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Report on Fully Affine Invariant Image Comparison

If a physical object has a smooth or piecewise smooth boundary, its images obtained by cameras in varying positions undergo smooth apparent deformations. These deformations are locally well approximated by affine transforms of the image plane. In consequence the solid object recognition problem has often been led back to the computation of affine invariant image local features. Such invariant features could be obtained by normalization methods, but no fully affine normalization method exists for the time being. As a matter of fact, the scale invariance, which actually means invariance to blur, is only dealt with by methods inspired from the scale space theory, like the SIFT method. By simulating zooms out, this method normalizes the four translation, rotation and scale (blur) parameters, out of the six parameters of an affine transform. Affine normalization methods like MSER normalize with respect to all six parameters of the affine transform, but this normalization is imperfect, not dealing rigorously with blur. The method proposed in this paper, affine SIFT (A-SIFT), simulates all image views obtainable by varying the two camera parameters left over by the SIFT method. Then it normalizes the other four parameters by simply using the SIFT method itself. The two additional parameters are the angles (a longitude and a latitude) defining the camera axis orientation. Mathematical arguments are developed to prove that the resulting method is fully affine invariant, up to an arbitrary precision. Against any prognosis, simulating all views depending on the two camera orientation parameters is feasible with no dramatic computational load. The method permits to reliably identify features that have undergone tilts of large magnitude, up to 30 and more, while state-of-the-art methods do not exceed tilts of 2.5 (SIFT) or 4.5 (MSER). The report puts in evidence the role of high transition tilts: while a tilt from a frontal to an oblique view exceeding 6 is rare, higher transition tilts are common as soon as two oblique views of an object are compared. Thus, a fully affine invariance is required for 3D scene analysis. This fact is substantiated by many experiments.