## Scale-Space Properties of Nonlinear Diffusion Filtering with a Diffusion Tensor (1994)

Venue: | Laboratory of Technomathematics, University of Kaiserslautern, P.O |

Citations: | 19 - 2 self |

### BibTeX

@TECHREPORT{Weickert94scale-spaceproperties,

author = {Joachim Weickert},

title = {Scale-Space Properties of Nonlinear Diffusion Filtering with a Diffusion Tensor},

institution = {Laboratory of Technomathematics, University of Kaiserslautern, P.O},

year = {1994}

}

### OpenURL

### Abstract

In spite of its lack of theoretical justification, nonlinear diffusion filtering has become a powerful image enhancement tool in recent years. The goal of the present paper is to provide a mathematical foundation for continuous nonlinear diffusion filtering as a scale-space transformation which is flexible enough to simplify images without loosing the capability of enhancing edges. By studying the Lyapunov functionals, it is shown that nonlinear diffusion reduces L p norms and central moments and increases the entropy of images. The proposed anisotropic class utilizes a diffusion tensor which may be adapted to the image structure. It permits existence, uniqueness and regularity results, the solution depends continuously on the initial image, and it satisfies an extremum principle. All considerations include linear and certain nonlinear isotropic models and apply to m- dimensional vector-valued images. The results are juxtaposed to linear and morphological scale-spaces. . Keywords....

### Citations

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Citation Context ...lic field is governed by mean curvature methods [2] and diffusion techniques, on which we will focus in the sequel. The simplest diffusion filtering eliminates noise by a linear and isotropic process =-=[30, 53, 26]-=-. Let the original image be represented by a real-valued function f 2 L 1 (IR m ). Then, one takes f as initial condition for the diffusion equation @ t u = \Deltau (1) and interprets the solution u(x... |

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Citation Context ...lic field is governed by mean curvature methods [2] and diffusion techniques, on which we will focus in the sequel. The simplest diffusion filtering eliminates noise by a linear and isotropic process =-=[30, 53, 26]-=-. Let the original image be represented by a real-valued function f 2 L 1 (IR m ). Then, one takes f as initial condition for the diffusion equation @ t u = \Deltau (1) and interprets the solution u(x... |

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Citation Context ...t (x \Gamma y) f(y) dy: (3) An obvious disadvantage of this proceeding is the fact that it does not only smooth noise, but also blurs important features such as edges. To avoid this, Perona and Malik =-=[35]-=- suggest filtering by nonlinear diffusion. They apply a nonuniform process (which they name anisotropic) that reduces the diffusivity at those locations having a larger likelihood to be edges. This li... |

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Citation Context ...the mean curvature flow (6), which can be written as @ t u = uswithsas above. Thus, one may construct a filter which diffuses linearly inside regions and smoothes by mean curvature motion along edges =-=[2]-=-. This approach also provides a contrast and a scale parameter, but it is not clear whether it permits edge enhancement. Anyway, 22 due to the linear diffusion, morphological grey scale invariance get... |

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Citation Context ...e have already seen that this is equivalent to linear diffusion filtering. Numerous theoretical results indicate that this diffusion filter is the only "reasonable" way to define a linear sc=-=ale-space [26, 54, 5, 28, 19, 1]-=-. Nevertheless, linear diffusion filtering does not only blur edges, but also dislocates them when moving from finer to coarser scales. So, edges which are identified at a coarse 3 scale do not give t... |

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Citation Context ...e have already seen that this is equivalent to linear diffusion filtering. Numerous theoretical results indicate that this diffusion filter is the only "reasonable" way to define a linear sc=-=ale-space [26, 54, 5, 28, 19, 1]-=-. Nevertheless, linear diffusion filtering does not only blur edges, but also dislocates them when moving from finer to coarser scales. So, edges which are identified at a coarse 3 scale do not give t... |

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Citation Context ...lty of finding an appropriate stopping time for the diffusion process and to give a theoretical justification for nonlinear diffusion filtering, several authors introduced an additional reaction term =-=[32, 16, 13, 43, 14]-=-. This aims to allow nontrivial steady states. Diffusion filtering can then be regarded as a steepest descent method for some energy functional. (This idea may also be extended to study reaction-diffu... |

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Citation Context ...s not imply that the number of local extrema is nonincreasing. It is not very difficult to give illustrative counterexamples where this is yet violated for the linear diffusion case in two dimensions =-=[27, 28]-=-. 3.3.2 The maximum--minimum principle From the nonenhancement of local extrema, it can be shown by using a reasoning of Illner [21] that for t 2st 1 ? 0 we get min y2 \Omega u(y; t 1 )su(x; t 2 )smax... |

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Citation Context ...ing results, the Perona--Malik idea has triggered numerous modifications. By means of spatial or temporal regularization, nonlinear diffusion filters were proposed which are more robust against noise =-=[11, 31, 13, 14, 50, 47, 48, 49]-=-. The isotropic nonlinear filter of Catt'e, Lions, Morel and Coll [11] is a well-investigated representative. These authors replace the diffusivity g(jruj) of the Perona--Malik model by g(jru oe j) wi... |

66 |
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Citation Context ...ecent decade. Elliptic methods are 1 frequently related to variational problems via the Euler equations (see e.g. [32, 44]), while shock filters are important representatives from the hyperbolic area =-=[34, 3]-=-. The parabolic field is governed by mean curvature methods [2] and diffusion techniques, on which we will focus in the sequel. The simplest diffusion filtering eliminates noise by a linear and isotro... |

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Citation Context ...rona--Malik filter possesses solutions which fulfil the smoothness requirements of the extremum principle. As a promising alternative to construct nonlinear diffusion scale-spaces, Salden and Florack =-=[40, 15]-=- propose to carry over the linear scale-space theory to the nonlinear case by considering nonlinear diffusion processes which result from special rescalings of the linear one. Unfortunately, the Peron... |

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Citation Context ...ver, when renouncing the linearity requirement, other possibilities appear: Morphological equations represent an important class of nonlinear scale-spaces. They include the dilation/erosion equations =-=[12, 9, 1, 4]-=- @ t u = \Sigmajruj; the mean curvature motion (curve shortening flow, Euclidean geometric heat flow) [2, 1, 24] @ t u = jruj div ` ru jruj ' ; (6) the affine invariant equation [1, 41] @ t u = jruj `... |

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Citation Context ...ing results, the Perona--Malik idea has triggered numerous modifications. By means of spatial or temporal regularization, nonlinear diffusion filters were proposed which are more robust against noise =-=[11, 31, 13, 14, 50, 47, 48, 49]-=-. The isotropic nonlinear filter of Catt'e, Lions, Morel and Coll [11] is a well-investigated representative. These authors replace the diffusivity g(jruj) of the Perona--Malik model by g(jru oe j) wi... |

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Citation Context ...uctures at finer scales. This property is specified by numerous authors in different ways, using concepts like causality [26], nonenhancement of local extrema [28], no creation of new level-crossings =-=[20, 28]-=-, comparison principle [1], maximum--minimum principle [35], decay of Euclidean absolute curvature [41], and decay of the number of extrema and zero-crossings of the curvature [41]. It is desiable tha... |

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Citation Context ...Since this would forbid contrast enhancing processes, we will not pursue this idea any further. In order to prove a maximum--minimum principle, we utilize Stampacchia's truncation method instead (cf. =-=[8]-=-, p. 211). Theorem 3 (Extremum principle). Consider the problem (P) and define a := ess inf x2\Omega f(x); (22) b := ess sup x2\Omega f(x): (23) Then the solution u of (P) verifies asu(x; t)sb on \Ome... |

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Citation Context ...portant class of nonlinear scale-spaces. They include the dilation/erosion equations [12, 9, 1, 4] @ t u = \Sigmajruj; the mean curvature motion (curve shortening flow, Euclidean geometric heat flow) =-=[2, 1, 24] @ t -=-u = jruj div ` ru jruj ' ; (6) the affine invariant equation [1, 41] @ t u = jruj ` div ` ru jruj " 1 3 (7) and combinations of these ideas [1, 25]. An exhaustive axiomatic treatment of these equ... |

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Citation Context ... do not give the correct location and must be traced back. A modification of the affine invariant flow avoiding the shrinkage problem for the evolution of a curve is proposed by Sapiro and Tannenbaum =-=[42]-=-. Nevertheless, for grey-scale images, the practical use of this modification is yet under research. Since the fundamental equation of shape analysis involves no additional parameters and offers numer... |

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Citation Context ...g it with a Gaussian. The resulting matrix J ae (ru oe ) := K aes(ru oe\Omega ru oe ) (ae ? 0) is called moment tensor, scatter matrix or structure tensor. It is a useful tool for texture description =-=[39]-=- and local structure analysis of spatio--temporal image sequences [23]. As an application to image restoration, Nitzberg and Shiota [31] used the quadratic form induced by the structure tensor to dete... |

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Citation Context ...ing results, the Perona--Malik idea has triggered numerous modifications. By means of spatial or temporal regularization, nonlinear diffusion filters were proposed which are more robust against noise =-=[11, 31, 13, 14, 50, 47, 48, 49]-=-. The isotropic nonlinear filter of Catt'e, Lions, Morel and Coll [11] is a well-investigated representative. These authors replace the diffusivity g(jruj) of the Perona--Malik model by g(jru oe j) wi... |

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Citation Context ...ver, when renouncing the linearity requirement, other possibilities appear: Morphological equations represent an important class of nonlinear scale-spaces. They include the dilation/erosion equations =-=[12, 9, 1, 4]-=- @ t u = \Sigmajruj; the mean curvature motion (curve shortening flow, Euclidean geometric heat flow) [2, 1, 24] @ t u = jruj div ` ru jruj ' ; (6) the affine invariant equation [1, 41] @ t u = jruj `... |

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Citation Context ...o allow nontrivial steady states. Diffusion filtering can then be regarded as a steepest descent method for some energy functional. (This idea may also be extended to study reaction-diffusion systems =-=[45, 46, 36, 14]-=-.) An example for a diffusion filter with a reaction term is the Nordstrom model [32] @ t u = div (g(jruj) ru) + ff(f \Gamma u) (ff ? 0): However, this alteration just shifts the problem of finding a ... |

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Citation Context ...e have already seen that this is equivalent to linear diffusion filtering. Numerous theoretical results indicate that this diffusion filter is the only "reasonable" way to define a linear sc=-=ale-space [26, 54, 5, 28, 19, 1]-=-. Nevertheless, linear diffusion filtering does not only blur edges, but also dislocates them when moving from finer to coarser scales. So, edges which are identified at a coarse 3 scale do not give t... |

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Citation Context ...nce [1]. Therefore, it is sometimes named fundamental equation of shape analysis [29]. The morpholgical equations (6) and (7) can also be classified in a unique way by means of group invariances, see =-=[33]-=- and the references therein. Morphological transformations possess the property that the filtering result depends only on the level lines of the image and, therefore, they are invariant under any nond... |

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Citation Context ... faster decay of g near . They lead to steeper edges with a longer "lifetime", while smaller values diminish staircasing at edges. A detailed investigation of this family of diffusivities is=-= given in [6]-=-. All figures in the present paper were obtained with ff := 5, in order to demonstrate that the choice of ff is not critical and that this is not a crucial parameter. In practise, anisotropic diffusio... |

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Citation Context ...ma Conversely, if u(t) = Mf (a.e.) on\Omega holds for all t 2 [0; T ], it is evident that V (0) = V (T ). (b) To proof the behaviour of u(t) for t ! 1, we can adapt the methods of Illner and Neunzert =-=[22]-=- for directed linear diffusion to our (undirected) nonlinear case. (i) Choosing r(s) = s 2 and differentiating the corresponding Lyapunov function V (t) gives V 0 (t) = \Gamma2 Z \Omega hru; D(ru oe )... |

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Citation Context ... (7) is the unique equation within this axiomatic framework fulfilling the additional requirement of projection invariance [1]. Therefore, it is sometimes named fundamental equation of shape analysis =-=[29]-=-. The morpholgical equations (6) and (7) can also be classified in a unique way by means of group invariances, see [33] and the references therein. Morphological transformations possess the property t... |

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Citation Context ...rtening flow, Euclidean geometric heat flow) [2, 1, 24] @ t u = jruj div ` ru jruj ' ; (6) the affine invariant equation [1, 41] @ t u = jruj ` div ` ru jruj " 1 3 (7) and combinations of these i=-=deas [1, 25]-=-. An exhaustive axiomatic treatment of these equations may be found in [1]. It can be shown that for m = 2, the affine invariant equation (7) is the unique equation within this axiomatic framework ful... |

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Citation Context ...sor may be constructed from an isotropic model, let us consider a Lipschitz continuous scalar diffusivity function g 2 C 1 ([0; 1); (0; 1]) which is represented in [0; 1) by a convergent power series =-=[48, 38]-=-: g(s) = 1 X k=0 a k s k : Since the tensor product J 0 (ru oe ) := ru oe\Omega ru oe (16) gives a positive semidefinite matrix there exists a unique positive semidefinite matrix p J 0 with p J 0 \Del... |

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Citation Context ...f techniques based on partial differential equations (PDEs) has been proposed in the recent decade. Elliptic methods are 1 frequently related to variational problems via the Euler equations (see e.g. =-=[32, 44]-=-), while shock filters are important representatives from the hyperbolic area [34, 3]. The parabolic field is governed by mean curvature methods [2] and diffusion techniques, on which we will focus in... |

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2 |
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Citation Context ...o a given order. Feature vectors play an important role for tasks like texture segmentation. Isotropic vector-valued diffusion models were proposed by Gerig, Kubler, Kikinis, Jolesz [17] and Whitaker =-=[51, 52]-=- in the context of medical imagery. Our results extend their work to the anisotropic case by taking into account the direction of the structures. The simplest idea how to apply diffusion filtering to ... |

1 |
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Citation Context ...is yet violated for the linear diffusion case in two dimensions [27, 28]. 3.3.2 The maximum--minimum principle From the nonenhancement of local extrema, it can be shown by using a reasoning of Illner =-=[21]-=- that for t 2st 1 ? 0 we get min y2 \Omega u(y; t 1 )su(x; t 2 )smax y2 \Omega u(y; t 1 ) 8 x 2 \Omega : Instead of doing this in the sequel, we are interested a more general result which does also ap... |

1 |
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Citation Context ...e\Omega ru oe ) (ae ? 0) is called moment tensor, scatter matrix or structure tensor. It is a useful tool for texture description [39] and local structure analysis of spatio--temporal image sequences =-=[23]-=-. As an application to image restoration, Nitzberg and Shiota [31] used the quadratic form induced by the structure tensor to determine the shape of their anisotropic Gaussian kernel. What makes the s... |

1 |
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Citation Context ...f techniques based on partial differential equations (PDEs) has been proposed in the recent decade. Elliptic methods are 1 frequently related to variational problems via the Euler equations (see e.g. =-=[32, 44]-=-), while shock filters are important representatives from the hyperbolic area [34, 3]. The parabolic field is governed by mean curvature methods [2] and diffusion techniques, on which we will focus in... |

1 |
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Citation Context ...rona--Malik filter possesses solutions which fulfil the smoothness requirements of the extremum principle. As a promising alternative to construct nonlinear diffusion scale-spaces, Salden and Florack =-=[40, 15]-=- propose to carry over the linear scale-space theory to the nonlinear case by considering nonlinear diffusion processes which result from special rescalings of the linear one. Unfortunately, the Peron... |

1 |
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Citation Context ...quations [12, 9, 1, 4] @ t u = \Sigmajruj; the mean curvature motion (curve shortening flow, Euclidean geometric heat flow) [2, 1, 24] @ t u = jruj div ` ru jruj ' ; (6) the affine invariant equation =-=[1, 41] @ t -=-u = jruj ` div ` ru jruj " 1 3 (7) and combinations of these ideas [1, 25]. An exhaustive axiomatic treatment of these equations may be found in [1]. It can be shown that for m = 2, the affine in... |

1 |
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Citation Context ...lty of finding an appropriate stopping time for the diffusion process and to give a theoretical justification for nonlinear diffusion filtering, several authors introduced an additional reaction term =-=[32, 16, 13, 43, 14]-=-. This aims to allow nontrivial steady states. Diffusion filtering can then be regarded as a steepest descent method for some energy functional. (This idea may also be extended to study reaction-diffu... |

1 |
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Citation Context ...o a given order. Feature vectors play an important role for tasks like texture segmentation. Isotropic vector-valued diffusion models were proposed by Gerig, Kubler, Kikinis, Jolesz [17] and Whitaker =-=[51, 52]-=- in the context of medical imagery. Our results extend their work to the anisotropic case by taking into account the direction of the structures. The simplest idea how to apply diffusion filtering to ... |