Regularization, scale-space, and edge detection filters

Abstract

Computational vision often needs to deal with derivatives of digital images. Such derivatives are not intrinsic properties of digital data; a paradigm is required to make them well-defined. Normally, a linear filtering is applied. This can be formulated in terms of scale-space, functional minimization, or edge detection filters. The main emphasis of this paper is to connect these theories in order to gain insight in their similarities and differences. We take regularization (or functional minimization) as a starting point, and show that it boils down to Gaussian scale-space if we require scale invariance and a semi-group constraint to be satisfied. This regularization implies the minimization of a functional containing terms up to infinite order of differentiation. If the functional is truncated at second order, the Canny-Deriche filter arises.

Morphological Counterparts of Linear Shift-Invariant Scale-Spaces

“…This coincides with the result from [15] stated in (44). Although this formula only holds for α > 1 2 , we can compute the pointwise limit…”

“…More recently, linear scale-spaces based on pseudodifferential operators have attracted attention, such as the Poisson scale-space [20], α-scale-spaces [19], summed α-scale-spaces [14], and relativistic scale-spaces [11]. Also regularisation methods and related concepts can be interpreted as scalespaces by considering their Euler-Lagrange equations, both in the linear and the nonlinear setting [50,44,53,10,13]. Since Gaussian scale-space can be described by a linear diffusion equation, it is natural to generalise it also to nonlinear diffusion scale-spaces [49,60].…”

Scale space methods for climate model analysis

“…There are many other possible applications of these techniques, since many types of analysis can benefit from considering individual scales separately. Scale space provides the basis for feature detection algorithms commonly used to detect blobs [Lindeberg, 1998;Lowe, 1999], edges [Nielsen et al, 1997;Perona and Malik, 1990], and corners [Zhang et al, 1995] in image data, and to track features through time in video applications. Additionally, the flexibility of the scale space framework allows for the construction of useful model diagnostics.…”

Grasping reveals visual misjudgements of shape

“…Calculating the tangential velocity involves taking a derivative. This was achieved by convolving the data with the derivative of a Gaussian kernel (see Nielsen et al 1997;Witkin 1983). The width of the kernel was three frames to either side, corresponding to a smoothing window of about 30 ms.…”