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7 An Information Infinitesimal Decomposed in Both Scale and Spatial Coordinates

One serious drawback of the patch-wise method of spatial information decomposition is poor spatial localization. Particularly for large scales, patches have to be large to have the contrast signal between successive resolutions not drown in noise.

Some objects, like lines, are however well localized in spatial coordinates, but not as sharply in scale space, and a good saliency map (expected information gain), need to handle this

In this section we develop an information infinitesimal defined for a point in space instead of the previous patch.

Rewrite Kullback contrast between two diffused probability distributions as:

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Using the definition of the Kullback contrast and rewriting with the diffusion equation and the properties we listed we get K as:

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We identify tex2html_wrap_inline711 as an information infinitesimal, fulfilling the same requirements as previously required by the Kullback contrast i.e. tex2html_wrap_inline713 and tex2html_wrap_inline715 .

We study this doubly decomposed information infinitesimal for the line information in the tinytown image. To get reasonable signal to noise ratio, despite the (for our digitized image) discrete derivative approximation, we study total information at a particular resolution, i.e. the prior distribution Q uniform over the image, rather than successive differences between resolution lengths we used in the previous experiments, since these differences are very small. With this simplification tex2html_wrap_inline719 .

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Figure 13: A point wise information infinitesimal is used to select regions to attend to, which are also analyzed for the scale space information distribution.

Fig. 13 shows the use of point wise information infinitesimal in an attention selection process. The upper left image shows the highest peaks in s(r,x), projected onto the image in spatial coordinates. These are found by increasing a threshold until only a a few regions get selected. Using a lower threshold of half the maximum difference in tex2html_wrap_inline723 (right top image) we get larger and many more regions, but only the regions also found with the high threshold are selected. The larger selected regions are likely to contain whole objects of interest. For computer processing rectangular regions are preferable. The irregular selected regions are approximated by rectangular cutouts, shown highlighted in the center image. The scale space information distributions for each one of the rectangles is summed up using the patch-wise algorithm in section 5, and are shown in the diagrams around the center image.

For most subsequent image processing and/or interpretation the rectangular ROI's (regions of interest) together with their scale properties to be the appropriate output. Standard algorithms can be run directly on the ROI's, tuning appropriate parameters to optimize performance for the relevant scale range.

Other algorithms could be designed to operate directly on the tex2html_wrap_inline725 bounded regions generated by thresholding or region growing of conspicuous regions in the 3-D information infinitesimal map.


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