My previous blog post reviewed the concept of the Hausdorff distance (which more descriptively could be called farthest distance.) Despite its usefulness in matching geometric data, there are surprisingly few open-source implementations, and seemingly no efficient ones for linear and polygonal data. This even includes CGAL and GRASS, which are usually reliable for provising a wide spectrum of geospatial operations.
This lack of high-quality Hausdorff extends to the JTS Topology Suite. JTS has provided the DiscreteHausdorffDistance class for many years, That code is used in many geospatial systems (via JTS, and also its GEOS port), including widely used ones such as PostGIS, Shapely, and QGIS. However, that imlementation has significant performance problems, as well as other usability flaws (detailed here).
So I'm excited to announce the release of a new fast, general-purpose algorithm for Hausdorff distance in JTS. It's a class called DirectedHausdorffDistance. It has the following capabilities:
- handles all linear geometry types: points, lines and polygons
- supports a distance tolerance parameter, to allow computing the Hausdorff distance to any desired accuracy
- can compute distance tolerance automatically, providing a "magic-number-free" API
- has very fast performance due to lazy densification and indexed distance computation
- can compute the pair of points on the input geometries at which the distance is attained
- provides prepared mode execution (caching computed indexes)
- allows determining farthest points which lie in the interior of polygons
- handles equal or nearly identical geometries efficiently
- provides the isFullyWithinDistance predicate, with short-circuiting for maximum performance
Hausdorff distance VS shortest distance between linestrings
The choice of name for the new class is deliberate. The core of the algorithm evaluates the directed Hausdorff distance from one geometry to another. To compute the symmetric Hausdorff distance simply involves choosing the largest of the two directed distances DHD(A, B) and DHD(B, A) This is provided as the function DirectedHausdorffDistance.hausdorffDistance(a,b).
Indexed Shortest Distance
The Hausdorff distance depends on the standard (Euclidean) shortest distance function (as evident from the mathematical definition:
DHD(A,B) = max a ∈ A dist(a,B) A key performance improvement is to evaluate shortest distance using the IndexedFacetDistance class. For point sets this optimization alone produces a significant boost.
For example, take a case of two random sets of 10,000 points. DiscreteHausdorffDistance takes 495 ms. DirectedHausdorffDistance takes only 22 ms - 22x faster. (It's also worth noting that this is similar performance to finding the shortest distance points using IndexedFacetDistance.nearestPoints).
Lazy Densification
The biggest challenge in computing the Hausdorff distance is that it can be attained at geometry locations which are not vertices. This means that linework edges must be densified to add points at which the distance can be evaluated. The key to making this efficient is to make the computation "adaptive" by performing "lazy densification". This avoids densifying edges where there is no chance of the farthest distance occurring.
Densification is done recursively by bisecting segments. To optimize finding the location with maximum distance the algorithm uses the branch-and-bound pattern. The edge segments are stored in a priority queue, sorted by a bounding function giving the maximum possible distance for each segment. The segment maximum distance is the largest of the distances at the endpoints. The segment maximum distance bound is the segment maximum distance plus one-half the segment length. (This is a tight bound. To see why, consider a segment S of length L, at a distance of D from the target at one end and distance D + e at the other. The farthest point on S is at a distance of L + 2D + e = L/2 + D + e/2. This is always less than L/2 + D + e, but approaches it in the limit.)
Proof of Maximum Distance Bound
The algorithm loops over the segments in the priority queue. The first segment in the queue always has the maximum distance bound. If this is less than the current maximum distance, the loop terminates since no greater distance will be found. If the segment distance to the target geometry is greater than the current maximum distance, it is saved as the new farthest segment. Otherwise, the segment is bisected, subsegment endpoint distances are computed, and both are inserted back into the queue.
Search for DHD using line segment bisection
By densifying until bisected segments drop below a given length the directed Hausdorff distance can be determined to any desired accuracy, The accuracy distance tolerance can be user-specified, or it can be determined automatically. This provides a "magic-number-free" API, which is significantly improves ease of use.
Performance comparison
Comparing the performance of DirectedHausdorffDistance to DiscreteHausdorffDistance is unfair, since the latter implementation is so inefficient. However, it's the one currently in use, so the comparison is relevant.
There are two possible situations. The first is when the directed Hausdorff distance is attained at vertices (which is often the case when the geometry vertices are already dense; i.e. segment lengths are short relative to the distance). As an example we will use two polygons of 6,426 and 19,645 vertices.
DiscreteHausdorffDistance with no densification (a factor of 1) takes 1233 ms. DirectedHausdorffDistance takes 25 ms - 49x faster. (In practice the performance difference is likely to be more extreme. There is no way to decide a priori how much densification is required for DiscreteHausdorffDistance to produce an accurate answer. So usually a higher amount of densification will be specified. This can severely decrease performance.)
The second situation has the Hausdorff distance attained in the middle of a segment, so densification is required. The query polygon has 468 vertices, and the target has 65.
Handling (mostly) equal geometries
The old Hausdorff distance algorithm had an issue reported in this post on GIS Stack Exchange. It asks about the slow performance of a case of two nearly-identical geometries which have a very small discrepancy. In the end the actual problem seemed to be due to the overhead of handling large geometries in PostGIS. However, testing it with the new algorithm revealed a significant issue.
Two nearly-identical geometries, showing discrepancy location
The problem is that the maximum distance bounding function depends on both the segment distance and the segment length. When the segment distance is very small (or zero), the distance bound is dominated by the segment length, so subdivision will continue until all segments are shorter than the accuracy tolerance. This lead to a large number of subsegments being generated during search, particularly when the tolerance is small (as required in the above case).
The solution is to check subsegments with zero distance to see if they are coincident with a segment of the target geometry. If so, there is no need to bisect the segment further, since subsegments must also have distance zero. With this check in place, identical (and nearly so) cases executes as fast as more general cases of the same size. Equally importantly, this detects very small discrepancies regardless of the accuracy tolerance.
For the record, the GIS-SE case now executes in about 45 ms, and detects the tiny discrepancy of 9 orders of magnitude smaller than the input geometry.
For the record, the GIS-SE case now executes in about 45 ms, and detects the tiny discrepancy of 9 orders of magnitude smaller than the input geometry.
The Hausdorff distance of ~0.00099
Handling Polygonal Input
If the Hausdorff distance is attained at a point lying on an edge then densifying the linework is sufficient. But for polygonal query geometries the farthest point can occur in the interior of the area:
The Directed Hausdorff distance is attained at an interior point of the query polygon
To find the farthest interior point the adaptive branch-and-bound approach can be used in the area domain. Conveniently, JTS already implements this in the MaximumInscribedCircle and LargestEmptyCircle classes (see this blog post.) In particular, LargestEmptyCircle supports constraining the result to lie inside an area, which is exactly what is needed for the Hausdorff distance. The target geometry is treated as obstacles, and the polygonal element(s) of the query geometry are the constraints on the location of the empty circle centre.
Directed Hausdorff Distance with multiple area constraints and heterogeneous obstacles
Currently most Hausdorff distance algorithms operate on point sets, with a very few supporting linear geometry. There seem to be none which compute the Hausdorff distance for polygonal geometries. While this might seem an uncommon use case, in fact it's essential to support another new capability of the algorithm: computing the isFullyWithinDistance predicate for polygonal geometries.
isFullyWithinDistance
Distance-based queries often require determining only whether the distance is less than a given value, not the actual distance value itself. This boolean predicate can be evaluated much faster than the full distance determination, since the computation can short-circuit as soon as any point is found which confirms being over the distance limit. It also allows using other geometric properties (such as envelopes) for a quick initial check. For shortest distance, this approach is provided by Geometry.isWithinDistance (and supporting methods in DistanceOp and other classes.).
The equivalent predicate for Hausdorff distance is called isFullyWithinDistance. It tests whether all points of a geometry are within a specified distance of another geometry. This is defined in terms of the directed Hausdorff distance (and is thus an asymmetric relationship):
isFullyWithinDistance(A,B,d) = DHD(A,B) <= d
The DirectedHausdorffDistance class provides this predicate via the isFullyWithinDistance(A,B,dist) function. Because the new class supports all types of input geometry (including polygons), the predicate is fully general. For even faster performance in batch queries it can be executed in prepared mode via the isFullyWithinDistance(A,dist) method. This mode caches the spatial indexes built on the target geometry so they can be reused.
For a performance example, consider a dataset of European boundaries (countries and islands) containing about 28K vertices. The boundary of Germany is used as the target geometry.
If isFullyWithinDistance is run with a distance limit of 20, it takes about 60 ms.
There's no direct comparison for DiscreteHausdorffDistance, but if that class is used to compute the directed Hausdorff distance with a conservative densification factor of 0.1, the time is about 1100 ms. Another point of comparison is to run a shortest distance query. This takes only 21 ms - but it's doing much less work.
A better implementation for ST_DFullyWithin
Another way to implement isFullyWithinDistance is to compute the buffer(d) of geometry B and test whether it covers A:
isFullyWithinDistance(A,B,d) = B.buffer(d).covers(A)
This is how the ST_DFullyWithin function in PostGIS works now. It's a reasonable design choice given the current lack of a performant Hausdorff distance implementation. However, there are a few problems with using buffer:
- Buffers of complex geometry can be slow to compute, especially for large distances
- There's a chance of robustness bugs affecting the computed buffer
- Buffers are linearized approximations, so there is a likelihood of false negatives for query geometries which lie close to the buffer boundary
- image of buffer quantization causing failures
Now, the DirectedHausdorffDistance implementation of isFullyWithinDistance can make this function faster, more accurate, more robust and cacheable. (And of course, the ST_HausdorffDistance function can benefit as well.)
Summary
The JTS DirectedHausdorffDistance class provides fast, cacheable, easy-to-use computation of Hausdorff distances and the isFullyWithinDistance predicate for all JTS geometry types. This is a major improvement over the old JTS DiscreteHausdorffDistance class, and essentially fully replaces it. More generally, it fills a notable gap in open-source geospatial functionality. It will allow many systems to provide a high-quality implementation for Hausdorff distance.
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