The magnitude of the gradient tells us how quickly the image is changing, while the direction of the gradient tells us the direction in which the image is changing most rapidly.
Note that the slope of the line may be negative; this tells us that y is decreasing as x increases. For example: The larger the magnitude of the slope, the steeper the line is, i.e. the more it approaches the vertical. In other words, if the line is near vertical then y is changing very fast relative to x. Slope (or Gradient). So far, we have learned that the equation for any line can be written in the form y = mx + b. We have also found that the value of b is given by the y-intercept. Now let's have a look at what m represents. Let (x 1, y 1) and (x 2, y 2) be two distinct points on the line given by y = mx + b So, the gradient of the line is 3. This means that the cost of transporting documents is $3 per km plus a fixed charge of $5, i.e. it costs $5 for the courier to arrive and $3 for every kilometre travelled to deliver the documents. The gradient is the rate of change of one variable with respect to another. Gradient of a Straight Line. The gradient of a straight line is the rate at which the line rises (or falls) vertically for every unit across to the right. Gradient of the 2-d function f(x, y) = xe is plotted as blue arrows over the pseudocolor plot of the function. Consider a room in which the temperature is given by a scalar field, T, so at each point (x, y, z) the temperature is T(x, y, z) . (We will assume that the temperature does not change over time.) At each point in the room, the gradient of T at that point will show the direction the temperature rises most quickly. The gradient of H at a point is a vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector. The greater the slope the steeper the line. Consider the linear function: y = a + bx b is the slope of the line. Slope means that a unit change in x, the independent variable will result in a change in y by the amount of b. slope = change in y/change in x = rise/run Slope shows both steepness and direction. Algebra If y is a linear function of x, then the coefficient of x is the slope of the line created by plotting the function. Therefore, if the equation of the line is given in the form y = mx + b \, then m is the slope. A) The slope is 0.9. It tells us for each 1 kg increase in weight, the expected increast in... show more For a sampel of males, it is found that the least squares regression line connecting their weight w kg and their height h cm is given by w = 0.9h - 80. It tells us for each 1 kg increase in weight, the expected increast in height is 0.9 cm. B) The slope is 0.9. It tells us that for each 1 cm increase in height, the expected increase in weight is 0.9 kg. The magnitude of the gradient tells us how quickly the image is changing, while the direction of the gradient tells us the direction in which the image is changing most rapidly. To illustrate this, think of an image as like a terrain, in which at each point we are given a height, rather than an intensity. We will see that the gradient allows us to answer this question. Let’s consider how the image changes as we move from position (x,y) by a small amount, ∆, in an arbitrary direction θ, which will take us to the position (x+ ∆ cos θ, y + ∆ sin θ). At (x,y) the image intensity is I(x,y) . In other words, the slope of the line tells us the rate of change of y relative to x. If the slope is 2, then y is changing twice as fast as x; if the slope is 1/2, then y is changing half as fast as x, and so on. Slope (or Gradient). So far, we have learned that the equation for any line can be written in the form y = mx + b. We have also found that the value of b is given by the y-intercept. Now let's have a look at what m represents. Let (x 1, y 1) and (x 2, y 2) be two distinct points on the line given by y = mx + b A higher slope value indicates a steeper incline. The slope is defined as the ratio of the rise divided by the run between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. Algebra If y is a linear function of x, then the coefficient of x is the slope of the line created by plotting the function. Therefore, if the equation of the line is given in the form y = mx + b \, then m is the slope. You can't learn about linear equations without learning about slope. The slope of a line is the steepness of the line. There are many ways to think about slope. Slope is the rise over the run, the change in 'y' over the change in 'x', or the gradient of a line. Check out this tutorial to learn about slope!
http://mathsfirst.massey.ac.nz/Algebra/StraightLinesin2D/Slope.htm http://www.mathsteacher.com.au/year10/ch03_linear_graphs/02_gradient/line.htm https://en.wikipedia.org/wiki/Gradient https://answers.yahoo.com/question/index?qid=20090704032713AAXVtOH https://au.answers.yahoo.com/question/index?qid=20091105112532AApHEgL http://www.cs.umd.edu/~djacobs/CMSC426/ImageGradients.pdf http://mathsfirst.massey.ac.nz/Algebra/StraightLinesin2D/Slope.htm https://answers.yahoo.com/question/index?qid=20090704032713AAXVtOH http://www.virtualnerd.com/pre-algebra/linear-functions-graphing/rate-of-change-slope/define-slope/slope-definition
what does the gradient tell us