偏微分方程在图像处理中的应用：平滑、恢复与边缘检测

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"基于PDE的图像处理" 在图像处理领域，偏微分方程（PDE）的应用占据了重要地位，特别是在图像平滑、恢复以及边缘检测等方面。约翰·威克特，作为该领域的专家，深入研究了图像处理，包括分割、活动轮廓模型和光学流动等。本章主要关注PDE方法在图像平滑、恢复和边缘检测中的应用，并以非正式的方式进行阐述，同时引用了大量的原始文献，以供读者找到完整的证明和数学细节。图像平滑是PDE方法的一个关键应用，通过消除噪声并保持图像的重要特征来改善图像质量。平滑过程通常涉及扩散过程，其中PDE用于控制信息的传播，以区别于重要结构和噪声。非线性扩散方法在此过程中扮演重要角色，它能够更精确地保留边缘，避免过度平滑。图像恢复，又称为去噪，涉及到从失真或损坏的图像中重建原始图像。PDE可以用来模拟图像退化过程，并逆向操作以恢复图像。例如，著名的罗恩-查理斯(Richardson-Lucy)算法和菲涅尔(Fourier)域的恢复方法都是基于PDE的。这些方法考虑了图像退化模型，如模糊和加噪声，从而提供更准确的恢复结果。边缘检测是图像分析的关键步骤，用于识别和定位图像中的边界。PDE方法通过扩散和梯度下降策略来检测边缘，例如Canny边缘检测器就是一种基于PDE的方法。这些方法试图找到图像强度梯度的最大值，同时抑制噪声引起的假边缘。此外，PDE在其他图像处理任务中也有广泛的应用，如形状从阴影中恢复、主动轮廓模型（如蛇形模型）、光学流估计以及立体视觉等。这些模型通常涉及到能量最小化的问题，可以通过PDE来求解。PDE方法的优势在于它们能够处理复杂的图像结构，同时保持计算上的可行性。本章的目的在于为读者提供一个PDE方法的入门，对比各种方法之间的相似性、差异和问题。通过这种方式，读者不仅能理解这些方法的工作原理，还能看到作者的研究是如何与现有工作相联系的，以及如何解决后续章节中提出的一些挑战。对于每种方法，都简述了基本思想和理论背景，鼓励读者进一步探索这些方法的细节及其在实际问题中的应用。

CHAPTER 1. PARTIAL DIFFERENTIAL EQUATIONS

Frohlich and Weickert [92], and Benhamouda [28]. All results p oint in the same

direction: the solution dep ends strongly on the regularizing eect of the dis-

cretization. Finer discretizations are less regularizing and reveal a larger danger

staircasing eects

, where the number of inection p oints of a smo othed step

edge increases.

Scale-space

interpretation

Perona and Malik renounced the assumption of Ko enderink's linear scale-space

axiomatic [143] that the smo othing treats all spatial p oints and scale levels

equally. Instead of this, they required that region b oundaries should b e sharp

and should coincide with the semantically meaningful b oundaries at each reso-

lution level

(immediate localization)

and that intra-region smo othing should b e

preferred to inter-region smo othing

(piecewise smoothing)

. These prop erties are

of signicant practical interest, as they guarantee that structures can b e detected

easily and corresp ondence problems can b e neglected. Their experiments demon-

strated that the Perona{Malik lter satises these requirements fairly well [192].

In order to establish a smo othing scale-space prop erty for this nonlinear dif-

fusion pro cess, it was argued that the Perona{Malik lter satises a maximum{

minimum principle, provided that it p ossesses C

solutions [192]. However, this

scale-space reasoning is not very satisfactory since it is unlikely that the Perona{

Malik lter exhibits smo oth solutions.

As an alternative to construct nonlinear diusion scale-spaces, Salden [210],

Florack [88, 90] and Eb erly [79] prop ose to carry over the linear scale-space theory

to the nonlinear case by considering nonlinear diusion pro cesses which result

from sp ecial rescalings of the linear one.

Unfortunately, the Perona{Malik lter

turns out not to b elong to this class [88, 90].

Alvarez, Guichard, Lions and Morel [11] have developed a nonlinear scale-

space axiomatic which comprises the linear scale-space theory as well as nonlinear

morphological processes (which we will discuss in 1.4 and 1.5). Their smo othing

axiom is a monotony assumption

(comparison principle)

requiring that the scale-

space is order-preserving:



)





(1.31)

This prop erty is closely related to a maximum{minimum principle and to

stability of the solution [11, 155]. However, the Perona{Malik mo del do es not t

into this framework as well since its nonmonotonic ux function prevents it from

satisfying the comparison principle (see also [193]).

A comparison of the scale-space reasonings of Florack and Alvarez et al. can b e found in

[117

1.2 NONLINEAR DIFFUSION FILTERING

1.2.2 Regularized

nonlinear mo dels

In spite of the ab ovementioned theoretical problems, it has b een observed that

practical implementations of the Perona{Malik pro cess work frequently very well.

This suggests that the numerical schemes provide implicit regularizations which

stabilize the pro cess. Hence, it seems natural to introduce the regularization di-

rectly into the continuous equation in order to b ecome more indep endent of the

numerical implementation [58, 181].

Hollig's nonuniqueness result [121] shows that the dynamics of the solution

may critically dep end on the sort of regularization. Hence, the regularization

should b e sp ecically adjusted to the desired goal of the forward{backward heat

equation [27].

One can apply spatial or temp oral regularization (and of course, a combination

of b oth). Below we shall discuss three examples which illustrate the variety of

p ossibilities and their tailoring towards a sp ecic task.

(a) The rst spatial regularization attempt is probably due to Posmentier who

observed numerically the stabilizing eect of averaging the gradient within

the diusivity [196].

A mathematically sound formulation of this idea is given by Catt

e, Lions,

Morel and Coll [58]. By replacing the diusivity

(

) of the Perona{

Malik mo del by

(



) with





they end up with

= div (

(



)

(1.32)

In [58] existence, uniqueness and regularity of a solution for

 >

0 has

b een established.

Other spatial regularizations of equations of Perona{Malik typ e have b een

prop osed by Weickert [248, 250] and will b e describ ed in 1.2.3. Whitaker

and Pizer [255] have suggested that the regularization parameter



should

b e a decreasing function in

. In addition, Li and Chen [148] have prop osed

to subsequently decrease the contrast parameter



We shall see that these spatial regularizations b elong to a class of well-

p osed problems which p ossess a large class of Lyapunov functionals which

guarantee that the solution converges to a constant steady-state.

From a practical p oint of view, spatial regularizations oer the advantage

that they make the lter insensitive to noise at scales smaller than



. There-

fore, when regarding (1.32) as an image restoration equation, it exhibits b e-

sides the contrast parameter



an additional

scale-parameter



. This avoids

a shortcoming of the genuine Perona{Malik pro cess which misinterprets

strong oscillations due to noise as edges which should b e preserved or even

enhanced.

CHAPTER 1. PARTIAL DIFFERENTIAL EQUATIONS

(b) P.-L. Lions proved in a private communication to Mumford that the one-

dimensional pro cess

(

)

;

(1.33)



(

) (1.34)

leads to a well-p osed lter (cf. [193]). We observe that

is intended as a

time-delay regularization of

where the parameter



0 determines

the delay. These equations arise as a sp ecial case of the spatio-temp oral

regularizations of Nitzb erg and Shiota [181], when neglecting any spatial

regularization. Mumford conjectures that this mo del gives piecewise con-

stant steady states [180]. In this case, the steady-state solution would solve

a segmentation problem.

dimensional forward{backward heat equation by considering the third-order

equation

((

)) +

 @

((

)) (1.35)

where  is strictly increasing and uniformly b ounded in IR, and



(

)

(

(

)) as

! 1

. This regularization was physically motivated by

intro ducing a relaxation time



into the diusivity.

For the corresp onding initial b oundary value problem with homogeneous

Neumann b oundary conditions they proved the existence of a unique gen-

eralized solution. They also showed that smo oth solutions may b ecome

discontinuous within

nite

time, b efore they nally converge to a piecewise

constant steady state.

1.2.3 Anisotropic

nonlinear mo dels

All nonlinear diusion lters that we have investigated so far utilize a scalar-

valued diusivity

which is adapted to the underlying image structure. Therefore,

they are isotropic and the ux

is always parallel to

. Nevertheless,

in certain applications it would b e desirable to rotate the ux towards the orien-

tation of interesting features. These requirements cannot b e satised by a scalar

diusivity anymore, a diusion tensor leading to anisotropic diusion lters has

to b e introduced.

First anisotropic ideas in image pro cessing date back to Graham [109] in

1962, followed by Newman and Dirilten [176], Lev, Zucker and Rosenfeld [146],

and Nagao and Matsuyama [174]. They used convolution masks that dep ended

on the underlying image structure. Related statistical approaches were prop osed

by Knutsson, Wilson and Granlund [141, 262].

1.2 NONLINEAR DIFFUSION FILTERING

Convolution with shap e-adapted anisotropic Gaussian kernels is studied by

Nitzb erg and Shiota [181], Lindeb erg and G

arding ([152], cf. also [153]), and

Yang et al. [266]. Unlike in the linear case, this is no more equivalent to suitable

diusion pro cesses. If one wants to relate these ideas to diusion ltering, one has

to iterate the convolution and carry out sophisticated scaling limits, cf. [181].

Anisotropic diusion lters usually apply spatial regularization strategies (an

exception is Cottet's time-delay regularization [67]). Below we study two typical

representatives of anisotropic diusion pro cesses. The rst one oers advantages

at noisy edges, whereas the second one is well-adapted to the pro cessing of one-

dimensional features.

(a)

nisotropic regularization of the Perona{Malik process

In the interior of a segment the nonlinear isotropic diusion equation (1.32)

b ehaves almost like the linear diusion lter (1.9), but at edges diusion

is inhibited. Therefore, noise at edges cannot b e eliminated successfully by

this pro cess.

To overcome this problem, a desirable metho d should prefer diusion along

edges to diusion p erp endicular to them.

Anisotropic mo dels do not only take into account the mo dulus of the edge

detector



, but also its direction. To this end, we construct the orthonor-

mal system of eigenvectors

of the diusion tensor

such that they

reect the estimated edge structure:

k r



; v



(1.36)

In order to prefer smo othing along the edge to smo othing across it, Weickert

[247, 250] prop osed to cho ose the corresp onding eigenvalues



and



(



) :=

(



)

;

(1.37)



(



) := 1

(1.38)

In general,

do es not coincide with one of the eigenvectors of

as long

 >

Hence, this mo del b ehaves really anisotropic. If we let the

regularization parameter



tend to 0, we end up with the isotropic Perona{

Malik pro cess.

Another anisotropic mo del which can b e regarded as a regularization of an

isotropic nonlinear diusion lter has b een describ ed in [248].

(b)

Anisotropic models for smoothing one-dimensional obje

cts

A second motivation for intro ducing anisotropy into diusion pro cesses

arises from the wish to pro cess one-dimensional features such as line-like

CHAPTER 1. PARTIAL DIFFERENTIAL EQUATIONS

structures. To this end, Cottet and Germain [66] constructed a diusion

tensor with eigenvectors as in (1.36) and corresp onding eigenvalues



(



) := 0

;

(1.39)



(



) :=





1 + (



=

)

(

 >

(1.40)

This is a pro cess diusing solely p erp endicular to



. For



0, we ob-

serve that

b ecomes an eigenvector of

with correp onding eigenvalue 0.

Therefore, the pro cess stops completely. In this sense, the Cottet{Germain

mo del is not intended as an anisotropic regularization of the Perona{Malik

equation.

In [66] the existence of weak solutions for this pro cess was proved.

Since the Cottet{Germain mo del diuses only in one direction, its success

critically dep ends on a correct estimate of the appropriate direction. We

shall see that this pro cess can b e improved when replacing



by a more

robust descriptor of lo cal orientation, the structure tensor.

1.2.4 Generalizations

Higher dimensions.

It is easily seen that many of the previous results

can b e generalized to higher dimensions. This may b e useful when considering

e.g. computerized tomography (CT) or magnetic resonance tomography (MRT)

image sequences arising from medical applications or when applying diusion

lters to the p ostpro cessing of uctuating higher-dimensional numerical data.

Three-dimensional diusion lters have been investigated by Gerig et al. [99] and

Rambaux and Garcon [202].

sophisticated structure descriptors.

The edge detector



en-

ables us to adapt the diusion to magnitude and direction of edges, but it cannot

distinguish b etween edges and corners or gives a reliable measure of lo cal ori-

entation. To this end, one can steer the smo othing pro cess by more advanced

structure descriptors such as higher-order derivatives [82] or tensor-valued ex-

pressions of rst-order derivatives [181, 179, 252]. The theoretical analysis in the

present work shall comprise the second p ossibility.

ector-valued mo dels.

Vector-valued images can arise either from devices

measuring multiple physical prop erties or from a feature analysis of one single

image. Examples for the rst category are colour images, multi-spectral Landsat

exp osures and multi-spin echo MRT images, whereas representatives of the second

class are given by statistical moments or the jet space induced by the image itself

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