Toyota is sponsoring the development of new algorithms for surface reconstruction and real-time localization in PCL, as part of the first PCL code sprint.
For a complete list of all the present and past PCL code sprints please visit http://www.pointclouds.org/blog.
It is with pleasure to share the successful completion of this Toyota Code Sprint in this final blog post. In this project, homography estimation based on multi-modal, multi-descriptor correspondence sets has been explored, and inspired the introduction of the multi-descriptor voting approach (MDv). The proposed MDv approach achieved a consistent accuracy in the 0.0X range, a level of consistency that is better than those based on single-type state of the art descriptors including SIFT. In the process, a framework for analyzing and evaluating single and multi-descriptor performance has been developed, and employed to validate the robustness of MDv, as compared with homography estimations based on a single descriptor type, as well as those based on RANSAC registration of best-K multi-descriptor correspondence sets. The code and dataset for this project are hosted on https://github.com/mult-desc/md, with dependencies on both PCL 1.7 and OpenCV 2.4.6.
Follows is an in-depth report detailing the project’s accomplishments, as well as design and validation considerations:Click here for a high resolution version of the report.
In the previous blog post I described my attempts to find a good balance between the contributions of the three terms (distance, normal, and color) to edge weight computation. As it often happens, as soon as I was done with the evaluation and the blog post, I realized that there is another type of information that could be considered: curvature. And indeed, it proved to have a very positive effect on the performance of the random walker segmentation.
Let me begin by exposing a problem associated with edge weights computed using the normal term alone (i.e. depending only on the angular distance between the normals of the vertices). Consider the following scene (left):
Voxelized point cloud (left) and a close-up view of the graph
edges in the region where the tall and round boxes touch
(right). The edges are colored according to their weights
(from dark blue for small weights to dark red for large
Most of the edges in the boundary region (right) are dark blue, however there are a number of red edges with quite large weights. This sort of boundary is often referred to as a “weak boundary” and, not surprisingly, has a negative effect on the performance of many segmentation algorithms. You can imagine that a boundary like this is a disaster for the flood-fill segmentation, because the flood will happily propagate through it. Luckily, the random walker algorithm is known for its robustness against weak boundaries:
Segmentations produced by the random walker algorithm
using three different choices of seeds (shown with red
In the first two cases one of the seeds is very close to the weak boundary, whereas another one is far away. In the third case there are multiple green seeds placed along the boundary, however a single purple seed is able to “resist” them from its remote corner.
This robustness has limits, of course. In the figure below the “table seed” is placed in the rear of the table, far from the boundaries with the boxes. The box segments managed to “spill” on the table through the weak boundaries:
Segmentation failure when a seed is placed too far from
a weak boundary
One way to address this problem is to make the sigma of the normal term smaller, therefore penalizing differences in normals’ orientations more. Enabling the distance term might also help, because the edges that contribute to the boundary weakness are often diagonal and therefore longer than the average. The figure below (left) demonstrates the graph edges in the same boundary region with new weights, computed using decreased normal term sigma (10% of the original one), and with the distance term enabled. (The overall edge color shift towards blue is due to it.)
Graph edges with weights computed with a smaller normal
term sigma and enabled distance term. Close-up view of
the region where the tall and round boxes touch (left)
and top-down view at the rear of the table (right).
Still, there are several edges with relatively large weights in the boundary region, but there are no large-weight paths connecting vertices on both sides of the boundary anymore. We got rid of the weak boundary, but this came at a price. Although the table itself is flat, the cloud that we get from the Kinect is not, and the further from the camera the more wavy it is. The image on the right shows a top-down view at the rear of the table, where the waves are particularly large. The edges that belong to the cavities between the “waves” were heavily penalized and virtually disappeared. Random walkers will have a hard time traversing this part of the table on their way to the boxes!
Having considered all of these I came to a conclusion that some additional geometrical feature is needed to improve the weighting function. Curvature was the first candidate, especially in the light of the fact that we anyways get it for free when estimating voxel normals (via PCA of the covariance matrix of the voxel neighborhood). I added one more exponential term to the weighting function:
where is simply a product of the voxel curvatures. Similarly to how it is done in the normal term, the product is additionally multiplied by a small constant if the angle between the voxels is convex (in order not to penalize convex boundaries).
The figure below demonstrates the edge weights computed using the new term alone:
Graph edges with weights computed using only the new
curvature term. Close-up view of the region where the
the rear of the table (right).
The boundary between the boxes is perfectly strong, whereas the weights in the rear of the table are not penalized too much. The seeding that resulted in a segmentation failure before no longer causes problems. I used the set of random seedings described in the last post to find the best sigmas for the extended weighting function. Then I generated 50 new seedings to test and compare the performance of the old (without the curvature term) and the new (with the curvature term) weighting functions. The figure below summarizes the distributions of under-segmentation errors:
The performance significantly improved both in terms of stability, quality, and number of failures (in fact, there are no failures at all).
The random walker segmentation algorithm requires that the data are modeled as a weighted graph, and the choice of edge weighting function has a great impact on the performance of the algorithm. In this blog post I will describe the weighting function and parameters that I ended up using.
Before talking about the weights of the edges between vertices, let’s discuss the vertices themselves. As mentioned in the previous blog posts, I have a pre-processing step where the input cloud is voxelized using OctreePointCloudAdjacency. Voxelization serves three purposes:
The voxels consequently become vertices of the graph, and each of them is connected with its neighbors by an edge. Each voxel has several properties: 3D position, normal orientation, and color, which may be used in edge weight computation.
As mentioned in the very first blog post on random walker segmentation, originally I used the edge weighting function from the following paper:
Later on I introduced several modifications. Now for a pair of vertices and the weight is defined as:
where , , and are the Euclidean, angular, and color differences between voxels, and the sigmas are used to balance their contributions. Compared to the weighting function of Lai et al., the color term was added, and scaling by mean values was removed.
I devised the following procedure in order to find appropriate values for the sigmas. I took a scene with known ground truth segmentation and generated 50 random proper seedings. Here by a “proper seeding” I mean a set of seeds where each seed belongs to a distinct ground truth segment, and each ground truth segment has a single seed (see example in the figure below on the left). For each of these seedings I ran random walker and computed the under-segmentation error (that was defined in one of the earlier blog posts, see example in the figure below on the right). Then I analyzed the distributions of errors resulted from different sigma values.
Voxelized point cloud with
one of the randomly
generated proper seeding
used in the experiments
Under-segmentation error of
the segmentation produced by
random walker from the
given seeds (erroneous voxels
Note that the ground truth itself is not perfect, because it is often impossible to tell apart the points at the boundary of two objects. Consequently, the ground truth segmentation is somewhat random at the boundaries, and we should not expect (or strive) any segmentation algorithm to produce exactly the same result. The under-segmentation error displayed above has 1039 erroneous voxels, and this is pretty much the best performance we could expect from a segmentation algorithm with this ground truth.
Let’s begin by examining the influence of the angular term. In this experiment I set Euclidean sigma to the value of voxel resolution and color sigma to 0.1. Below is a plot of under-segmentation error distributions for different choices of angular sigma (note that larger sigmas correspond to less influence):
Each distribution is visualized using a boxplot. The three main features are the position of the red bar (median of the distribution), size of the box (50% of the values fall inside the box), and the amount and positions of pluses (outliers). The first one gives an idea of the average performance of the algorithm. The second one expresses the segmentation stability with respect to the seed choice (with smaller box meaning better stability). The third one indicates segmentation failures. Indeed, a significant deviation of the under-segmentation error means that the output segmentation has large mis-labeled regions, which may be deemed as a failure.
Clearly, the median values of the distributions are almost the same. The differences are very small and due to the discussed properties of the under-segmentation error can not be used to draw conclusions of which sigmas are better. The box sizes, however, vary significantly. The sigmas from 0.1 to 1.1 yield the most stable performance. Also the number of failures is less for those sigmas. This evaluation does not provide enough information to chose any particular sigma in this range, so for now I settled on 0.2 (it is the second most stable, but yields less failures than 0.1).
In order to explore the influence of the color term, I set Euclidean sigma to the value of voxel resolution again and angular sigma to 0.2:
Note the last column, which shows the error distribution when the color term is removed completely. We see that sigmas from 0.1 to 0.15 provide slightly more stable results. Unfortunately it is not visible in the plot, but the number of pluses on the row is less for these sigmas. So I chose 0.15 as a result of this evaluation.
Speaking about the distance sigma, it turned out to have very small influence on the results. In most cases introduction of the distance term does not change the output at all. Still, in few cases it helps to avoid complete segmentation failure. It turned out that setting this sigma to the voxel resolution value gives the best results.
Finally, let me demonstrate the segmentations produced with the chosen sigmas. Among the 50 random seedings only 5 resulted in segmentation failure:
5 failed segmentations
Clearly, failures happened when a seed was placed either exactly on the boundary between two objects (#2), or on the outermost voxels of an object (#1, #3, #4, #5).
The remaining 45 seedings yielded good segmentations. Below are 15 of them (selected randomly):
15 succeeded segmentations
In the past weeks I decided to put the “spectral thing” on hold and turned back to the random walker segmentation. In this blog post I will talk about refactoring of the random walker algorithm and my experiments with different linear solvers. In the follow-up post I will explore how the segmentation results depend on seed placement, as well as discuss edge weighting functions and the choice of parameters for them.
First of all, I refactored and cleaned up my implementation. Recall that the random walker algorithm could be applied to cluster any kind of data as long as there is a way to model it using a weighted graph. I decided that it makes sense to have a generic templated implementation of the algorithm which would work with any weighted graph. An obvious choice to represent graphs in C++ is to use the primitives available in the Boost Graph Library (BGL). This library is very generic, feature-rich, and flexible, though at the expense of a rather steep learning curve. I took their implementation of the Boykov-Kolmogorov max-flow algorithm as an example of how to design the interface for a generic graph-based algorithm. In my case the public interface is just one templated function:
template<class Graph, class EdgeWeightMap, class VertexColorMap> bool randomWalkerSegmentation(Graph& g, EdgeWeightMap weights, VertexColorMap colors);
The user has to provide a graph, a property map that associates a weight to each edge of the graph, and a property map that contains initial vertex colors. (I adopted the term “colors” instead of “labels” because BGL has a pre-defined vertex property type with this name.) The output of the algorithm (i.e. label assignment) is written back to the color map. Internally the function instantiates a class, which does all the boring work of constructing and solving a system of linear equations, as well as interpreting its solution as a label assignment.
While this generic graph segmentation function might be useful for someone, the general audience will be interested in a class that implements a complete point cloud segmentation pipeline. This class should take care of converting an input cloud into a weighted graph, segmenting it, and turning the random walker output into a labeled point cloud. At the moment it is not clear for me how the first step should be designed. Indeed, there are multiple ways to represent a point cloud as a weighted graph, both in terms of topology and edge weights. Currently I voxelize the input cloud and use 26-neighborhood to establish edges between nodes. Alternatively, one may work with a full point cloud and use some fixed-radius neighborhood. One more option might be to generate a mesh and work with it. Exploration of these possibilities remains as a future work.
The second issue that I addressed recently was the performance of the algorithm. The main computational effort is spent on solving a sparse system of linear equations, where the number of equations is determined by the number of unlabeled vertices (i.e. basically the size of the whole point cloud). For example, the typical size of voxelized scenes from the OSD dataset that I use in my experiments is about 30000 vertices. Originally, I used the ConjugateGradient solver of Eigen, because it is “recommended for large symmetric problems”. The time needed to segment a typical point cloud with this solver is about 1 second on my three years old i5 laptop. I decided to try other options available in Eigen. In particular, I tested BiCGSTAB with Diagonal and IncompleteLUT preconditioner, SimplicialLLT, SimplicialLDLT, and SimplicialCholesky solvers. The figure below shows the runtime of these solvers with respect to the problem size. (Only one of the Simplicial*** solvers is plotted as they demonstrated very similar performance.)
The computation time depends linearly on the problem size for all solvers, however SimplicialLDLT has a much smaller growth rate. For a typical 30 thousand vertices problem it needs about 200 ms. What’s more, it can solve for multiple right-hand sides at the same time, whereas ConjugateGradient and BiCGSTAB can not. This means that as the number of labels (i.e. desired segments) grows, the computational time does not increase.
In fact, Eigen offers some more options such as CholmodSupernodalLLT, which is a wrapper for SuiteSparse package, and SparseLU, which uses the techniques from the SuperLU package. Unfortunately, the former complained that the matrices that I provide are not positive definite (though they actually are), and the latter is a very recent addition that is only available in Eigen 3.2 (which I do not have at the moment).
Taking into account the evaluation results I switched to the SimplicailLDLT solver in my random walker implementation.
I am pleased to announce that I just finished the second Toyota Code Sprint. The topic we tackled this time was superquadrics applied for Computer Vision tasks such as object modeling and object detection.
The code we developed was not yet integrated into PCL (it does use PCL for all the processing), but lies in a separate repository which you can find here: https://github.com/aichim/superquadrics .
An extensive report that presents some of the theory behind the concepts and algorithms we used in the project, as well as implementation details and results can be found at the end of this post.
At the end of this work, we present a set of ideas that can be used to extend the project in subsequent code sprints:
In the several previous posts I tried to provide some insight in how the spectral clustering technique may be applied to the point cloud processing domain. In particular, I have demonstrated different visualizations of eigenvectors and also did a manual analysis of one particular scene. In this blog post I will (finally) describe my algorithm that does automatic analysis of eigenvectors, which leads to unsupervised supervoxel clustering.
The input of the clustering algorithm is , a set of first eigenvectors of the graph Laplacian. Each eigenvector has elements that correspond to the supervoxels is the original problem space. The task of the algorithm is to determine the number of clusters that the data points form in the subspace spanned by the eigenvectors and, of course, assign points to these clusters.
The key insight drawn from previous examinations and discussions of the eigenvectors is that the clusters are linearly separable in one-dimensional subspaces spanned by the eigenvectors. In other words, for every pair of clusters there exists at least one eigenvector so that in its subspace these clusters are linearly separable. Based on this premise I built an algorithm which is a pretty straightforward instance of divisive hierarchical clustering approach. It starts with a single cluster that contains all the data points and recursively splits it in a greedy manner. The following pseudo-code summarizes the algorithm:
The interesting part are, of course, FindBestSplit and SplitQuality functions. But to get this straight, let me first define what a “split” is. A split is a tuple , where is the eigenvector along whose subspace the split occurs, and is a threshold value. The points that have their corresponding elements in the eigenvector less than go to the first cluster, and the remaining go to the second. For example, the figure below shows with a red line a split on the left and a split on the right:
Example splits in the subspaces spanned by the
Which of these two splits is better? Intuitively, the one on the left is more promising than the one on the right. But how to define the split quality? My initial approach was to use the difference between the points immediately above and below the splitting line as the measure. This worked to some extent, but was not good enough. Then I switched to a measure based on the relative densities of the bands above and below the splitting line. Consider the figure below:
Example splits with three bands highlighted. The
“split” band is shown in red, the “top” and “bottom”
bands are shown in yellow
The “split” band is the region of low density around the splitting line. The “top” and “bottom” bands are the high density regions immediately above and below the “split” band. I will omit the details of how these bands are computed, because it is likely that I modify the implementation in future. The quality of the split is defined as , where is the density of the corresponding band.
With this quality measure at hand, the FindBestSplit function simply iterates over all available one-dimensional subspaces (i.e. over all eigenvectors) and finds the split with the highest quality.
The performance of the algorithm is excellent on simple scenes:
Spectral supervoxel clustering of simple table-top scenes
And is rather good (though definitely not perfect) on more cluttered ones:
Spectral supervoxel clustering of cluttered table-top scenes
For example, the green book in the first scene is split into two clusters. In the second scene two small boxes in the center of the image are erroneously merged into one cluster. I think these issues are closely related with the split quality measure and the threshold associated with it. Definitely, there are ways to improve these and I plan to work on it the future.
Before exposing the clustering algorithm as promised in the last blog post, I decided to motivate it by showing how the eigenvectors may be analyzed “by hand”. Hopefully, this will also provide more intuition about what these eigenvectors actually are and how they are related with the data.
Just to remind, the problem I am trying to solve is about segmenting a set of supervoxels into meaningful components. Here a meaningful component means a subset of supervoxels that are close to each other in Euclidean sense, and are separated from the rest by a sharp change in orientation. If we view each supervoxel as a point then the problem is about clustering points in a -dimensional space. (Currently since supervoxels have 3 Euclidean coordinates plus 3 coordinates of the normal vector, however additional dimensions, e.g. color, may be added later.) The difficulty of the problem comes from the fact that the components may have arbitrary irregular shape in these dimensions. Therefore I want to map these points so some other space where the components will correspond to tight clusters, perhaps even linearly separable. The current idea is to use a subspace spanned by the first few eigenvectors of graph Laplacian of the original data.
In the last blog post I provided a visualization of the eigenvectors in the original problem space. For convenience, here are the first four eigenvectors again:
The first 4 eigenvectors in the original problem space
I want to perform clustering in a subspace though, so it is helpful to develop an intuition about how the data look like in it. The figure below demonstrates the data points (that is, supervoxels), projected on each of the first four eigenvectors. (Here and in what follows the data is whitened, i.e. de-meaned and scaled to have unit variance. Additionally, the values are sorted in increasing order.)
Data points in subspaces spanned by the first 4 eigenvectors
Obviously, in each of these subspaces (except for the second) the data is linearly separable in two clusters. What is not obvious, however, is how many clusters there will be in the combined subspace. The next figure shows data points in subspaces spanned by two different pairs of eigenvectors:
Data points in subspaces spanned by first and third (left)
and fourth and third eigenvectors (right)
Now it becomes evident that there are at least three clusters. Will there be more if we consider the subspace spanned by all these three eigenvectors? It turns out there will, see the point cloud below:
Unfortunately, we have just approached the limit in terms of how many dimensions could be conveniently visualized. Though for this particular data set it is enough, there won’t appear more clusters if we consider additional eigenvectors. Summarizing, in the subspace spanned by the first four eigenvectors the data points form four tight and well-separated (linearly) clusters. And these clusters actually correspond to the three boxes and the table in the original problem space.
Now the only thing left is to develop an algorithm which would do this kind of analysis automatically!
Correspondence rejection classes implement methods that help eliminate correspondences based on specific criteria such as distance, median distance, normal similarity measure or RanSac to name a few. Couple of additional filters I’ve experimented with include a uniqueness measure, and Lowe’s ratio measure as in “Distinctive image features from scale invariant keypoints”, D.G. Lowe, 2004. I’ve also explored the tradeoffs in implementing the filters within CorresondenceEstimation itself, or as external CorrespondenceRejection classes. The former is computationally more efficient if the rejection process is done in one pass, while the latter allows for scene-specific squential filter banks.
Follows is a quick reference guide of the available correspondence rejection classes with remarks extracted from the source code.
I concluded the last blog post by noting that it seems to be possible to segment objects based on analysis of the eigenvectors of Laplacian constructed from the point cloud. This time I will provide a visual interpretation of eigenvectors and then describe the problem of their analysis.
Let me start with a quick note on eigenvector computation. As mentioned before, they are obtained through eigendecomposition of Laplacian that represents the surface. In the beginning I used SelfAdjointEigenSolver of Eigen library. It runs in time (where is the number of points), which obviously does not scale well. Later I switched to SLEPc. It can limit computation only to a desired number of first eigenpairs and therefore does the job much faster, but still seems to have polynomial time. Therefore I decided to execute supervoxel clustering as a pre-processing step and then compute the distances over the supervoxel adjacency graph, which has a dramatically smaller size than the original point cloud.
Now let’s turn to the eigenvalues themselves. The figure below demonstrates the voxelized point cloud of a simple scene (left) and supervoxel adjacency graph (right), where the adjacency edges are colored according to their weights (from dark blue for small weights to dark red for large weights):
Voxelized point cloud (voxel
size 0.006 m)
Supervoxel adjacency graph
(seed size 0.025 m), colored
according to edge weight
Eigendecomposition of Laplacian of this weighted adjacency graph yields a set of pairs . Each eigenvector has as many elements as there are vertices in the graph. Therefore it is possible to visualize an eigenvector by painting each supervoxel according to its corresponding element in the vector. The figure below shows the first 9 eigenvectors which correspond to the smallest eigenvalues:
First 9 eigenvectors of the graph Laplacian
The first eigenvector clearly separates the scene into two parts: the third box and everything else. In the second eigenvector the table is covered with gradient, but the first and third boxes have (different) uniform colors and, therefore, stand out. The third eigenvector highlights the second box, and so on.
I have examined quite a number of eigenvectors of different scenes, and I think the following common pattern exists. First few eigenvectors tend to break scene in several regions with uniform colors and sharp edges. In the next eigenvectors gradients begin to emerge. Typically, a large part of the scene would be covered with gradient, and a smaller part (corresponding to a distinct component of the graph) would be have some uniform color.
The overall goal is to figure out the number of distinct components of the graph (that is, objects in the scene) and segment them out. As I admitted before, it is clear that the eigenvectors capture all the information needed to do this, so the question is how to extract it. In fact, this problem has already received a lot of attention from researchers under the name of “Spectral Clustering”. (Yeah, I made quite a detour through all these distance measures on 3D surfaces before I came to know it). The standard approach is described in the following paper:
In a nutshell, the original problem space typically has many dimensions (in our case the dimensions are Euclidean coordinates, normal orientations, point colors, etc.). The clusters may have arbitrary irregular shapes, so they could neither be separated linearly, nor with hyper-spheres, which renders standard techniques like K-means inapplicable. The good news are, in the subspace spanned by the first few eigenvectors of Laplacian of the graph (constructed form the original data) the data points form tight clusters, and thus K-means could be used. This effect is evident in the eigenvectors that I demonstrated earlier. Unfortunately, the number of clusters still needs to be known. There exist literature that addresses automatic selection of the number of clusters, however I have not seen any simple and reliable method so far.
In the next blog post I will describe a simple algorithm that I have developed to analyze the eigenvectors and demonstrate the results.
With my current work on optimizing correspondence estimation across the uv/xyz domains, it is worth providing a topology of the available correspondence estimation classes in PCL. For a highlevel treatment of the registration API, please refere to the registration tutorial.
Correspondence estimation attempts to match keypoints in a source cloud to keypoints in a target cloud, based on some similarity measure, feature descriptors in our case. Although applying scene relevant descriptor parameters and correspondence thresholds may reduce erronous matches, outliers persist with impact on pose estimation. This is due to the implied assumption that for each source keypoint, a corresponding target keypoint exists. The difficulty in estimating model or scene-specific descriptor parameters is another factor.
Follows is a quick reference guide of the available correspondence estimation classes with remarks extracted from the source code.
Last time I wrote about distance measures on 3D surfaces, though I did not give any details about how they are computed. In this blog post I will give a formal definition, followed by two important properties that simplify the computation and provide insights that might help to solve the ultimate goal: identification of distinct objects in a scene.
Given a mesh (or a point cloud) that represents a surface, a discretization of the Laplace-Beltrami operator (LBO) is constructed. This discretization is a sparse symmetric matrix of size , where is the number of vertices (points) in the surface. The non-zero entries of this matrix are the negated weights of the edges between adjacent vertices (points) and also vertex degrees. This matrix is often referred to as Laplacian. Eigendecomposition of Laplacian consists of pairs , where are eigenvalues, and are corresponding eigenvectors.
The diffusion distance is defined in terms of the eigenvalues and eigenvectors of Laplacian as follows:
The biharmonic distance bears a strong resemblance to it:
Here means -th element of eigenvector . Both distances have a similar structure: a sum over all eigenpairs, where summands are differences between corresponding elements of eigenvectors scaled by some function of eigenvalues. There are two properties of these distances that I would like to stress.
Firstly, the summands form a decreasing sequence. The figure below illustrates this point with eigenvalues of Laplacian of a typical point cloud:
In the left image the first hundred of eigenvalues (except to which is always zero) are plotted. Note that the values are normalized (i.e. divided) by . The magnitudes of eigenvalues increase rapidly. On the right the multipliers of both diffusion and biharmonic distances are plotted (also computed with normalized eigenvalues). The biharmonic distance multiplier is plotted for several choices of the parameter . Clearly, only a few first terms in the summation are needed to approximate either of the distances. This has an important consequence that there is no need to solve the eigenproblem completely, but rather is suffices to find a limited number of eigenpairs with small eigenvalues.
Secondly, the distance between two points and depends on the difference between their corresponding elements in eigenvectors and . The figure below demonstrates the (sorted) elements of a typical eigenvector:
One may see that there are groups of elements with the same value. For example, there are about one hundred elements with value close to . The pair-wise distances between the points that correspond to these elements will therefore be close to zero. In other words, plateaus in eigenvector graphs correspond to sets of incident points, and such sets may be interpreted as objects.
Summing up, it seems like it should be possible to identify distinct objects in a point cloud by analyzing the eigenvectors of Laplacian (even without explicitly computing any of the distance measures). Moreover, only a few first eigenvectors are relevant, so it is not necessary to solve the eigenproblem entirely.
In the recent weeks I have been developing an approach which would allow automatic selection of seed points. I decided to proceed with the methods which use graph-based representation of point clouds and, as the matter of fact, are closely related with random walks on those graphs. I still do not have any solid results, though there are some interesting outputs that I would like to share in this blog post.
In the domain of mesh segmentation, or more generally 3D shape analysis, there is a fundamental problem of measuring distances between points on a surface. The most trivial and intuitive is geodesic distance, which encodes the length of the shortest path along the surface between two points. It has a number of drawbacks, the most important being its sensitivity to perturbations of the surface. For example, introducing a hole along the shortest path between two points, or a small topological shortcut between them may induce arbitrary large change in the distance. This is an undesired property, especially considering the noisy Kinect data that we work with.
A more sophisticated distance measure, diffusion distance, is based on the mathematical study of heat conduction and diffusion. Suppose we have a graph that represents a shape (it may be constructed exactly the same way as for the Random Walker segmentation). Imagine that a unit amount of heat is applied at some vertex. The heat will flow across the edges and the speed of its diffusion will depend on the edge weights. After time has passed, the initial unit of heat will be somehow distributed among the other vertices. The Heat Kernel () encodes this distribution. More specifically, is the amount of heat accumulated after time at vertex if the heat was applied at vertex . Based on this kernel the diffusion distance between each pair of points is defined. Importantly, the distance depends on the time parameter and captures either local or global shape properties.
Another distance measure, commute-time distance, is the average time it takes a random walker to go from one vertex to the other and come back. Finally, biharmonic distance was proposed most recently in:
This distance measure is non-parametric (does not depend on e.g. time) and is claimed to capture both local and global properties of the shape.
The figure below demonstrates biharmonic, commute-time, and heat diffusion distance maps computed with respect to the point marked with a red circle:
Left column: voxelized point cloud (top), biharmonic distance
(middle), commute-time distance (bottom).
Right column: heat diffusion distance for several choices of
the time parameter.
In each image the points with smallest distance are painted in dark blue, and the points with largest distances are dark red. The absolute values of distances are very different in all cases.
I think these distance maps could be used to infer the number of distinct objects in the scene. Indeed, the points that belong to the same object tend to be equidistant from the source point, so different objects correspond to different blobs of homogeneous points. Finding objects thus is the same as finding modes of the distribution of distances, which could be accomplished with Mean-Shift algorithm.
Speaking about particular choice of distance measure, biharmonic distance and heat diffusion distance with large time parameter intuitively seem to be better than others, however this is a subject for a more careful examination.
This is a follow-up post to the segmentation using random walks. It turned out that my initial implementation had several bugs which significantly worsened the performance. In this blog post I will describe them and show the outputs produced by the fixed algorithm.
The segmentation results that I demonstrated last time expose two (undesirable) features: vast regions with random label assignment and “zones of influence” around seed points. My first intuitive explanation was that since the edge weights are always less than 1, a random walker can not get arbitrary far from its starting point, therefore if a seed is reasonably far, the probability of getting there is very small. Surprisingly, I did not find any mentions or discussions of this effect in the literature. Moreover, while carefully re-reading “Random Walks for Image Segmentation” I mentioned that the algorithm outputs for each point and label pair not the probability that a random walker started from that point will reach the label, but rather the probability that it will first reach that label (i.e. earlier than other labels).
I decided to visualize edge weights and the probabilities I get with my implementation. In the following experiment I labeled three points, one on the table, and the other two on the boxes (see the left figure):
Color point cloud
with three labeled
a random walker
will first reach the
top right label
The figure in the middle shows all edges along which random walkers move. Each edge is visualized by its middle point colored according to edge weight (using “jet” color map where 0 is dark blue and 1 is dark red). The weights do make sense, as the edges in the planar regions are mostly reddish (i.e. weight is close to 1), whereas on the surface boundaries they are bluish (i.e. weight is close to 0). Moreover, it is clear that concave and convex boundaries have different average weights.
The figure on the right shows all points colored according to the probability that a random walker started from that point will first reach the labeled point on the box in the back. The probability images for the other labels are similar, thus for the majority of points the probability of first reaching either label is close to zero. Clearly, this can not be right, because some label has to be reached first anyways, and therefore the probabilities at each point should sum up to one. This observation triggered a long search for bugs in my implementation, but in the end I discovered and fixed two issues.
As I mentioned before, the solution for random walker segmentation is obtained by solving a system of linear equations, where coefficients matrix consists of edge weights and their sums, arranged in a certain order. The system is huge (there are as many equations as there are unlabeled points), but sparse (the number of non-zero coefficients in a row depends on the number of edges incident to a point). The first issue was due to a bug (or a feature?) of OctreePointCloudAdjacency class that I used to determine neighborhoods of points. In a nutshell, it is a specialized octree, where leaves store pointers to their direct neighbors (in terms of 26-connectivity). For some reason, a leaf always stores a pointer to itself in the list of neighbors. This caused bogus self-edges in my graph which themselves did not show up in the matrix (because diagonal elements are occupied by sums of weights of incident edges), however treacherously contributed to those sums, thus invalidating them.
The second issue was more subtle. For the system to have a solution it has to be non-singular. In terms of the graph it means that it either has to be connected, or should contain at least one seed in every connected component. From the beginning I had a check to enforce this requirement, however I did not take into account that the weighting function may assign zero weight to an edge! In the pathological case all the edges incident to a point may happen to be “weightless”, thus resulting in an all-zero row in the system.
Below are the results obtained using the random walker algorithm after I fixed the described issues:
Random walk segmentation
with overlaid seed points
Probabilities that a random
walker will first reach each
of the labels
As I did it previously, I labeled a single point in each object. (Actually, a single point in each disjoint part of each object. That is why there are multiple labeled points on the table.) The scene is perfectly segmented. On the right is an animated GIF with a sequence of frames which show probabilities (potentials) for each label. In most cases the separation between object and background is very sharp, but in two cases (the small box on top of the book and standing box on the right) some non-zero probabilities “spill” outside the object. Nevertheless, this does not affect the final segmentation since the potentials for the correct labels are higher.
I am very happy with the results, however one should remember that they were obtained using manual selection of labels. This code sprint is aimed at an automatic segmentation, so next I plan to consider different strategies of turning this into a fully autonomous method.
In the last few weeks I have been working on two things in parallel. On one hand, I continued to improve supervoxel segmentation, and a description of this is due in a later blog post. On the other hand, I started to look into an alternative approach to point cloud segmentation which uses random walkers. In this blog post I will discuss this approach and show my initial results.
The random walker algorithm was proposed for interactive image segmentation in:
The input is an image where several pixels are marked (by the user) with different labels (one for each object to be segmented out) and the output is a label assignment for all the remaining pixels. The idea behind the algorithm is rather intuitive. Think of a graph where the nodes represent image pixels. The neighboring pixels are connected with edges. Each edge is assigned a weight which reflects the degree of similarity between pixels. As it was mentioned before, some of the nodes are marked with labels. Now take an unlabeled node and imagine that a random walker is released from it. The walker randomly hops to one of the adjacent nodes with probability proportional to the edge weight. In the process of this random walk it will occasionally visit labeled nodes. The one that is most likely to be visited first determines the label of the node where the walker started.
Luckily, one does not have to simulate random walks from each node to obtain the probabilities of arriving at labeled nodes. The probabilities may be calculated analytically by solving a system of linear equations. The matrix of coefficients consists of edge weights and their sums, arranged in a certain order. The author did a great job explaining why this is so and how exactly the system should be constructed, so those who are interested are referred to the original paper.
Originally, the algorithm was applied to segment 2D images, however it could be used for any other data as long as there is a way to model it using a weighted graph. For us, of course, 3D point clouds are of a particular interest. The following paper describes how random walker segmentation could be applied for meshes or point clouds:
The authors construct the graph from a point cloud is the following way. Each point in the cloud becomes a node in the graph and the normal vector is computed for it.
For a pair of nodes and two distances are defined:
In the angular distance is a coefficient which depends on the relative concavity between points. For the convex case it is set to , effectively discounting the difference, whereas for the concave case it is equal to .
Using K-nearest neighbors search the neighborhood of a node is established. An edge is created between the node and each other node in the neighborhood. The weight of the edge depends on the two distances and is defined as:
and are used to balance the contributions of different distances. and in the denominators stand for the mean values of the distances over the whole point cloud.
In my implementation I decided to voxelize point cloud like it is done in SupervoxelSegmentation. I also reused the OctreePointCloudAdjacency class contributed by Jeremie Papon to establish the neighborhood relations between voxels. The weight is computed exactly as it is proposed by Lai et al. Finally, I use the Sparse module of Eigen to solve the linear system.
I mentioned before, but did not stress attention on the fact that this segmentation approach is semi-automatic. This means that the user has to provide a set of labeled points. My current implementation either accepts user input or generates labeled points uniformly using the same approach as SupervoxelSegmentation does for seed generation. There are smarter ways of producing initial labeling and I plan to consider this issue later.
Here are the some initial results that I got using random walk segmentation algorithm. The big red dots show the locations of labeled points:
Voxelized point cloud (voxel
size 0.006 m)
Random walk segmentation
(with one manually labeled
point per object)
Random walk segmentation
(with multiple manually
labeled points in each object)
Random walk segmentation
(labeled points are uniformly
In the first case (top right) I manually labeled each object in the scene with one point. Many of the object boundaries are correctly labeled, however there are vast regions with totally random labels. This is especially evident in the upper right corner, where there is just one seed. A circular region around it is correctly labeled with single color, however the points outside of it are all randomly colored. According to my understanding, this happens because the edge weights are always less than 1, so a random walker can not get arbitrary far from the starting point. Thus if there are no labeled points in the vicinity of a point, then all the computed probabilities are nearly zero and label assignment happens randomly (because of numerical imprecision).
In the second case (bottom left) I manually labeled each object in the scene with multiple points guided by an intuition about how large the “zones of influence” are around each labeled point. The resulting segmentation is rather good.
In the third case (bottom right) the labeled points were selected uniformly from a voxel grid with size 10 cm. As a result I got an over-segmentation that resembles the ones produced by SupervoxelSegmentation.
I think the initial results are rather promising. In the future I plan to work on the weighting function as it seems to be the key component for the random walk segmentation. I would like to understand if and how it is possible to vary the size of the “zone of influence” around labeled points.
I concluded the previous blog post with a claim that the improved refinement procedure yields a better segmentation. That conclusion was based purely on a visual inspection and rang a bell reminding that it is time to start thinking about quantitative evaluation metrics and prepare a tool set to preform evaluations. In this blog post I will describe my first steps in this direction.
Let us start with a prerequisite for any evaluation, ground truth data. So far I have been using scenes from Object Segmentation Database (OSD). This dataset contains a ground truth annotation for each point cloud, however I am not completely satisfied with its quality. Consider the following point cloud (left) and the provided annotation (middle):
Color point cloud
Some of the points in the close vicinity of the object boundaries are wrongly annotated. For example, almost all points on the left edge of the standing box are annotated as belonging to the table. I hypothesize that the annotation was obtained automatically using some color image segmentation algorithm. (Thus it is not really “wrong”, it is correct in terms of the color data.) As we are working with 3D point clouds, the spatial relations derived from the depth data should have larger weight than color and the annotation should be revised to respect geometrical boundaries of the objects. Unfortunately, there is no tool in PCL that would allow to load a point cloud and edit the labels. It costed me quite a few hours of work to put together a simple editor based on PCLVisualizer that could do that. An example of refined annotation is shown in the figure above (right).
The authors of SupervoxelSegmentation used two metrics to evaluate how nicely the supervoxels adhere to the object boundaries: under-segmentation error and boundary recall. These metrics were introduced in:
I decided to begin with the under-segmentation error. Formally, given a ground truth segmentation into regions and a supervoxel segmentation into supervoxels , the under-segmentation error for region is defined as:
Simply put, it takes a union of all the supervoxels that overlap with a given ground truth segment and measures how much larger its total size is than the size of the segment. The authors of SupervoxelSegmentation use a slightly modified version which summarizes the overall error in a single number:
where is the total number of voxels in the scene. In practice, this error sums up the sizes of all the supervoxels that cross the ground truth boundaries.
In my opinion, this definition biases error with the average supervoxel size. Consider the supervoxel segmentation (left):
(seed size 0.1 m)
cross ground truth
according to the
Clearly, the segmentation is rather good in terms of the border adherence. The only failure is in the right corner of the lying box, where the green supervoxel “bleeds” on the table. The middle image shows which voxels will be counted towards the error, i.e. those supervoxels that cross the ground truth boundaries. In fact, every supervoxel in the close vicinity of an edge is counted. The reason is simple: the edge between two surfaces in a typical point cloud obtained with an RGB-D camera is ambiguous. The ground truth segmentation is therefore random to some extent and the chance that the supervoxel boundary will exactly coincide with the ground truth segment boundary is close to zero. Consequently, the error always includes all the supervoxels around the boundary, and the larger they are on average, the larger the error is. The “real” error (“bled” green supervoxel) is completely hindered with it.
I decided to use a modified definition. Whenever a supervoxel crosses ground truth boundary(ies), it is split in two (or more) parts. I assume that the largest part is correctly segmented, and only count the smaller parts as erroneous. The figure above (right) demonstrates which voxels would be counted towards the error in this case. Still, there is some “noise” around boundaries, but it does not hinder the “real” error. I also do not like that the error in the original definition is normalized by the total point cloud size. I think that the normalization should be related to the amount of objects or the amount (length) of object boundaries. I will consider this options in future, but for now I just count the number of erroneous pixels and do not divide it by anything.
I would like to conclude with the results of evaluating simple and improved supervoxel refinement procedures using the described error metric. The figure below shows the voxels considered as erroneous (in red) in supervoxel segmentations obtained with 2 rounds of simple refinement (left) and 2 rounds of improved refinement (right):
(seed size 0.1 m, with
(seed size 0.1 m, with
The under-segmentation error without refinement is 15033. Simple refinement reduces it to 11099 after the first iteration and 10755 after the second. Improved refinement reduces it to 9032 and 8373 respectively. I think the numbers do agree with the intuition, so I will continue using this metric in future.