## Regularization, Scale-Space, and Edge Detection Filters (0)

Venue: | Journal of Mathematical Imaging and Vision |

Citations: | 39 - 8 self |

### BibTeX

@ARTICLE{Nielsen_regularization,scale-space,,

author = {Mads Nielsen and Luc Florack and Rachid Deriche},

title = {Regularization, Scale-Space, and Edge Detection Filters},

journal = {Journal of Mathematical Imaging and Vision},

year = {},

volume = {7},

pages = {291--308}

}

### Years of Citing Articles

### OpenURL

### 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. 1 Introduction Given a digital signal in one or more dimensions, we want to define its derivatives in a well-posed way. This can...

### Citations

3391 | A computational approach to edge detection
- Canny
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(Show Context)
Citation Context ...n must be differentiable. In edge detection many methods have been used for finding robust image derivatives. One method is to define a sample edge as well as criteria for an optimal detection. Canny =-=[7]-=- used a noisy step edge to find the optimal linear detection filter according to criteria of signal-to-noise-ratio, localization, and uniqueness of detection. Deriche [8] used these criteria on an inf... |

1395 | Scale-space and edge detection using anisotropic diffusion - Perona, Malik - 1990 |

697 |
The structure of images
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(Show Context)
Citation Context ...ition is linear. In the following, we describe these three methods, their motivation, and discuss their relations. Gaussian scale-space [2] can be motivated from different points of views. Koenderink =-=[3]-=- introduced it as a one parameter family of blurred images satisfying the causality criterion: every isophote in scale-space must be upwards convex. This requirement expresses in a precise sense that ... |

565 |
Scale-space filtering
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(Show Context)
Citation Context ...he method must be linear since differentiation by its very definition is linear. In the following, we describe these three methods, their motivation, and discuss their relations. Gaussian scale-space =-=[2]-=- can be motivated from different points of views. Koenderink [3] introduced it as a one parameter family of blurred images satisfying the causality criterion: every isophote in scale-space must be upw... |

503 | Scale-Space Theory in Computer Vision
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- 1994
(Show Context)
Citation Context ...m the grid details. The recursivity constraint ensures that a filtered signal smoothed to some degree (eventually by the digitizer) can be smoothed to a higher degree by a filter from the same family =-=[4,10]-=-. The recursivity constraint can be expressed as the semi-group constraint. Definition5. (Semi-group property) A semi-group G is a set of compositions:sG \Theta G ! G which is associative. Because reg... |

429 |
Théorie des Distributions
- Schwartz
- 1966
(Show Context)
Citation Context ...er, the Canny-Deriche filter arises. 1 Introduction Given a digital signal in one or more dimensions, we want to define its derivatives in a well-posed way. This can be done in a distributional sense =-=[1]-=- by convolving the signal with a smooth test function. Instead of taking the derivatives of the signal, we take the derivatives of the convolved signal by differentiation of the smooth filter prior to... |

244 | Using Canny’s criteria to derive a recursively implemented optimal edge detector. Int
- Deriche
- 1987
(Show Context)
Citation Context ...r an optimal detection. Canny [7] used a noisy step edge to find the optimal linear detection filter according to criteria of signal-to-noise-ratio, localization, and uniqueness of detection. Deriche =-=[8]-=- used these criteria on an infinite domain to find an optimal filter. In the following sections, regularization is reviewed and the relation to Gaussian scale-space is derived. Multi-dimensional regul... |

99 | Fast algorithm for low-level vision - Deriche - 1990 |

66 |
The syntactical structure of scalar images
- Florack
- 1993
(Show Context)
Citation Context ...e must be upwards convex. This requirement expresses in a precise sense that coarse scale details must have a cause at finer scales and essentially singles out the normalized Gaussian filter. Florack =-=[4]-=- defines a visual front-end to be linear, spatially isotropic, spatially homogeneous, separable and scale invariant, leading to the Gaussian as well. Since the Gaussian is smooth, all derivatives of t... |

66 |
From Images to Surfaces
- Grimson
- 1981
(Show Context)
Citation Context ...o the Gaussian filter. 3 Truncated smoothness operators To be consistent with scale-space, Tikhonov regularization must be of infinite order. Nevertheless, low-order regularization is often performed =-=[13]-=-. We might perceive of low-order regularization as a regularization using a truncated Taylorseries of the smoothness functional. Here we list the linear filters h z found by Fourier inversion. The not... |

66 | Biased anisotropic diffusion - A unified regularization and diffusion approach to edge detection - Nordström - 1990 |

37 |
A regularized solution to edge detection
- Poggio, Voorhees, et al.
- 1985
(Show Context)
Citation Context ...separable and scale invariant, leading to the Gaussian as well. Since the Gaussian is smooth, all derivatives of the images are well-defined in Gaussian scale-space. In Tikhonov regularization theory =-=[5,6]-=-, we search for a differentiable function, which in some precise sense is closest to our signal. The measure of difference between the solution and the signal is a functional of the solution. In this ... |

30 |
T.: An extended class of scale-invariant and recursive scale space filters
- Pauwels, Gool, et al.
- 1995
(Show Context)
Citation Context ...orms p. 2 The above filters were derived from the class of regularization filters, but constitute the full class of all analytic scale invariant semi-group filters as it was derived by Pauwels et al. =-=[11]-=-. Here the existence of an infinitesimal generator is the starting point. Combined with scale invariance the above class of filters arises. Since the semi-group constraint is implied by the existence ... |

26 |
The Analytical Theory of Heat
- Fourier
(Show Context)
Citation Context ...rable, and having derivatives up to order N all well-defined and square integrable. 2.1 Scale invariant regularization For an action to be scale invariant it must be dimensionless, i.e. without units =-=[9]-=-. Any physical law is in general scale invariant. In order for a filter to be physically meaningful it must thus be dimensionless. In the following we use dimensional analysis to insure scale invarian... |

12 |
Recursive Regularization Filters: Design, Properties and Applications
- Unser, Aldroubi, et al.
- 1991
(Show Context)
Citation Context ... the order of regularization. Proof By following the argumentation in the continuous case (and substituting the Fourier transform by the z-transform) or the discrete formulation given by Unser et al. =-=[17]-=-, we find that the minimization is implemented by convolution with the filter h(z) = 1 1 + P N i=1sisd i (z)sd i (z \Gamma1 ) where the hat indicates the z-transform. The transform of the difference o... |

9 |
Optimal filter for edge detection methods and results
- Castan, Zhao, et al.
- 1990
(Show Context)
Citation Context ...o exist when we consider only first order characteristics. In the spatial domain, we find the filter h 0 (x) = p ff x jxj e \Gammajxj= p ff This is the edge detection filter proposed by Castan et al. =-=[16]-=-. The derivative of this filter is not well-defined, and we see that we indeed have projected into a Sobolev space of first order. The above results can be generalized to optimal detection of higher o... |

5 |
Arseninn, "Solution of ill-posed problems
- Tikhonov, Y
- 1977
(Show Context)
Citation Context ...separable and scale invariant, leading to the Gaussian as well. Since the Gaussian is smooth, all derivatives of the images are well-defined in Gaussian scale-space. In Tikhonov regularization theory =-=[5,6]-=-, we search for a differentiable function, which in some precise sense is closest to our signal. The measure of difference between the solution and the signal is a functional of the solution. In this ... |

3 |
Regularization and scale-space
- Nielsen, Florack, et al.
- 1994
(Show Context)
Citation Context ... t) = h(s)sh(t) where the parameter-concatenation is the p-norm addition 3 s \Phi p t = (s p + t p ) 1=p 3 This is only the most general parameter concatenation when the filters must be dimensionless =-=[14]-=-. We call the filter-parameter the scale parameter, since it resembles the scale parameter in scale-space theory. Proposition7. (Scale invariant semi-group regularization) If a one-parameter regulariz... |

2 |
Ueber die vollen invariantsystemen
- Hilbert
(Show Context)
Citation Context ...e with a spatially varying scale. So far, we have discussed 1-dimensional regularization. In more dimensions mixed derivatives show up complicating the picture. However, imposing Cartesian invariance =-=[12]-=- (translation, rotation, and the scaling given implicit by the scaling nature of regularization) on the functional E leads to: Proposition8. (High-dimensional Cartesian invariant regularization) The m... |