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-adapted interest point operators.
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 formula_1 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 formula_2, let formula_3 and let formula_4 be a positive definite 2×2 matrix. Then, a non-uniform Gaussian kernel can be defined as
and given any input image formula_6 the affine Gaussian scale-space is the three-parameter scale-space defined as
Next, introduce an affine transformation formula_8 where formula_9 is a 2×2-matrix, and define a transformed image formula_10 as
Then, the affine scale-space representations formula_12 and formula_13 of formula_6 and formula_10, respectively, are related according to
provided that the affine shape matrices formula_17 and formula_18 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 formula_20 as well as local shape matrix formula_4 and an integration shape matrix formula_22, introduce an "affine-adapted multi-scale second-moment matrix" according to
it can be shown that under any affine transformation formula_24 the affine-adapted multi-scale second-moment matrix transforms according to
Again, disregarding somewhat messy technical details, the important message here is that "given a correspondence between the image points formula_26 and formula_27, the affine transformation formula_9 can be estimated from measurements of the multi-scale second-moment matrices formula_29 and formula_30 in the two domains.
An important consequence of this study is that if we can find an affine transformation formula_9 such that formula_30 is a constant times the unit matrix, then we obtain a "fixed-point that is invariant to affine transformations" (Lindeberg 1994 section 15.4; Lindeberg and Garding 1997). For the purpose of practical implementation, this property can often be reached by in either of two main ways. The first approach is based on "transformations of the smoothing filters" and consists of:
The second approach is based on "warpings in the image domain" and implies:
This overall process is referred to as "affine shape adaptation" (Lindeberg and Garding 1997; Baumberg 2000; Mikolajczyk and Schmid 2004; Tuytelaars and van Gool 2004; Ravela 2004; Lindeberg 2008). In the ideal continuous case, the two approaches are mathematically equivalent. In practical implementations, however, the first filter-based approach is usually more accurate in the presence of noise while the second warping-based approach is usually faster.
In practice, the affine shape adaptation process described here is often combined with interest point detection automatic scale selection as described in the articles on blob detection and corner detection, to obtain interest points that are invariant to the full affine group, including scale changes. Besides the commonly used multi-scale Harris operator, this affine shape adaptation can also be applied to other types of interest point operators such as the Laplacian/Difference of Gaussian blob operator and the determinant of the Hessian (Lindeberg 2008). Affine shape adaptation can also be used for affine invariant texture recognition and affine invariant texture segmentation.
Closely related to the notion of affine shape adaptation is the notion of "affine normalization", which defines an "affine invariant reference frame" as further described in Lindeberg (2013a,b, 2021:Appendix I.3), such that any image measurement performed in the affine invariant reference frame is affine invariant.