

Histogram Preserving Image Transformations:
The topleft image is transformed with Hamiltonian transformations to produce the remaining three images. These images are severely distorted and yet have exactly the same histogram as the original image. The histogram of an image is invariant if and only if the image is transformed with a Hamiltonian transformation.



Pose Estimation of Polyhedral Objects:
The class of histogram preserving image transformations has been used to identify image projection models that preserve the histogram (up to a scale). The projections include the weakperspective and paraperspective projections. Based on this observation, the histogram of a polyhedral object can be expressed as the sum of the histograms of the projections of its visible faces. This representation can be used to estimate the pose of a polyhedral object.



Effect of Shape on the Multiresolution Histogram:
The images on the left are parameterized superquadrics. Their histograms are approximately the same. The plot shows the rate of change of the histogram with image resolution plotted as a function of the superquadric parameter. The rate of change is minimum for a circle and increases for shapes with sharp corners.



Effect of Texture on the Multiresolution Histogram:
The images on the left are textures with texels placed with an increasing degree of randomness. Their histograms are approximately the same. The plot shows the rate of change of the histogram with image resolution plotted as a function of the randomness of the texel placement. The rate of histogram change decreases with the randomness.



Texture Recognition Using Multiresolution Histograms:
Several textures from the Brodatz database under different rotations are seen in this picture. The multiresolution histogram was used to match textures. Matching works well because the multiresolution histogram captures spatial information and at the same time is invariant to rotations of the image.



Texture Recognition under Illumination and Viewpoint Changes:
The CURET database includes 3D textures imaged from different viewpoints and under different illuminations. The multiresolution histogram was used to match different instances of these textures. This experiment demonstrates that the multiresolution histogram is robust to illumination and viewpoint changes as well.
