Shape context is a feature descriptor used in object recognition. Serge Belongie and Jitendra Malik proposed the term in their paper "Matching with Shape Contexts" in 2000.
Theory The shape context is intended to be a way of describing shapes that allows for measuring shape similarity and the recovering of point correspondences. The basic idea is to pick n points on the contours of a shape. For each point pi on the shape, consider the n − 1 vectors obtained by connecting pi to all other points. The set of all these vectors is a rich description of the shape localized at that point but is far too detailed. The key idea is that the distribution over relative positions is a robust, compact, and highly discriminative descriptor. So, for the point pi, the coarse histogram of the relative coordinates of the remaining n − 1 points,
h i ( k ) = # { q ≠ p i : ( q − p i ) ∈ bin ( k ) } {\displaystyle h_{i}(k)=\#\{q\neq p_{i}:(q-p_{i})\in {\mbox{bin}}(k)\}}
is defined to be the shape context of p i {\displaystyle p_{i}} . The bins are normally taken to be uniform in log-polar space. The fact that the shape context is a rich and discriminative descriptor can be seen in the figure below, in which the shape contexts of two different versions of the letter "A" are shown.
(a) and (b) are the sampled edge points of the two shapes. (c) is the diagram of the log-polar bins used to compute the shape context. (d) is the shape context for the point marked with a circle in (a), (e) is that for the point marked as a diamond in (b), and (f) is that for the triangle. As can be seen, since (d) and (e) are the shape contexts for two closely related points, they are quite similar, while the shape context in (f) is very different. For a feature descriptor to be useful, it needs to have certain invariances. In particular it needs to be invariant to translation, scaling, small perturbations, and, depending on the application, rotation. Translational invariance comes naturally to shape context. Scale invariance is obtained by normalizing all radial distances by the mean distance α {\displaystyle \alpha } between all the point pairs in the shape although the median distance can also be used. Shape contexts are empirically demonstrated to be robust to deformations, noise, and outliers using synthetic point set matching experiments. One can provide complete rotational invariance in shape contexts. One way is to measure angles at each point relative to the direction of the tangent at that point (since the points are chosen on edges). This results in a completely rotationally invariant descriptor. But of course this is not always desired since some local features lose their discriminative power if not measured relative to the same frame. Many applications in fact forbid rotational invariance e.g. distinguishing a "6" from a "9".
Use in shape matching A complete system that uses shape contexts for shape matching consists of the following steps (which will be covered in more detail in the Details of Implementation section):
Randomly select a set of points that lie on the edges of a known shape and another set of points on an unknown shape. Compute the shape context of each point found in step 1. Match each point from the known shape to a point on an unknown shape. To minimize the cost of matching, first choose a transformation (e.g. affine, thin plate spline, etc.) that warps the edges of the known shape to the unknown (essentially aligning the two shapes). Then select the point on the unknown shape that most closely corresponds to each warped point on the known shape. Calculate the "shape distance" between each pair of points on the two shapes. Use a weighted sum of the shape context distance, the image appearance distance, and the bending energy (a measure of how much transformation is required to bring the two shapes into alignment). To identify the unknown shape, use a nearest-neighbor classifier to compare its shape distance to shape distances of known objects.
Details of implementation
Step 1: Finding a list of points on shape edges The approach assumes that the shape of an object is essentially captured by a finite subset of the points on the internal or external contours on the object. These can be simply obtained using the Canny edge detector and picking a random set of points from the edges. Note that these points need not and in general do not correspond to key-points such as maxima of curvature or inflection points. It is preferable to sample the shape with roughly uniform spacing, though it is not critical.
Step 2: Computing the shape context This step is described in detail in the Theory section.
Step 3: Computing the cost matrix Consider two points p and q that have normalized K-bin histograms (i.e. shape contexts) g(k) and h(k). As shape contexts are distributions represented as histograms, it is natural to use the χ2 test statistic as the "shape context cost" of matching the two points:
C S = 1 2 ∑ k = 1 K [ g ( k ) − h ( k ) ] 2 g ( k ) + h ( k ) {\displaystyle C_{S}={\frac {1}{2}}\sum _{k=1}^{K}{\frac {[g(k)-h(k)]^{2}}{g(k)+h(k)}}}
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