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A robust image transform

Since contrast determines the ability to find unique correspondences, we motivate our approach by considering the sources of contrast within a local image window that contains an occlusion boundary. We define the ``foreground'' to be the scene layer on which the central point of the window resides; points on all other layers are considered ``background''. We desire a transform which ignores background contrast but is sensitive to contrast energy from the occluding boundaries of the foreground layer.

In general one does not know a priori whether contrast within a particular window is entirely within the foreground layer, is due to the occlusion boundary between foreground and background, or is entirely within the background layer. When contrast is in the foreground layer, an ideal template would model it fully, both in magnitude and sign. When the contrast is due to an occlusion edge, it is reasonable only to define a template based on the contrast energy, since the sign of contrast is arbitrary with changing background. When contrast is in the background layer, it should be ignored in an ideal template.

Figure 3:   Construction of the Radial Cumulative Similarity (RCS) transform. (a) Color window, (b) central color $\bf C$ (in box at lower-left) and map of local similarity S. Bright pixels indicate similar value as central color. (c) neighborhood of cumulative similarity, N, where each pixel reflects the likelihood the ray from the center point has uniform color.
(a)\psfig {figure=finger/,width=1.25in}(b)\psfig {figure=finger/,width=1.25in}(c)\psfig {figure=finger/,width=1.25in} 
We define a robust local image representation that approximates this ideal, without any prior knowledge of the occlusion location. Our representation is comprised of a central image-attribute value (typically color) and of a local contrast neighborhood of this attribute, attenuated to discount background influence. Many different diffusion functions could be used to attenuate background influence; in this paper we explore radial cumulative probability functions. The local neighborhood is defined by estimating the contrast energy of the attribute relative to the center value, interpreting this energy probabilistically, and computing the cumulative likelihood that the attribute is unchanged along the ray from the template center to a particular neighborhood point.

Formally, given a discrete color image intensity function ${\bf I}(x,y)$we compute a local robust representation:

\begin{displaymath}{\cal R}_{{\bf I},x,y} ~=~ \{ {\bf C}_{{\bf I},x,y}, N_{{\bf I},x,y}(i,j) \} \end{displaymath}
where $-M_n \leq i,j \leq M_n$. Our representation is comprised of two terms, a central value and a neighborhood function; the central value is simply the image attribute averaged over the center point or a small central area:
\begin{displaymath}{\bf C}_{{\bf I},x,y} ~=~ \frac{1}{(2M_c+1)^2}\sum_{i,j=-M_c}^{i,j\leq M_c} {\bf A}({\bf I},x+i,j-i) .\end{displaymath}
where ${\bf A}({\bf I},x,y)$ is an image attribute function and can be defined to be any local image property. In this paper we explore attribute functions which return the color or hue vector corresponding to the pixel at the given location. We typically keep the central region small, with Mc = 0 or 1. The neighborhood is defined over window coordinates $-M_n \leq i,j \leq M_n$ using the similarity of other image attribute values to the central value:
\begin{displaymath}S_{{\bf I},x,y}(i,j) ~=~ e^{-{\bf E}_{{\bf I},x,y}(i,j)^T{\bf E}_{{\bf I},x,y}(i,j)} .\end{displaymath}
\begin{displaymath}{\bf E}_{{\bf I},x,y}(i,j) ~=~ ({\bf C}_{{\bf I},x,y} - {\bf A}({\bf I},x+i,y+j)) \end{displaymath}
Note that $- \log S$ is a local contrast energy function, and is thus independent of contrast sign.

When tracking a single feature of known size, we could simply use $S_{{\bf I},x,y}(i,j)$ over a fixed (possibly non-rectangular) window cropped to resolve the entire feature and the occlusion boundary. This would yield a template which captures both the foreground and occlusion contrast, and was insensitive to contrast sign. However, when automatically tracking features for image analysis/synthesis, or when computing dense correspondence for stereo or motion, we rarely have the luxury of knowledge of appropriate window size.

For fully automatic processing, we define a function which substantially attenuates the influence of exterior pixels. We define our neighborhood function by propagating the attribute similarity function S outward along a ray from the center of the window, so that once we encounter a dissimilarity (i.e., contrast energy) we attenuate the influence of any contrast found farther out along that ray. We are essentially making the assumption that the most proximate contrast is due either to surface contrast or occlusion contrast; background contrast must lie beyond an occurrence of occlusion contrast. Our algorithm reflects the conservative assumption that, in the absence of any prior knowledge of occlusion location, correspondence judgments are best made on the most proximate contrast.

Our neighborhood function is the cumulative product of S, computed radially from the center point:

\begin{displaymath}N_{{\bf I},x,y}(i,j) ~=~ \prod_{(k,l) \in {\it r}_{i,j}} S_{{\bf I},x,y}(k,l) \end{displaymath}
where ${\it r}_{i,j}$ is the set of points that lie along the ray from (0,0) to (i,j), inclusive. Other possible neighborhood functions include pixel-fill or diffusion operators; these would also capture non-convex local similarity structure.

We call the representation ${\cal R}$ the Radial Cumulative Similarity (RCS) transform, since it reflects the radial homogeneity of a given attribute value. Figure 3 illustrates the computation of color RCS for a image window containing a fingertip. The substantial benefit of the RCS transform is invariance to sign of contrast at an occluding boundary, as well as invariance to background contrast. As an example Figure 4 shows the RCS transform for the marked locations in Figure 2; despite dissimilar background structure and occlusion contrast sign reversal, the transformed pairs are substantially similar.

Figure 4:  The RCS transform is stable despite occlusion boundaries of different contrast sign. (a,b) show the RCS transform of the marked locations in Figure 2(b,f), while (c,d) show the RCS transform of Figure 2(d,h).
(a)\psfig {figure=eye/,width=1in}(b)\psfig {figure=eye/,width=1in}(c)\psfig {figure=mouth1/,width=1in}(d)\psfig {figure=mouth1/,width=1in} 

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Next: Finding correspondences Up: A radial cumulative similarity Previous: Introduction 
Trevor Darrell