Affine shape adaptation

Affine shape adaptation is a methodology for iteratively adapting the shape of the smoothing kernels in an affine group of smoothing kernels to the local image structure in neighbourhood region of a specific image point. Equivalently, affine shape adaptation can be accomplished by iteratively warping a local image patch with affine transformations while applying a rotationally symmetric filter to the warped image patches. Provided that this iterative process converges, the resulting fixed point will be affine invariant. In the area of computer vision, this idea has been used for defining affine invariant interest point operators as well as affine invariant texture analysis methods. Affine shape adaptation is a methodology for iteratively adapting the shape of the smoothing kernels in an affine group of smoothing kernels to the local image structure in neighbourhood region of a specific image point. Equivalently, affine shape adaptation can be accomplished by iteratively warping a local image patch with affine transformations while applying a rotationally symmetric filter to the warped image patches. Provided that this iterative process converges, the resulting fixed point will be affine invariant. In the area of computer vision, this idea has been used for defining affine invariant interest point operators as well as affine invariant texture analysis methods. The interest points obtained from the scale-adapted Laplacian blob detector or the multi-scale Harris corner detector with automatic scale selection are invariant to translations, rotations and uniform rescalings in the spatial domain. The images that constitute the input to a computer vision system are, however, also subject to perspective distortions. To obtain interest points that are more robust to perspective transformations, a natural approach is to devise a feature detector that is invariant to affine transformations. Affine invariance can be accomplished from measurements of the same multi-scale windowed second moment matrix μ {displaystyle mu } as is used in the multi-scale Harris operator provided that we extend the regular scale space concept obtained by convolution with rotationally symmetric Gaussian kernels to an affine Gaussian scale-space obtained by shape-adapted Gaussian kernels (Lindeberg 1994 section 15.3; Lindeberg and Garding 1997). For a two-dimensional image I {displaystyle I} , let x ¯ = ( x , y ) T {displaystyle {ar {x}}=(x,y)^{T}} and let Σ t {displaystyle Sigma _{t}} be a positive definite 2×2 matrix. Then, a non-uniform Gaussian kernel can be defined as and given any input image I L {displaystyle I_{L}} the affine Gaussian scale-space is the three-parameter scale-space defined as Next, introduce an affine transformation η = B ξ {displaystyle eta =Bxi } where B {displaystyle B} is a 2×2-matrix, and define a transformed image I R {displaystyle I_{R}} as Then, the affine scale-space representations L {displaystyle L} and R {displaystyle R} of I L {displaystyle I_{L}} and I R {displaystyle I_{R}} , respectively, are related according to provided that the affine shape matrices Σ L {displaystyle Sigma _{L}} and Σ R {displaystyle Sigma _{R}} are related according to Disregarding mathematical details, which unfortunately become somewhat technical if one aims at a precise description of what is going on, the important message is that the affine Gaussian scale-space is closed under affine transformations. If we, given the notation ∇ L = ( L x , L y ) T {displaystyle abla L=(L_{x},L_{y})^{T}} as well as local shape matrix Σ t {displaystyle Sigma _{t}} and an integration shape matrix Σ s {displaystyle Sigma _{s}} , introduce an affine-adapted multi-scale second-moment matrix according to