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
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 & Garding 1997).
Then, a non-uniform Gaussian kernel can be defined as and given any input image
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.
, introduce an affine-adapted multi-scale second-moment matrix according to it can be shown that under any affine transformation
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
An important consequence of this study is that if we can find an affine transformation
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 & Garding 1997; Baumberg 2000; Mikolajczyk & Schmid 2004; Tuytelaars & van Gool 2004; Ravela 2004; Lindeberg 2008).
In the ideal continuous case, the two approaches are mathematically equivalent.
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.