Shape matching is an attempt to operationalize our intuitive notion of similarity between shapes: if we consider for example the two handwritten letters in Figure 1, they appear rather similar to a human observer, while being radically different if compared as vectors of pixels.
One way to address this issue could be to design a notion of distance between two shapes : then, two shapes that match pretty well will be close in terms of this distance. More precisely, we will design three distances all based on the following shape matching procedure.
Description of the Shape Matching procedure
In the approach we investigate here, we will, as suggested in , treat an object as a point set and we assume that the shape of an object is captured by a finite subset of its points. In practice, we uniformly sample the internal and external contour of the object using an edge detector (see Figure 2). This gives us a set , of points.
Then, we face a correspondence problem: for each point on the first shape, we want to find the “best” matching point on the second shape. This matching can be efficiently performed by using rich local descriptors of the shape, such as shape context, introduced in . The idea is the following :
Given a point on the shape, we consider the set of vectors originating from a point to all other sample points on a shape. These vectors express the configuration of the entire shape with respect to the reference point (see Figure 3). Obviously, as the number of points increase, the description of the shape becomes more precise. But the full set of vectors is much too detailed to be used as a shape descriptor, therefore we choose a more compact descriptor: we consider the densities of these vectors in the log-polar space.
More specifically, for each point on the shape, the shape context of is defined by the histogram (see Figure 4):
where the bins are taken uniformly in the log-polar space.
Then, the problem is to compare the shape context of two points on the first shape and on the second shape. We use the test statistic to determine wether or not two histograms (i.e. shape context) are significantly different:
This statistic can be interpreted as the cost of matching and .
Then, we draw a bipartite graph, with nodes the sampled points of the first and second shape. Then, every points and are linked by an edge with weight if we cannot reject the hypothesis that their two shape context are the same (see Figure 5). This gives us for every point on the first shape, a set of potential match on the second one, and respectively (see Figure 5).
Finally, we want to obtain the best bipartite graph matching, namely, we want to minimize the total cost matching,
with the constraint that the matching be one-to-one, is a permutation on the neighbors of .
Construction of Shape Distances
Given two arbitrary shapes, we can now obtain potential correspondences between points on these two shapes by using the shape matching procedure described above. For example, correspondences between points on the two words in Figure 6 a) are given by the best bipartite matching in Figure 6 b).
|a) The two words on which the shape matching procedure is performed|
Then, we can use this finite set of correspondences to estimate a transformation of the plane , allowing us to map any point on the first shape to a point on the second shape, and therefore extending the discrete set of correspondences to the continuous level.
To this end, we use two thin plate spline transformations (see ) for each coordinates:
More precisely, we choose a source and a target between the two shapes and then we solve two 1D interpolation problems.
Roughly speaking, we seek for the transformation that minimizes the bending energy necessary to map points on the source shape to their corresponding matches on the target shape.
In the particular example of the two words in Figure 6, we can see on the subsequent video the effect of applying the transformation to the first word.
With this mapping, we can already imagine two distances between two arbitrary shapes and . The first one is a generalization to the continuous level of the total cost matching introduced previously:
with the thin plate spline transformation and the cost of matching two points introduced in the previous post. We can also compute another distance, corresponding to the amount of deformation necessary to align the two shapes. This distance is naturally set equal to the bending energy :
Finally, we introduce a last distance capturing the appearance information (texture and intensity information of the two grayscale images). To this end, we apply the thin plate spline transformation to the source image (see Figure 7) and compute the difference (pixels by pixels, see Figure 8) between the resulting warped image and the target image.
Then, we compute the squared mean and the standard deviation of the resulting image (interpreted as a matrix), and set the third distance equal to the sum of these two quantities.
Figure 8: Top: Difference between the two original images, Bottom: Difference between the two images after transformation of the source image
Construction of a Similarity Measure
From the three distances previously introduced, we wish to design a similarity measure, expressing how close is a shape from another. To this end, we first build a single distance that separates the better the data.
For this, we consider a training set of 42 words. In this training set, we know which words are semantically the same and which words are semantically different. Then, we compute the three distances between each pair of words. For each pair, we obtain a point of which coordinates correspond to the respective values of the three distances. Additionally, we color this point in red if the two words compared are semantically the same and in blue if the two words compared are semantically different. We obtain the following plot:
The idea is now to find the plane that separates the better the two groups. Then, by projecting onto the vector normal to this plane, we obtain a unique distance that separates the best the two groups. This can be done using Fischer discrimination, and we obtain the following distance:
From this unified notion of distance, we introduce the similarity between two shapes and :
If two shapes are very close, then their distance will be close to 0, and the similarity close to 1. Conversely, shapes very different will have a similarity close to 0. Note that the similarity measure is always well defined because we never compare strictly identical shapes.
Clustering Analysis with k-Nearest Neighbors
This measure of similarity will allow us to perform clustering analysis using the -nearest neighbors methodology. To this end, we build an undirected fully-connected graph: each node correspond to a shape/word and edges between the nodes are weighted by the similarity function between the two nodes (shapes). Such a graph is called a similarity graph. An example of it is presented on Figure 10. In this example, we compared a database of 42 handwritten words, with 11 semantically different words.
Then, the idea of the -nearest neighbors is to first reduce the number of edges in the similarity graph: two vertices and are connected if is among the nearest neighbors of , with a parameter to choose. Once the number of edges reduced, the goal is to separate the graph in components (clusters), so that the the sum of the weights of the edges we have to cut to disconnect these components is minimized.
If we apply this procedure to our similarity graph, with and we obtain the clusters shown on Figure 11. On this figure, each color corresponds to a cluster, and nodes have been colored according to their respective cluster. We additionally added an uncertainty measure, represented by the following color code:
- Green: The nodes which only neighbors are members of their cluster: these nodes being only connected with their pairs, it is really unlikely that they have been misclassified.
- Pink: The nodes which have more neighbors in their cluster than outside of it: these nodes are strongly connected with their cluster, but still they could relate with some other nodes outside of it, so we might wonder if they have been correctly classified.
- Red: The nodes which have more neighbors outside their cluster than inside: these nodes are better connected with the rest of the graph than to the cluster they belong to. Therefore we have to be very careful in the trust we put in these nodes: it is very likely that they have been misclassified and that they should belong in reality to another cluster.
The results of the clustering analysis are quiet good: 86% (on average) of the words have been correctly grouped together. More precisely, 98% of the green nodes have been correctly classified, against 87% for the pink nodes and only 35% for the red (so the uncertainty measure seems to be a good indicator).
A Semi-Automatic Transcription tool (Demo)
We use the preceding procedures to develop a semi-automatic transcription tool of ancient texts.
Let say we have digitized an old handwritten book. Then, to be able to extract the information out of it, we need to transcript it, so that a computer could recognize the words. Usually, a specialist is paid to realize manually this tedious task. But his work could be considerably facilitated by the clustering analysis developed earlier: as soon as he types the transcription of a word, it automatically propagates to all the members of the cluster to which the word belongs. In the demo we present here, the actual numbers of words the user has to type in order to transcript all the 42 words is reduced by 70% on average. Of course errors might occur, but we can notify the user of potential mis-transcriptions using the uncertainty measure previously introduced. In the demo we propose here, uncertain words are transparent and represented into dashed-line boxes (see figure 12). An alternative version representing uncertainty with a color code is also available here (red: high uncertainty, blue: normal uncertainty, green: low uncertainty).
Demo of the tool on a database of 42 words:
For optimal performances, please begin transcription by words in plain black boxes and correct potential errors in dashed-line boxes.
The results obtained on the training set of 42 words are quite encouraging: the number of words to type during the transcription process is reduced on average by 70% and 86% of the words have been correctly classified by the clustering analysis. However, the current version is static, and doesn’t update the clusters when the user is manually correcting a mis-transcripted word. It would be interesting to implement such a dynamic behavior, in order to improve even more the accuracy and efficiency of the tool. On the long range, we could even imagine a collaborative database storing the transcriptions of each user of the tool, so that the procedure would become more and more efficient and eventually tend to a fully automatic tool when the number of transcripted words in the database becomes big enough. Finally, there is still a lot to do in terms of computation: the running time of the procedure explodes with the number of words to process. It is necessary to optimize the code in C++ and maybe reduce the number of operations by applying pre-filters to the words, so that we do not compare words that are obviously different.
 S. Belongie,J. malik, J. Puzicha, Shape Matching an Object Recognition using Shape Contexts, IEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 24, April 2002.
 Daniel Kressner, Computational Linear Algebra, Lecture notes, EPFL, Spring 2014.
 Trevor Hastie, Robert Tibshirani, Jerome Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Second Edition, February 2009.