Overview
Since the pioneering work by Witkin (1983) and Koenderink (1984) on the notion of “scale-space representation”, a large number of different scale-space formulations have been stated, based on different types of assumptions (usually referred to as scale-space axioms). The main subject of this chapter is to provide a synthesis between these linear scale-space formulations and to show how they are related. Another aim is to show how the scale-space formulations, which were originally expressed for continuous data on spatial domains without preferred directions, can be extended to discrete data as well as to spatio-temporal domains with preferred directions. Connections will also be pointed out to approaches based on non-uniform smoothing.
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References
See also chapter 1 by ter Haar Romeny (1996) and chapter 8 by Salden (1996) in this volume.
This derivation has been shortened substantially to save space. More detailed arguments showing how the assumption of scale invariance narrows down the class of smoothing kernels are presented in different forms in (Florack et al., 1992b; Pauwels et al., 1995; Lindeberg, 1994a ).
In a closely related work in chapter 7 in this volume, Nielsen (1996) arrives at filters of the same form from a slightly different starting point. He considers the problem of deriving optimal smoothing filters, and formulates an optimization problem in Euclidean norm. Thereby, the solution can be expressed as a linear filter, with the filter coefficients determined by a linear system of equations. Combined with shift-invariance, this gives rise to a convolution structure, and by requiring the filters to form a semi-group and to be associated with a scale parameter of dimension length raised to some power, it then follows that the filter must have a scale invariant Fourier transform which is additive under some self-similar reparametrization of the scale parameter. In other words, the Fourier transform must be of the form (6.37). In the spatial domain, this corresponds to regularization involving infinite orders of differentiation. If on the other hand, the regularization functional is truncated at lower orders of differentiation, then a larger class of regularization filters is obtained, including the recursive filters studied by (Deriche, 1987). It is interesting to note that these are also scale-space kernels in the sense that they are guaranteed to not increase the number of local extrema in a signal.
If further information is available about the image formation process, e.g.,such that the continuous signal can be reconstructed exactly from the sampled data, then the discrete signal can be treated as equivalent to the original continuous signal, and an equivalent discrete scale-space model be expressed for the continuous scale-space representation of the reconstructed continuous signal. Chapter 9 by Aström and Heyden (1996) in this volume exploits this idea based on the sampling theorem and the assumption of an ideally sampled band limited signal.
For the corresponding transformation kernels from the input signal f to a linear combination of scale-space derivatives, however, the semi-group structure is replaced by a cascade smoothing property. This means that any transformation kernel h(t) corresponds to the result of convolving some fixed kernel h0 with a Gaussian kernel, i.e., h(. t) = ho g(t) Hence, these kernels satisfy h(t)= g(s) h(t) where g denotes the Gaussian kernel.
Moreover, as shown in chapter 12 by Griffin (1996), the classification of what scale-space singularities (bifurcations) can occur with increasing scale, transfers from the rotationally symmetric Gaussian scale-space to the affine Gaussian scale-space (see (Koenderink and Doom, 1986a; Lindeberg, 1992; Lindeberg, 1994e) as well as chapter 11 by Damon (1996), chapter 10 by Johansen (1996) chapter 13 by Kalitzin (1996). Examples of image representations depending on this deep structure of scale-space can be found in (Lindeberg, 1993a) and in chapter 14 by Olsen (1996).
This essentially corresponds to the duality between transformations of image operators and image domains described in detail in chapter 5 by Florack (1996) in this volume.
Compared to the affine invariant level curve evolution scheme proposed by (Alvarez et al., 1993) and (Sapiro and Tannenbaum, 1993), given as a one-parameter solution to a non-linear differential equation (see equation (6.73) in section 6.5.2), an obvious disadvantage of the affine Gaussian scale-space is that it gives rise to a three-parameter variation. The advantage is that commutative properties can still be preserved within a family of linear transformations.
Note the difference in terminology in (Alvarez et al., 1993), where the semi-group structure is referred to as causality.
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© 1997 Springer Science+Business Media Dordrecht
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Lindeberg, T. (1997). On the Axiomatic Foundations of Linear Scale-Space. In: Sporring, J., Nielsen, M., Florack, L., Johansen, P. (eds) Gaussian Scale-Space Theory. Computational Imaging and Vision, vol 8. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8802-7_6
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