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 \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 (; ). For a two-dimensional image I, let \bar{x} = (x, y)^T and let \Sigma_t be a positive definite 2×2 matrix. Then, a non-uniform Gaussian kernel can be defined as :g(\bar{x}; \Sigma) = \frac{1}{2 \pi \sqrt{\operatorname{det} \Sigma_t}} e^{-\bar{x} \Sigma_t^{-1} \bar{x}/2} and given any input image I_L the affine Gaussian scale-space is the three-parameter scale-space defined as :L(\bar{x}; \Sigma_t) = \int_{\bar{xi}} I_L(x-\xi) \, g(\bar{\xi}; \Sigma_t) \, d\bar{\xi}. Next, introduce an affine transformation \eta = B \xi where B is a 2×2-matrix, and define a transformed image I_R as :I_L(\bar{\xi}) = I_R(\bar{\eta}). Then, the affine scale-space representations L and R of I_L and I_R, respectively, are related according to :L(\bar{\xi}, \Sigma_L) = R(\bar{\eta}, \Sigma_R) provided that the affine shape matrices \Sigma_L and \Sigma_R are related according to :\Sigma_R = B \Sigma_L B^T. 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 \nabla L = (L_x, L_y)^T as well as local shape matrix \Sigma_t and an integration shape matrix \Sigma_s, introduce an
affine-adapted multi-scale second-moment matrix according to :\mu_L(\bar{x}; \Sigma_t, \Sigma_s) = g(\bar{x} - \bar{\xi}; \Sigma_s) \, \left( \nabla_L(\bar{\xi}; \Sigma_t) \nabla_L^T(\bar{\xi}; \Sigma_t) \right) it can be shown that under any affine transformation \bar{q} = B \bar{p} the affine-adapted multi-scale second-moment matrix transforms according to :\mu_L(\bar{p}; \Sigma_t, \Sigma_s) = B^T \mu_R(\bar{q}; B \Sigma_t B^T, B \Sigma_s B^T) B. Again, disregarding somewhat messy technical details, the important message here is that
given a correspondence between the image points \bar{p} and \bar{q} , the affine transformation B can be estimated from measurements of the multi-scale second-moment matrices \mu_L and \mu_R in the two domains. An important consequence of this study is that if we can find an affine transformation B such that \mu_R is a constant times the unit matrix, then we obtain a
fixed-point that is invariant to affine transformations (; ). 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: • estimating the second-moment matrix \mu in the image domain, • determining a new adapted smoothing kernel with covariance matrix proportional to \mu^{-1}, • smoothing the original image by the shape-adapted smoothing kernel, and • repeating this operation until the difference between two successive second-moment matrices is sufficiently small. The second approach is based on
warpings in the image domain and implies: • estimating \mu in the image domain, • estimating a local affine transformation proportional to \hat{B} = \mu^{1/2} where \mu^{1/2} denotes the square root matrix of \mu, • warping the input image by the affine transformation \hat{B}^{-1} and • repeating this operation until \mu is sufficiently close to a constant times the unit matrix. This overall process is referred to as
affine shape adaptation (; ; ; ; ; ). 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 (). 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 (,, :Appendix I.3), such that any image measurement performed in the affine invariant reference frame is affine invariant. == See also ==