapped as pattern recognition system. The day to day decisions that we take right from there boiling down to very complex systems where you have to ultimately decide certain threshold values and by the end of the day what is at the heart of these systems is to take decisions. You have to take those decisions, you have to choose, you have to make your choices and that actually formulates the crux of pattern recognition. We are going to see one application quickly using wavelet transform, but predominantly this will be left as an assignment to the viewers of this lecture and I am just going to lay down the framework. Let us say we have two letters correspondingly letter L, letter C and also letter U. So, these are the three letters that we have with us and let us say I am going to map L matrix. Let us say it is an image, it is a 4 by 4 matrix and I am going to map it so that it indeed looks like L. So, it indeed looks like L. I am going to map C also so that it looks almost like C and correspondingly U. So, this indeed looks like a U and what needs to be done from hence onwards is calculate W of L using the same philosophy. Calculate W of C and also calculate W of U and then calculate mean squared error. The concept of mean squared error is well known. We can find out the difference between the corresponding elements for example, element 1 1 of the first matrix with the second matrix, the same element 1 1 and we can find out the difference and take the square of it and in the end find out the average. So, that is why it is called as mean squared error. You take the error which is the difference, square it up and finally find out the average value or the mean value. So, find out the mean squared error between L and C, C and U and then U and L and you will realize that indeed L and C are much, much closer than the distance between L and U and probably C and U. So, this is one exercise that is left to the viewers of this video lecture. With this we come to the end of this particular lecture and what we have seen in today's lecture is the decomposition of the signal into 2 series namely scaling series and the wavelet series gives us great depth. We can think of using them individually because the phi series is low pass filter can be used for applications like denoising and psi series is a high pass filter and can be used for applications like detecting certain patterns and together they have even more beautiful applications. I will leave it to the viewers to explore more and more applications of wavelet transform. We will stop here. Thank you.


Hey there lads and ladies. It is Petrifying Pumpkins here and I am back with a super quick, super exciting firewall zero hour news update. This is all about DLC number two which is coming out tomorrow. First of all I just want to give a quick thanks to Madison74PSN who sent me this link which links to the firewall zero hour official community over on Facebook where a member from Australia has posted up a screenshot of the store where their store seems to have updated to show the new contractor. We get a look at her face. We see that her name is Nash and we get a description here and it says a brand new contractor joins the fight. Nash brings with her the new binary scale which lets you take two C4 charges into a mission. Now we did eventually guess that she would be bringing two C4 charges so it looks like we were on the money in that particular one after some hints. But what's also interesting here is that this user says that she costs nearly six Australian dollars and when I convert that to euros that's like nearly four euros so it's not particularly expensive. However I feel like a lot of you will say that she should be free but I think as long as her skill binary is free then there's no real need for her to be free she doesn't offer anything else you know and technically she that kind of does count as just a cosmetic just a skin you know as long as her skill as long as the thing that makes her unique is added to the game for everyone else to equip I don't have a problem with that but I might have a problem with it if it isn't we will have to wait and see tomorrow and so that's kind of it for this video las and ladies I'd be interested to see what you think I I think I can already predict what some of you are gonna say but let's have that discussion down in the comments below thank you very much for watching as always if you like this video if you want to keep up to date with firewall zero hour and some psv or news and whatever then subscribe to this channel and consider doing all that usual moist shies I'll see you guys in the next one bye for now


narcissistic relationships versus healthy relationships it's good to know what you don't want in a relationship knowing what you don't want can help you to define what you do want which is why when we've been in a relationship with a narcissist for a long period of time we will often know what we don't want it's good to have that contrast it's good to see that striking difference between a narcissist and someone who is emotionally healthy there are many desirable and attractive qualities and traits that sets the healthy person apart from the narcissist but you may not be aware of what those qualities and traits are until you have been involved with a narcissist sometimes you have to be in a really bad relationship before you can appreciate one that is amazing when you have been with the narcissist for several months or years you will really appreciate someone who is emotionally healthy and that is why I would like to describe the narcissistic relationship first so that you can then appreciate everything good about the healthy relationship a relationship with a narcissist is one-sided all of the effort is coming from you while the narcissist just takes everything that you have to give it will be very unequal one balanced and this will cause trouble and distress for you if you dare to point out how unfair the relationship is if you stop doing what the narcissist wants and expects you to do they might call you selfish or claim that you are the narcissist simply because they are not getting their way they will forget everything that you have done for them up until that point because just that one time you wouldn't let them have their way it causes a narcissist injury and makes them really angry where they may even seek to punish you in return for that perceived injury they may insult you or express disapproving comments or judgments about you if you don't do what they want they will become very impatient they will become quickly irritated or annoyed and you may feel as though you are walking on eggshells they fail to understand your point of view because they're self-absorbed and lack empathy they only care about their own interests or needs most of your conversations involve them talking about what interests them what they need from you or what they want you to do for them any time that you try to talk about your own interests or needs the narcissist doesn't want to hear it or they will just hijack the conversation and start talking about something that interests them they are too preoccupied with their own interests or needs to consider you and this leaves you feeling alone and invalidated you feel as though you're not being heard and you wonder how this person could even say that they know you or love you when they never took the time to hear you out they never took the time to listen to you in the way that you listen to them so of course they don't know you of course they don't love you they only love the image that they have in their minds which really has nothing to do with you it has everything to do with what you can do for them or how you can make them feel but it has nothing to do with you as a person because they never took the time to get to know you they never had an interest in getting to know you because the sole purpose of them being with you is so that you can provide something to them they're in survival mode they need to use you for their own survival and this is something you will notice when you have a conversation with them that is why the conversation always revolves around their interests or needs because they are in need they are in need of your time energy and attention but no amount that you give to them will ever be enough you can't just attend to them we'll hear them out for a couple of hours and then they'll start to consider you know it doesn't work that way with an narcissist they will want your life to revolve around them catering to their needs 24 hours a day and when it comes down to you needing their attention most of the time they won't even be listening or it will just remind them of something that they need you to attend to they really have no interest in you as a person they don't love you they love what you can do for them and they love how you can make them feel but they don't love you they can't love you love is about giving and accepting when all the narcissist seems to do with you is take disapprove deny refuse they're always in opposition to you it's not love it's the opposite of love and the opposite of love is not hate it's indifference which can be defined as them having a lack of interest concern and sympathy for you they don't want to know or learn something about you you're not a matter of interest or importance to them and that is why there is no support in the form of shared feelings or opinions that is why there is no understanding because really not only do they not love you they don't even care about you that is why they feel no pity or sorrow for your misfortune because they're only concerned with their own interests or needs when love is about giving love is about acceptance there is no acceptance with the narcissist all you do is argue or disagree day after day because you're not even compatible there is a serious incompatibility between your opinions principles or interests and it gets to a point where you're afraid to even express your own opinions you're afraid to express how you feel because you never know how they're going to react you don't know if it's going to start another argument because you're dealing with someone who is like a loose cannon they behave in an unpredictable and uncontrolled way that is likely to cause danger or harm and you never know what might set them off this is because they lack emotional stability which causes them to experience rapid often exaggerated changes in mood it causes you to feel like you're always walking on eggshells because you never know how they might react it's an unstable chaotic environment of complete confusion and disorder a healthy relationship it's the opposite of all of this when you are with someone who is emotionally healthy you will feel secure you will feel safe and unthreatened because you're in a safe and threatening environment there will be things that you have in common things that you both value and appreciate but then there will also be differences whether our differences there will be boundaries and mutual respect there will be an understanding that you don't share all of the same likes and interests or the same beliefs or values and that's not going to be a problem as long as you feel equally respected and appreciated they will be there for you they will be there to provide you with support and comfort especially when you have problems they will provide encouragement or emotional help they will listen to you they will understand you they will anticipate your needs there's just something you get from an emotionally healthy person that is difficult to describe or express but it is highly pleasing highly satisfying and it is what causes you to want to establish a relationship with them a relationship is usually based on shared feelings interests or experiences but you don't get that with a narcissist it makes you wonder how you even got together because most of the time you spend together is in disagreement or opposition they expect you to see it as a privilege just to be in their presence they're with you every day so what more could you want from them of course they don't consider that with that logic anyone could qualify to be in this relationship what makes them so special what are they given to you that makes them deserve to be with you the truth is the narcissist's idea of love is very different to our idea of love they see love as what you can give to them while we see love as what we can give to others and that is why when you're doing whatever they want you to do they will tell you that they love you how do you feel in this relationship do you feel loved and appreciated do you feel like this person really values you or do you feel annoyed and distressed do you feel like you don't exist in this relationship do you feel like they're taking advantage of you do you feel disconnected do you feel afraid if you are with a narcissist you are going to be feeling like you're living in the opposite way of how you want to live however you feel right now you want to feel the exact opposite you don't want to feel uncertain about yourself you want to feel confident you don't want to feel anxious you want to feel peace you don't want to feel indifference you want to feel love think about how you feel right now it will help you to imagine how you really want to feel when you define what you don't want it helps you to become aware of what you do want it's about interest love is about acceptance it's about peace patience and contentment when you are in a healthy relationship you will feel their empathy when you look into their eyes you will feel that they really care about you and they may even tell you that they do but they won't just say it with words they will prove it with their actions and their actions and words will be very consistent they won't change depending on their feelings or circumstances if someone really loves and cares for you that's not something that just suddenly goes away when you are in a healthy relationship and you make a fault or mistake they will not hold it against you they will not use it as a reason for no longer treating you with the respect that you deserve when you are in a healthy relationship and you love someone you will accept their faults and mistakes you will accept their flaws and imperfections and love them anyway because love is acceptance love is about not wanting to change anything if you really love someone why would you ever want them to change the narcissist does not love you they never did and they never will they are not capable of experiencing love you cannot experience love without high empathy if you want to know how much empathy a person has pay attention to how they treat the people around you are they kind to people in general are they kind to animals are they kind to people who can't do anything for them this is what love is about it's about accepting the world accepting everyone and everything and not wanting to change anything but this is how we must feel about ourselves first before we can feel that way about anyone else we must love and accept ourselves before we can love and accept anyone else the narcissist does not love themselves it's an act they do not accept themselves that is why they created a false self and that is why they will never be capable of loving you or anyone else love is about being vulnerable it is not about being afraid but when you're with a narcissist you're too afraid to be vulnerable so how could you ever love them when they take you for granted and they don't treat you the way you deserve you don't accept them as they are you want them to change and that is not love when you are in a healthy relationship and there is love there is both give and take they're interested in you they want to learn more about you the supportive and understanding there's no pretending there's just a natural desire to be together you feel comfortable with them you feel loved you feel cared for and protected as though they are encouraging your progress and development you will never experience love with a narcissist but one day you may fall in love with someone else it all starts with loving yourself remind yourself every day of everything you love and appreciate about yourself remind yourself of what makes you valuable and work on those qualities work on becoming the best version of yourself and one day you will meet someone who is doing the same thing healing and developing themselves hoping that one day they would meet someone just like you thank you for watching i hope this video resonated with you please like comment share and subscribe click the bell icon to receive notifications for my future videos if you're delighted to donate my paypal link is in the video description coaching inquiries you can email me at archivalcoachinadjuma.com thank you for watching and i'll talk to you soon