- Scale-space axioms
In
image processing andcomputer vision , ascale-space framework can be used to represent an image as a family of gradually smoothed images. This framework is very general and a variety of scale-space representations has been described in the literature. A typical approach for choosing a particular type of scale-space representation is to establish a set of scale-space axioms, describing basic properties of the desired scale-space representation and often chosen so as to make the representation useful in practical applications. Once established, the axioms narrow the possible scale-space representations to a smaller class, typically with only a few free parameters.A set of standard scale space axioms, discussed below, leads to the linear Gaussian scale-space, which is the most common type of scale space used in image processing and computer vision.
Scale space axioms for the linear scale-space representation
The linear
scale-space representation L(x, y, t) = (T_t f)(x, y) = g(x, y, t)*f(x, y) of signal f(x, y) obtained by smoothing with the Gaussian kernel g(x, y, t) satisfies a number of properties 'scale-space axioms' that make it a special form of multi-scale representation:
*"linearity":T_t(a f + b h) = a T_t f + b T_t hwhere f and h are signals while a and b are constants,
*"shift invariance':T_t S_{(Delta x, Delta_y)} f = S_{(Delta x, Delta_y)} T_t fwhere S_{(Delta x, Delta_y)} denotes the shift (translation) operator S_{(Delta x, Delta_y)} f)(x, y) = f(x-Delta x, y - Delta y)
*the "semi-group structure:g(x, y, t_1) * g(x, y, t_2) = g(x, y, t_1 + t_2)with the associated "cascade smoothing property":L(x, y, t_2) = g(x, y, t_2 - t_1) * L(x, y, t_1)
*existence of an "infinitesimal generator" A:partial_t L(x, y, t) = (A L)(x, y, t)
*"non-creation of local extrema" (zero-crossings) in one dimension,
*"non-enhancement of local extrema" in any number of dimensions:partial_t L(x, y, t) leq 0 at spatial maxima and partial_t L(x, y, t) geq 0 at spatial minima,
*"rotational symmetry":g(x, y, t) = h(x^2+y^2, t) for some function h,
*"scale invariance:hat{g}(omega_x, omega_y, t) = hat{h}(frac{omega_x}{varphi(t)}, frac{omega_x}{varphi(t)})for some functions varphi and hat{h} where hat{g} denotes the Fourier transform of g,
*"positivity"::g(x, y, t) geq 0 ,
*"normalization"::int_{x=-infty}^{infty} int_{y=-infty}^{infty} g(x, y, t) , dx , dy = 1 .The Gaussian kernel is also separable in Cartesian coordinates, i.e. g(x, y, t) = g(x, t) , g(y, t). Separability is, however, not counted as a scale-space axiom, since it is a coordinate dependent property related to issues of implementation. In addition, the requirement of separability in combination with rotational symmetry per se fixates the smoothing kernel to be a Gaussian.
In the
computer vision ,image processing andsignal processing literature there are many other multi-scale approaches, usingwavelets and a variety of other kernels, that do not exploit or require the same requirements asscale-space descriptions do; please see the article on relatedmulti-scale approaches . There has also been work on discrete scale-space concepts that carry the scale-space properties over to the discrete domain; see the article onscale-space implementation for examples and references.ee also
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scale space
*scale space implementation References
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