Saturday, 20 June 2020

JTS OverlayNG - Tolerant Topology Transformation

This is another in a series of posts about the new OverlayNG algorithm being developed for the JTS Topology Suite. (Previous ones are here and here).  Overlay is a core spatial function which allows computing the set-theoretic boolean operations of intersection, union, difference, and symmetric difference over all geometry types. OverlayNG introduces significant improvements in performance, robustness and code design.

JTS has always provided the ability to specify a fixed-precision model for computing geometry constructions (including overlay).  This ensures that output coordinates have a defined, limited precision.  This can reduce the size of data transfers and storage, and generally leads to cleaner, simpler geometric output.  The original overlay implementation had some issues with robustness, which were exacerbated by using fixed-precision.  One of the biggest improvements in OverlayNG is that fixed-precision overlay is now guaranteed to be fully robust.  This is achieved by using an implementation of the well-known snap-rounding noding paradigm. 

Geometric algorithms which operate in a fixed-precision model can encounter situations called topology collapse.  This happens when line segments and points become coincident due to vertices or intersection points being rounded.  The OverlayNG algorithm detects occurrences of topology collapse and transforms them into valid topology in the overlay result.

Topology collapse during overlay with a fixed precision model

As a bonus, handling topology collapse during the overlay process also allows it to be tolerated when present in the original input geometries.  This means that some kinds of "mildly" invalid geometry (according to the OGC model) are acceptable as input.  Invalid geometry is transformed to valid geometry during the overlay process.

Specifically,  input geometry may contain the following situations, which are invalid in the OGC geometry model:
  • A ring which self-touches at discrete points (the so-called "inverted polygon" or "exverted hole")
  • A ring which self-touches in one or more line segments
  • Rings which touch other ones along one or more line segments 
Note that this does not extend to handling polygons that overlap, rather than simply touch.  These are "strongly invalid", and will trigger a TopologyException during overlay.

An interesting use for this capability is to process individual geometries.  By simply computing the union of a single geometry the geometry is transformed into an OGC-valid geometry.  In this way OverlayNG functions as a (partial) "MakeValid" operation.  
A polygon which self-touches in a line transforms to a valid polygon with a hole

A polygon which self-touches in a point transforms to a valid polygon with a hole

A collection of polygons which touch in lines transforms to a valid polygon with a hole

Moreover, some spatial systems use geometry models which do not conform to the OGC semantics.  Some systems (such as ArcGIS) actually specify the use of inverted polygons and exverted holes in their topology model.  And in days of yore there were systems which were unable to model holes explicitly, and so used a "connected hole" topology hack (AKA "lollipop holes".) This represented  holes as an inversion connected by a zero-width corridor to the polygon shell. Both of these models are accepted by OverlayNG. Thus it provides a convenient way to convert from these non-standard models into OGC-valid topology. 

This is one more reason why overlay is the real workhorse of spatial data processing!

Thursday, 18 June 2020

JTS OverlayNG - Noding Strategies

In a previous post I unveiled the exciting new improvement in the JTS Topology Suite called OverlayNG.  This new implementation provides significant improvements to the core function of spatial overlay.  Overlay supports computing the set-theoretic boolean operations of intersection, union, difference, and symmetric difference over all geometric types.

One of the design goals of JTS is to create modular, reusable data structures and processes to implement spatial algorithms.  This increases development velocity and testability, and makes algorithms easier to understand.  In spatial algorithms it is not always obvious how to identify appropriate abstractions for reuse, so this is an on-going effort of design and refactoring.

After the implementation of spatial overlay in the very first release of JTS, it became clear that overlay can be split into the following phases:

  1. Noding, in which an set of possibly-intersecting linestrings is converted to an arrangement in which linestrings touch only at endpoints
  2. Topology Analysis, during which the topology graph of the noded arrangement is determined
  3. Result Extraction, in which the geometric components of the desired result are extracted from the topology graph

It also became clear that the Noding phase is critical, since it determines the overall performance and robustness of the overlay computation.  Moreover, tradeoffs between these two qualities can be made by using different noding strategies.  For instance, the "classic" JTS noding approach is fast, but susceptible to robustness issues.  Alternatively, noding using the well-known snap-rounding paradigm is slower, but can be made fully robust. 

To encapsulate this concept, JTS introduced the Noder API.  Since it post-dated the original overlay code, using it in overlay had to await a reworking of that codebase.  The OverlayNG project provided this opportunity.  OverlayNG allows supplying a specific Noder class to be used during overlay. 

One of the main goals of the OverlayNG project was to develop a noder to provide fully robust noding.  This would eliminate the notorious TopologyException errors which bedevil the use of overlay.  The effort has paid off with the development of not one, but two new noders.  The Snapping Noder has very good performance and (with the addition of some heuristics, and so far as is known) provides robust full-precision evaluation.  And the Snap-Rounding noder provides guaranteed robustness as well the ability to enforce a fixed-precision model for output.

So now OverlayNG can be run with the following suite of noders, depending on use case.  The images show the result of intersection and union on the following geometries:

Fast Full-Precision Noder

The MCIndexNoder noding strategy has been available since the early days of JTS. It has very good performance due to the use of monotone chains and the STRtree spatial index.  However, it is a relatively simple algorithm which due to numerical robustness issues does not always produce a valid noding.  In overlay it is always used in conjunction with a noding validator, so that noding failure can be detected and an alternative strategy used to perform the operation successfully.

Intersection and Union with full-precision floating noding

Snapping Noder

The SnappingNoder is a refinement of the MCIndexNoder which snaps existing input and computed intersection vertices together, if they are closer than a snap distance tolerance.  This dramatically improves the robustness of the noding, with only minor impact on performance. 

Noding robustness issues are generally caused by nearly coincident line segments, or by very short line segments.  Snapping mitigates both of these situations.  The choice of snap tolerance is a heuristic one.  Generally, a smaller snap distance has less chance of distorting the topology, but it need to be large enough to resolve intersection computation imprecision.  In practice, excellent robustness is provided by using a very small snap distance (e.g. a factor of 10^12 smaller than the geometry magnitude).

Snapping of course risks creating topology collapses, but OverlayNG  is designed to handle these correctly.  However, there are occasional situations where the snapped arrangement is too invalid to  be handled.  This can be detected, and with some simple heuristic adjustments (e.g. a more aggressive  snap distance) the overlay can be rerun.  This strategy has proven to be fully robust in all cases tried so far.

Intersection and Union with Snapping Noding (snap tolerance = 0.5)

Snap-Rounding Noder

The SnapRoundingNoder implements the well-known snap-rounding paradigm.  It provides fully robust noding by rounding and snapping linework to a fixed-precision grid.  This has the unavoidable effect of rounding every output vertex to the precision grid. This may or may not be desirable depending on the situation.  A useful side effect is that it provides an effective means of reducing the precision of geometries in a topologically valid way. 

In the early stages of OverlayNG design and development I expected that snap-rounding would be required to ensure fully-robust overlay, in spite of the downside of fixed-precision output.  But the development of the SnappingNoder and accompanying heuristics means that this noder need only be used when control over overlay output precision is desired. 

Using an appropriate precision model is a highly worthwhile goal in spatial data management, since it reduces the amount of memory needed to represent data, and improves robustness and portability.  This is unfortunately often neglected, mostly due to lack of tools available to enforce it.  Hopefully this capability will encourage users to maintain a precision model which is better matched to the true precision of their data.

Intersection and Union with Snap-Rounding noding (precision scale = 1)

Segment Extracting Noder

This is a special-purpose noder which is really more of a "non-noder". It simply extracts every line segment in the input.  It is used on geometry collections which form valid, fully-noded, non-overlapping polygonal coverages.  When used with OverlyNG, this has the effect of dissolving the duplicate line segments and producing the union of the input coverage. By taking advantage of the structure inherent in the coverage model the SegmentExtractingNoder offers very fast performance.  It can also operate on fully-noded linear networks. 

Union of a Polygonal Coverage with SegmentExtractingNoder

The support for pluggable noding and the development of a suite of fast and/or robust noders constitutes the biggest advance of the OverlayNG code.  It finally allows JTS to provide fully robust noding and true support for a fixed-precision model!  This has been a dream of mine for more than a decade.  It's good to think that the end of the era of TopologyException issues is in sight!

Sunday, 17 May 2020

JTS Overlay - the Next Generation

In the JTS Topology SuiteOverlay is the general term used for the binary set-theoretic operations intersection, union, difference and symmetric difference.  These operations accept two geometry inputs and construct a geometry representing the operation result.  Along with spatial predicates and buffer they are the most important functions in the JTS API.

Intersection of MultiPolygons

Overlay operations are used in many kinds of spatial processes. Any system aspiring to provide full-featured geometry processing simply has to provide overlay operations.  In fact, many geometry libraries exist solely to provide implementations of overlay. Notable libraries include the ESRI Java API, Clipper and wagyu.  Some of these provide overlay only for polygons, which is the most difficult case to compute.

Overlay in JTS 

The JTS overlay algorithm supports the full OGC SFS geometry model, allowing any combination of polygons, lines and points as input.  In addition, JTS provides an explicit precision model, to allow constraining output to a desired precision.  The overlay algorithm is also available in C++ in GEOS.  There it provides overlay operations in numerous systems, including PostGIS, QGIS, Shapely, and r-sf.  This codebase has had an long lifespan; it was developed back in 2001 for the very first release of JTS, and while there have been improvements over the years the core of the design has remained unchanged.

However, there are some long-standing issues with JTS overlay.  The most serious one is that in spite of much valiant effort over the years, overlay is not fully robust.  The constructive nature of overlay operations makes them particularly susceptible to the robustness issues which are notorious in geometric algorithms using floating-point numerics.  It can happen that running an overlay operation on seemingly innocuous, valid inputs results in the dreaded TopologyException being thrown.  There is a steady trickle of issue reports about this in JTS, and even more for GEOS (such as here, here and here...).

Another issue is that the codebase is complex, and thus hard to debug and modify. Partly this is because of the diversity of inputs and the explicit precision model.  To support this the JTS overlay algorithm has a rich and detailed semantics.  But some of the complexity is due to the original design of the code. This makes it difficult to incorporate new ideas for improvements in performance and robustness.

Next Generation Overlay

So for many years it's been on my mind that JTS overlay needs a thorough overhaul. I chipped away at the problem over time, but it was clear that it was going to be a major effort.  Now, thanks to the support of my employer Crunchy Data, I've at last been able to focus on a complete rewrite of the JTS overlay module.  It's called OverlayNG.

The basic algorithm remains the same:
  1. Extract the input linework, and node it together
  2. Build a topology graph from the noded linework
  3. Compute a full topological labelling of the graph
  4. Extract the resultant polygons, lines and points from the graph
This algorithm is time-tested and is able to handle the complexities of multiple geometry types and topology collapse. The new codebase benefits from 20 years of experience to become simpler and more modular, with increased testability, and potential for reuse.

OverlayNG has the following improvements:
  • A snap-rounding noder is available to make overlay fully robust.  This eliminates the possibility of TopologyExceptions (when an appropriate precision model is used).
Intersection operation with Snap-rounding 
and Topology Collapse removal
  • Snap-rounding allows full support for specifying the output precision model.  The precision model can be specified independently for each overlay call, which is more flexible and easier to use.  The use of snap-rounding also provides fully valid precision reduction for geometries.  This makes it feasible for the first time to fully operate in a fixed-precision regime.
Precision Reduction turned all the way up to 11
  • Significant performance optimizations are included (notably, one which makes polygon intersection much faster in many cases)
Intersection of a MultiPolygon with a grid (7x faster with OverlayNG)
  • Pluggable noding allows providing different noding strategies.  One use is to run OverlayNG with the original floating-point noder, which is faster than snap-rounding (but of course has the robustness issues noted above).  Another is to use a special-purpose noder to provide very fast polygonal coverage union.
Union of a polygonal coverage (10x faster with OverlayNG)
  • modular and cleaner codebase allows easier testability, maintenance, enhancement and reuse.  A winged-edge graph model is used for the topology graph. This is simpler and less memory intensive.
  • The rebuild gives an opportunity to make some semantic improvements: 
    • Empty results are returned as empty atomic geometries of appropriate type, rather than awkward-to-handle empty GeometryCollections
    • Linear output is merged node-to-node.  This gives union a more natural and useful  semantic
A benefit of the new codebase is that it is easier to enhance and extend.  For example, it should be straightforward to finally provide a SplitPolygon function for JTS.  Another potential extension is overlay for Polygonal Coverages.

Code that is so widely used needs to be thoroughly tested against real-world workloads.  Initially OverlayNG will be released as a separate API in JTS.  This allows it to be used along with the original overlay.  It can be used as a fallback for cases which fail in the original overlay process.  Once the new code has been proved out in real world use, it is likely to become the standard overlay code path. Also, the code will be ported to GEOS soon, where we're hoping it will provide significant benefits to the many systems that use GEOS.

I'll be posting more articles about aspects of OverlayNG soon. The code is almost ready to release, after some final testing. In the meantime, the pre-release code is available in a Git branch. It would be great to get as much beta-testing as possible before final release, so try it out and log some feedback!

Tuesday, 14 April 2020

Maximum Inscribed Circle and Largest Empty Circle in JTS

There is often a need to find a point which is guaranteed to lie in the interior of a polygon.  Uses include placing cartographic labels, and using the point as a proxy for polygon containment or overlap (such as in polygon overlay).

There are several ways to compute a "centre point" for a polygon.  The simplest is the polygon centroid, which is the Center of Mass of the polygon area.  This has a straightforward O(N) algorithm, but it has the significant downside of not always lying inside the polygon!  (For instance, the centroid of a "U" shape lies outside the shape).  This makes it non-useful as an interior point algorithm.

JTS provides the InteriorPoint algorithm, which is guaranteed to return a point in the interior of a polygon.  This works as follows:
  • Determine a horizontal scan line on which the interior point will be located. To increase the chance of the scan line having non-zero-width intersection with the polygon the scan line Y ordinate is chosen to be near the centre of the polygon's Y extent but distinct from all of vertex Y ordinates.
  • Compute the sections of the scan line which lie in the interior of the polygon.
  • Choose the widest interior section and take its midpoint as the interior point.
This works perfectly for finding proxy points, and usually produces reasonable results as a label point.  

However, there are polygons for which the above algorithm finds points which lie too close to the boundary for a label point.  A better choice would be the point that lies farthest from the edges of the polygon.  The geometric term for this construction is the Maximum Inscribed Circle. The farthest point is the center of the circle. 

Comparison of center points for a not-so-typical polygon

In the geographic domain this is romantically termed the Pole of Inaccessibility.  
Pole Of Inaccessibility in Canada

This point occurs at a node of the medial axis of the polygon, so in theory all that is needed is to compute the medial axis and test the set of node points.  However, medial axis algorithms are notoriously difficult to implement, and can be expensive to compute.  So it's appealing to look for a simple and fast way to compute a good approximation to the Maximum Inner Circle center.  There have been various approaches to this, including a geodetic grid-based approach by Garcia-Castellanos & Lombardo, and one by Martinez using random point distributions.  Recently Mapbox released a clever implementation which uses successive refinement of a grid along with a branch-and-bound technique to reduce the amount of searching needed.  

JTS now has a version of this algorithm, called MaximumInscribedCircle.  It significantly improves performance by using spatial indexing techniques for both polygon interior testing and distance computation.  This makes it very fast to find the MIC even for large, complex polygons.  Performance is key for computing label points, since it is likely to be used for many polygons on a typical map.

Grid refinement to find Maximum Inscribed Circle

An interesting property of the MIC is that its radius is the distance at which the negative buffer of the polygon disappears (becomes empty).  Thus the MIC radius length is a measure of the "narrowness" of a polygon.   This is often useful for purposes of simplification or data cleaning, to remove narrow polygonal artifacts in data.

Sequence of negative buffers containing the Maximum Inscribed Circle center

And, as the infomercials say, that's not all!  If you act today you also get a free implementation of  the Largest Empty Circle !  The Largest Empty Circle is defined for a set of geometric obstacles.  It is the largest circle that can be constructed whose interior does not intersect any obstacle and whose center lies in the convex hull of the obstacles.  The obstacles can be points, lines or polygons (although only the first two are currently implemented in JTS).  Classic use cases for the Largest Empty Circle are in logistics to find a location for a new chain store in a set of store locations; or to find the largest roadless area in environmental planning.

It turns out that the LEC can be computed by essentially the same algorithm as the MIC, with a few small changes.  And of course it also uses spatial indexing to provide excellent performance.
Largest Empty Circles for point and line obstacles

Maximum Inscribed Circle and Largest Empty Circle are now in JTS master, and will be released in the upcoming version 1.17.

Further Improvements

There are some useful enhancements that can be made:
  • For Maximum Inscribed Circle, allow a second polygonal constraint.  This supports finding a label point within a view window rectangle.
  • For Largest Empty Circle, allow a client-defined boundary polygon.  This allows restricting the circle to lie within a tighter bound than the convex hull
  • For both algorithms, it should be feasible to automatically determine a termination tolerance

Monday, 3 February 2020

Running commands in the JTS TestBuilder

The JTS TestBuilder is a great tool for creating geometry, processing it with JTS spatial functions, and visualizing the results.  It has powerful capabilities for inspecting the fine details of geometry (such as the Reveal Topology mode).  I've often thought it would be handy if there was a similar tool for PostGIS.  Of course QGIS excels at visualizing the results of PostGIS queries.  But it doesn't offer the same simplicity for creating geometry and passing it into PostGIS functions.

This is the motivation behind a recent enhancement to the TestBuilder to allow running external (system) commands that return geometry output.  The output can be in any text format that TestBuilder recognizes (currently WKT, WKB and GeoJSON).   It also provides the ability to encode the A and B TestBuilder input geometries as literal WKT or WKB values in the command.  The net result is the ability to run external geometry functions just as if they were functions built into the TestBuilder.


Running PostGIS spatial functions

Combined with the versatile Postgres command-line interface psql, this allows running a SQL statement and loading the output as geometry. Here's an example of running a PostGIS spatial function.  In this case a MultiPoint geometry has been digitized in the TestBuilder, and processed by the ST_VoronoiPolygons function.  The SQL output geometry is displayed as the TestBuilder result.

The command run is:

/Applications/ -qtA -c 
"SELECT ST_VoronoiPolygons('#a#'::geometry);"

Things to note:
  • the full path to psql is needed because the TestBuilder processes the command using a plain sh shell.  (It might be possible to improve this.)
  • The psql options -qtA  suppress messages, table headers, and column alignment, so that only the plain WKB of the result is output
  • The variable #a# has the WKT of the A input geometry substituted when the command is run.  This is converted into a PostGIS geometry via the ::geometry cast.  (#awkb# can be used to supply WKB, if full numeric precision is needed)

Loading data from PostGIS

This also makes it easy to load data from PostGIS to make use of TestBuilder geometry analysis and visualization capabilities.  The query can be any kind of SELECT statement, which makes it easy to control what data is loaded.  For large datasets it can be useful to draw an Area of Interest in the TestBuilder and use that as a spatial filter for the query.  The TestBuilder is able to load multiple textual geometries, so there is not need to collect the query result into a single geometry.

Loading data from the Web

Another use for commands is to load data from the Web, by using curl to retrieve a dataset.  Many web spatial datasets are available in GeoJSON, which loads fine into the TestBuilder. Here's an example of loading a dataset provided by an OGC Features service (pygeoapi):

Command Panel User Interface

The Command panel provides a simple UI to make it easier to work with commands.  Command text  can be pasted and cleared.  A history of commands run is recorded for the TestBuilder session.  Recorded commands in the session can be recalled via the Previous and Next Command buttons.
Buttons are provided to insert substitution variable text.

To help debug incorrect command syntax, error output from commands is displayed.

It can happen that a command executes successfully, but returns output that cannot be parsed.  This is indicated by an error in the Result panel.  A common cause of this is that the command produces logging output as well as the actual geometry text, which interferes with parsing.  To aid in debugging this situation the command window shows the first few hundred characters of command output.  The good news is that many commands offer a "quiet mode" to provide data-only output.

Unparseable psql output due to presence of column headers.  The pqsl -t option fixes this.

If you find an interesting use for the TestBuilder Command capability, post it in the comments!

Thursday, 21 November 2019

Variable-distance buffering in JTS

The operation of buffering a geometry is a core geospatial concept.  Standard buffers are computed using a fixed distance around the input geometry. JTS has provided a buffer implementation since its inception, and this is used in GEOS and all the other downstream projects as well. 

One way to generalize the buffer concept is to allow the buffer distance to vary along the geometry.  As often the case with geospatial concepts this construction has a few different names, including tapered buffer, cone buffer, varying buffer, and variable-distance buffer.  The classic use case for variable-distance buffers is generating polygons for the "cone of uncertainty" along predicted hurricane tracks:

Another use case is depicting rivers showing the width of reaches along the river course. 

I took a crack at prototyping "variable width buffers" a few years ago, in the JTS Lab.  Recently there were a couple of GIS StackExchange posts looking for this functionality in PostGIS and Shapely   They were good motivation to buff up the prototype and move it into JTS core.  But I was dismayed to realize that the output had some serious deficiencies, due to an overly-simplistic algorithm.  The idea in the Lab code was appealingly simple: compute buffer circles around each line vertex at the specified distance, and union (merge) them with trapezoids computed around the connecting line segment.  But this produced ugly discontinuities for large deltas in buffer distances. 

For such a seemingly simple concept there is surprisingly little prior art to be found.  Even the GIS That Shall Not Be Named does not seem to provide a native variable buffer function (although recently I found a couple of user contribs which do, to some extent).   There's an implementation in QGIS - but since it seems to be based on the original problematic JTS code that didn't really help.   So I had the fun of coding it up from scratch.  

The problem with the original code is that it should have used the outer tangent lines to the buffer circles at each vertex.  Wikipedia has a good discussion of this construction.  Even better, it provides an elegant mathematical algorithm, which worked perfectly when coded up.

The construction computes a single tangent segment.  The opposite segment is simply the reflection in the line between the circle centres.  The geometric math to handle this is now provided for reuse as  LineSegment.reflect().  Finally, it is notoriously tricky to produce buffer curves with high quality in the fine details.  In this case, generating the line segment buffer caps required care to avoid creating out-of-phase vertices, which would produce a "bumpy" result when merged.

This is now available in JTS as the VariableBuffer class.  In addition to the general-purpose API which allows specifying a distance at each vertex, it provides a couple of simpler functions which accept a start and end distance; and a start, middle and end distance.  The first is an easy way to produce the classic cone of uncertainty.  The latter is useful for buffering rings, or just creating interesting shapes.

The next step is to port this to GEOS. Then it can be exposed in PostGIS and all the other downstream projects like Shapely and R-SF, and maybe even QGIS.

Future Work

There's some enhancements that could be made:
  • Some or all of the buffer end cap and join styles provided by the classic buffer operation could be supported, as well as the ability to set the quadrant segment count
  • Variable buffering of polygons and multigeometries is straightforward to add.  
  • It might be possible to allow different distances for each side of the input line (although it may be tricky to get the joins right)
  • The approach of using union to merge the individual segment buffers works well. But it would improve performance to feed the entire generated variable buffer curve directly into the existing buffer generation code
More experimentally, variable buffering might provide a way to construct a decent approximation to a true geodetic buffer.  The vertex distances would be computed using true geodetic distance.  There might be some complications around long line segments, but perhaps these can be finessed (e.g. by densification). 

Wednesday, 21 August 2019

JtsOp - a CLI for JTS

Since inception JTS has provided two client tools to aid in using the library.  They are the TestBuilder and the TestRunner

  • The TestBuilder is a GUI tool with many powerful capabilities for loading, editing and visualizing geometry.  It also provides the ability to run numerous geometric functions which expose (and in some cases enhance) the JTS library functionality.
  • The TestRunner is a command-line tool which runs tests in the JTS XML test format. 

But there's a gap which these two tools don't fill.  It's often required to run JTS operations on geometry data for purposes of testing, debugging or timing operations.  This can be done in the TestBuilder, but being a GUI it's highly manual process, and tedious to repeat multiple times.  It is (just) possible to use the TestRunner for this, but that introduces the awkwardness of wrapping the input data in XML.  The only other option up until now was to write a Java program, which is overkill for quick tests, and not very accessible for some.

What's really needed is a JTS equivalent of the UNIX expr. It should have the ability to accept geometry inputs, run an operation on them, and output the results.  (Another comparison might be to a very small subset of  GDAL/OGR, focussed on geometry only - and of course running in Java).

The TestBuilder already provides a rich framework for most of this functionality, so it turned out to be simple to expose this as a command-line tool.  Behold - the jtsop command!

jtsop has the following capabilities:
  • read geometries from files or command-line
  • input formats include WKT, WKB, GeoJSON, GML and SHP
  • execute any TestBuilder operation on the geometry input (which includes all JTS Geometry methods)
  • output the result as WKT, WKB, GeoJSON, GML or SVG
  • report metrics for geometry and execution times
  • dynamically load and run geometry functions provided in external Java classes 

Compute the area of a WKT geometry and output it as text
jtsop -a some-geom.wkt -f txt area 

Compute the unary union of a WKT geometry and output as WKB
jtsop -a some-geom.wkt -f wkb Overlay.unaryUnion 

Compute the union of two geometries in WKT and WKB and output as WKT
jtsop -a some-geom.wkt -b some-other-geom.wkb -f wkt Overlay.Union

Compute the buffer of distance 10 of a WKT geometry and output as GeoJSON
jtsop -a some-geom.wkt -f geojson Buffer.buffer 10

Compute the buffer of a literal geometry and output as WKT
jtsop -a "POINT (10 10)" -f wkt Buffer.buffer 10

Output a literal geometry as GeoJSON
jtsop -a "POINT (10 10)" -f geojson

Compute an operation on a geometry and output only geometry metrics and timing
jtsop -v -a some-geom.wkt Buffer.buffer 10


jtsop is already proving its worth in the JTS development process.  Some other use cases come to mind:
  • Converting geometry between formats (e.g. WKT to GeoJSON)
  • Filtering and extracting subsets of geometry data (there are various selection functions available to do this)
  • Computing summary statistics of geometry data  

Further Work

There's some interesting enhancements that could be added:
  • provide options to refine the input such as spatial filtering or geometry type coercion
  • allowing chaining multiple operations together using a little DSL (this is possible via shell piping, but doing this internally would be more efficient).  This could use the pipe operator a la Elixir.
  • Include the Proj4J library to allow coordinate system conversions (this can be done now as a simple extension using the dynamic function loading, but it would be nice to have it built in)
  • output geometry as images (using the TestBuilder rendering pipeline)


Tuesday, 11 June 2019

Mandelbrot Set in SQL using SVG with RLE

A popular SQL party trick is to generate the Mandelbrot set using Common Table Expressions (CTEs) to implement the required iteration.   The usual demo outputs the image using ASCII art:

This is impressive in its own way...  but really it's like, so 70's, man.

Here in the 21st century we have better tooling for describing graphics using text - namely, Scalable Vector Graphics (SVG).  And best of all, it's built right into modern browsers (finally!).  So here's the SQL Mandelbrot set brought up to date with SVG output.

A straightforward conversion of the quert is relatively easy.  Simply render each cell pixel as an SVG rect element of size 1, and use a grayscale colour scheme. But  a couple of improvements produce a much better result:
  1. A more varied colour palette produces a nicer image
  2. Using one SVG element per cell result in a very large file, which is slow to render.  There's a lot of repeated pixels in the raster, so a more compact representation is possible  
The colour palette is easily improved with a bit of math (modulo cumbersome SQL syntax).  Here I use a two-ramp palette, sweeping through shades from black to blue, and then through tints to white.

A simple way of reducing raster size is to use Run-Length Encoding (RLE).  This works well with SVG because the rect element can simply be extended by increasing the width attribute.  The tricky part is using SQL to merge the rows for contiguous same-value cells .  As is often the case, a straightforward procedural algorithm requires some cleverness to accomplish in SQL.  It had me stumped for a while.  The solution seemed bound to involve window functions, but trying various combinations of the multitudinous options available didn't produce the desired result.  Then I realized that the problem is isomorphic to that of merging contiguous date ranges.  That is a high-value SQL use case, and there's numerous solutions available.  The two that stand out are (as discussed here):
  • Start-of-Group - this approach uses a LAG function to flag where the group value changes, followed by a running SUM to compute a unique index value for each group (run, in this case).  Group rows are then aggregated on the index
  • Tabibitosan - this is a clever and efficient approach, but is harder to understand and less general
The solution presented uses Start-of-Group, for clarity.  RLE reduces the number of SVG elements to about 12,000 from 160,000, and file size to 1 MB from 11 MB, and hence much faster loading and render time in a web browser.

Here's the output image, with the SQL query producing it below (also available here).

Here's how the query works:
  1. x is a recursive query producing a sequence of integers from 0 to 400 using standard SQL
  2. z is a recursive query creating the Mandelbrot set on a 400x400 grid.  A scale and offset maps the grid cell ordinates into the complex plane, centred on the Mandelbrot set.  The query computes successive values of the set equation for each cell.  A cell is terminated when it is determined that the equation limit is unbounded.   
  3. itermax selects the maximum iterations for each cell.  This result set contains the final result of the Mandelbrot computation
  4. runstart finds and flags the start of each RLE "run" group for each row of the raster
  5. runid computes an id for each run in each row
  6. rungroup groups all the cells in each run and finds the start and end X index
  7. plot assigns a colour to each run, based on the iteration limit i
  8. the final SELECT outputs the SVG document, with rect elements for each run

x(i) AS (
    SELECT i + 1 FROM x WHERE i ≤ 400
z(ix, iy, cx, cy, x, y, i) AS (
    SELECT ix, iy, x::FLOAT, y::FLOAT, x::FLOAT, y::FLOAT, 0
        (SELECT -2.2 + 0.0074 * i, i FROM x) AS xgen(x, ix)
        (SELECT -1.5 + 0.0074 * i, i FROM x) AS ygen(y, iy)
    SELECT ix, iy, cx, cy, 
      x*x - y*y + cx AS x, y*x*2 + cy, i + 1
    FROM z
    WHERE x*x + y*y < 16.0
    AND i < 27
itermax (ix, iy, i) AS (
    SELECT ix, iy, MAX(i) AS i
    FROM z
    GROUP BY iy, ix
runstart AS (
    SELECT iy, ix, I,
        THEN 0 ELSE 1 END AS runstart
    FROM itermax
runid AS (
    SELECT iy, ix, I,
        SUM(runstart) OVER (PARTITION BY iy ORDER By ix) AS run
    FROM runstart
rungroup AS (
    SELECT iy, MIN(ix) ix, MAX(ix) ixend, MIN(i) i
    FROM runid
    GROUP BY iy, run
plot(iy, ix, ixend, i, b, g) AS (
    SELECT iy, ix, ixend, i,
        WHEN i < 18 THEN (255 * i / 18.0 )::integer
        WHEN i < 27 THEN 255
        ELSE 0 END AS b,
        WHEN i < 18 THEN 0
        WHEN i < 27 THEN (255 * (i - 18) / (27 - 18 ))::integer
        ELSE 0 END AS g
    FROM rungroup
    ORDER BY iy, ix
SELECT '<svg viewBox="0 0 400 400" '
  || ' style="stroke-width:0" xmlns="">' 
  || E'\n'
  || string_agg(
      '<rect style="fill:rgb(' 
      || g || ',' || g || ',' || b || ');"  '
      || ' x="' || ix || '" y="' || iy
      || '" width="' || ixend-ix+1 || '" height="1" />', E'\n' )
  || E'\n' || '</svg>' || E'\n' AS svg
FROM plot;

Monday, 25 March 2019

PostgreSQL's Linux moment

My perspicacious colleague Paul Ramsey says "Postgres is having its Linux moment". There is certainly a buzz around PostgreSQL, evidenced by datapoints such as:
Reasons for this include:
  • the shift from proprietary to open source as the software model of choice
  • the rise of cloud-based DB platforms, where the flexibility, power and cost (free!) of Postgres makes it an obvious choice.  All the major players now have Postgres cloud offerings, including Amazon, Microsoft and Google.
And happily riding along is PostGIS, bundled with most if not all major PostgreSQL distros.  (Note how the Google blog post announcing cloud Postgres highlights a geospatial use case).  So it's an exciting time to be able to work on PostGIS at Crunchy Data.

Monday, 4 March 2019

Fast Geometry Distance in JTS

The second-most important criteria for a spatial algorithm is that it be fast.  (The most important is that it's correct!)  Many spatial algorithms have a simple implementation available, but with performance of O(n2) (or worse).  This is unacceptably slow for production usage, since it results in long runtimes for data of any significant size. In JTS a lot of effort has gone into identifying O(n2) performance hotspots and engineering efficient replacements for them.

One long-standing hotspot is the algorithm for computing Euclidean distance between geometries.  The obvious distance algorithm is a brute-force O(MxN) comparison between the vertices and edges (facets) of the input geometries.  This is simple to implement, but very slow for large inputs.  Surprisingly, there seems to be little in the computational geometry literature about more efficient distance algorithms.  Perhaps because of this, many geometric libraries provide only the slow brute force algorithm - including JTS (until now). 

Happily, it turns out there is a faster approach to distance computation.  It uses data structures and algorithms which are already provided in JTS, so it's relatively easy to implement. The basic idea is to build a spatial index on each of the input geometries, and then use a Branch-and-Bound search algorithm to efficiently traverse the index trees to find for the minimum distance between geometry facets.  This is a generalization of the R-tree Nearest Neighbour algorithm described in the classic paper by Rousssopoulos et al.  [1]. 

JTS has the STRtree R-tree index implementation (a packed R-tree using the Sort-Tile-Recursive algorithm). This has recently been enhanced with several kinds of nearest-neighbour searches.  In particular, it now supports a method to find the nearest neighbours between two different trees.  The IndexedFacetDistance class uses this capability to implement fast distance searching on the facets of two geometries.

Another benefit of this approach is that it allows caching the index of one geometry.  This further increases performance in the common case of repeated distance calculations against a fixed geometry.

The performance improvement is impressive.  Here's the timings for computing the distance from Antarctica to other world countries:

Data size
Data size
1 polygon
(19,487 vertices)
244 polygons
(366,951 vertices)
164 ms 136 s x 830

Branch-and-bound search also speeds up isWithinDistance queries.  Here's a within-distance selection query between another antipodean continent and a large set of small rectangles:

Data size
Data size
Time Time
1 polygon
(7,316 vertices)
100,000 polygons
(500,000 vertices)
53 ms 10.03 s x 19

A small fly in the algorithmic ointment is that Indexed Distance is not always better than the brute-force approach.  For small geometries (such as points or rectangles) a simple scan is actually faster, since it avoids the overhead of building indexes.  It may be possible to determine a tuning parameter that allows automatically choosing the fastest option.  Or the client can choose the faster approach, using knowledge of the use case.

Future Work

A few further ideas to build or investigate:
  • Implement a caching FastDistanceOp using IndexedFacetDistance and indexed Point-In-Polygon.  This can be used to add a fast distance() method to PreparedGeometry 
  • Investigate improving isWithinDistance by using the MINMAXDISTANCE metric for envelopes.  This allows earlier detection of index nodes satisfying the distance constraint.
  • Investigate alternative R-Tree packing algorithms (such as Hilbert packing or sequence packing) to see if they improve performance

[1] Roussopoulos, Nick, Stephen Kelley, and Frédéric Vincent. "Nearest neighbor queries."  ACM SIGMOD record. Vol. 24. No. 2. ACM, 1995.