图像分析与非线性平滑：中值滤波器与对比不变算子

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"gbmtut_jmorel - 偏微分图像处理的经典教材，专注于数字图像处理，由Frédéric Guichard和Jean-Michel Morel撰写，主要探讨非线性迭代平滑方法和对比度不变操作在图像分析中的应用。" 在图像处理领域，本书"Image Analysis and P.D.E.'s"由Frédéric Guichard和Jean-Michel Morel合著，是一本重要的参考资料，特别关注于偏微分方程（PDE）在图像分析中的应用。书中的核心观点是，通过适当的尺度迭代卷积，线性正定核可以收敛到高斯函数，这一过程与热方程的应用相类似。这在信号或图像的线性平滑方法中表现得尤为明显。书中进一步扩展了这一分析，将焦点转向非线性迭代平滑方法，特别是那些对比度不变的运算符。这些单调运算符具有一个关键特性，即它们对图像的对比度变化保持不变，因此非常适合用于图像分析。例如，中值滤波器就是这样的一个运算符，它能计算出与对比度变化无关的局部“平均值”。作者证明了所有单调且对比度不变的运算符在渐近行为上都有相似的性质。这一发现对于理解非线性平滑如何影响图像的结构和细节至关重要。通过这些运算符，图像的复杂特征可以被逐步提取，同时保持其基本结构不受对比度调整的影响。这本书深入探讨了这些概念，并可能涵盖以下几个关键知识点： 1. 迭代平滑：通过连续应用特定的非线性运算符来平滑图像，这种方法与线性滤波不同，因为它能够更好地保留图像边缘。 2. 对比度不变：运算符在改变图像的全局对比度时保持其效果不变，这对于处理光照变化、噪声或增强图像细节非常重要。 3. 单调性：运算符的单调性确保了平滑过程不会引入新的特征或丢失原有特征，这在保持图像结构完整性方面具有重要意义。 4. 中值滤波器：作为对比度不变运算符的一个实例，中值滤波器通过替换像素值为其邻域内的中值，有效地去除噪声，同时保护图像边缘。 5. 偏微分方程在图像处理中的应用：利用PDEs进行图像平滑和分析，如热方程的使用，可以模拟物理过程，如扩散或退火，以优化图像处理效果。 6. 渐近行为分析：研究运算符在长时间或大量迭代后的表现，了解其对图像特征的长期影响。通过学习这本书，读者将能够深入理解非线性图像平滑的理论基础，并掌握如何使用这些方法来解决实际的图像处理问题，包括降噪、边缘保护、特征提取等。这对于计算机视觉、图像处理和机器学习领域的研究人员和工程师来说是一份宝贵的资源。

CHAPTER 1. INTRODUCTION

Figure 1.6:

Gabor's deblurring. Gab or prop osed in 1960 to deblur an image by substracting its laplacian : this means

inverting the heat equation ! Left : original image, middle : three iterations of Gab or's algorithm, right : ten iterations. As

is well known, and can b e observed in the right image, the inverse heat equation blows up. A few iterations can, as we see

in the middle, nicely enhance the image.

Figure 1.7:

Gabor's deblurring again. Same deblurring exp eriment as in Figure 1.6, but applied on a much more blurred

image

iterated linear and nonlinear smoothing (that is, nonlinear PDE's) will be relevant to our main goal :

image analysis.

We can indeed improvethetime-reverse heat equation. The rst example, due to Rudin and Osher

in 87 and 92 prop oses an pseudoinverse, where the propagation term |Du| is tuned by the sign of the

laplacian.



sig n



The equation is called \sho ck lter". As we shall see, this equation propagates, with a constant speed, the

level lines of the image in the same direction as the reverse heat equation would do. It therefore enhances

the image. The equation is more or less equivalent to a good old nonlinear lter due to Kramer in the

seventies. Kramer's lter can be interpreted as a partial dierential equation, by the same kind of heuristic

arguments which Gab or developed to derive the heat equation. This equation is



sig nD

(

Du; Du

)

Thus, the laplacian is replaced by a directional second derivativeofthe image,

(

Du; Du

). We shall

later on interpret this dierential op erator as an \edge detector". Kramer's equation yields a slightly

better version of shock lter as is illustrated in Figure 1.8. Both deblurring equations work... to some

extent. They experimentally do not blow up and attain steady states ! The third deblurring metho d we

1.1. THE WHEN, WHY AND HOW OF P.D.E. MODELS IN IMAGE PROCESSING

Figure 1.8:

Image deblurring by sho ck lters and byavariational metho d. From left to right : blurred image, Rudin-Osher

shock lter which is a pseudoinverse of the heat equation attaining a steady state, Kramer's improved sho ck lter, also

attaining a steady state and the Rudin, Osher, Fatemi restoration metho d, obtained by deblurring with a controlled image

total variation. This last metho d is very eÆcient when the noise and blur models are known. It is currently b eing used by

the French Space Agency (CNES) for satellite image restoration (Chapter 16).

can mention here is, to our knowledge, the best version. It poses the deblurring problem as an inverse

problem. Given the observed image

,we try to nd a restored version

such that



is as close as

possible to

and the oscillation of

is nontheless b ounded :

restor ed

Arg min

(



(





)

The parameter



tunes the oscillation weallow for the restored version. If



is large, the restored version

will satisfy accurately the equation



, but maybevery oscillatory. If instead



is small, we get a

smooth but unaccurate solution. This parameter can b e computed in principle as a Lagrange multiplier.

The obtained restoration can be remarkable. We display the best result we can obtain with the blurred

Victor in Figure 1.8-right. This total variation restoration metho d also has fast wavelet packets versions.

It recently won a b enchmark in satellite image deblurring organised by the French Space Agency (CNES).

The original remark of Gabor, ab out image generation being related to the Laplacian of the image,

leads to the wavelet theory as well. Here is howitworks : if weconvolve the image with some smo othing

kernel and thereafter makethe dierence, we obtain a new image, actually a laplacian, which turns out

to be faded with resp ect to the original. In Figure 1.9, the last image on the right shows in black the

values of this laplacian image of Victor which dier signicantly from zero : In most natural images,

as here, this representation is sparse and adapted to compression. This is why one of the rst wavelet

representations, due to Burt and Adelson in 83, was called \Laplacian pyramid". It boils down to the

iteration of a convolution followed by subsampling. Weonlykeep the dierences between images smo othed

at dierent scales, i.e. their laplacians. The ob jective is a compressed representation, but to the price of

alossof invariance due to the multiscale subsampling.

In image analysis, the heat equation has had a very dierent use: Marr, Hildreth, Canny, Witkin,

Koenderink prop osed in the eighties to analyse an image by applying the heat equation. As Rudin and

Florack noticed, this is related to distribution theory. Indeed, details of the images, like b oundaries, corners

and other singularities cannot be computed without some previous smoothing because they are derivatives

of a nonsmooth function. And this smo othing has to be multiscale because the image is multiscale !

The heat equation is easily proved to be the only good candidate to the task if image analysis has to

be linear. What derivatives should be computed in an image ? The early research in computer vision

F. Guichard, J-M. Morel,

Image Analysis and PDE's 15

CHAPTER 1. INTRODUCTION

Figure 1.9:

The \laplacian pyramid" of Burt and Adelson. From left to right : original image, image blurred by gaussian

convolution, then dierence b etween the original image and the blurred version, whichsimulates the laplacian of the original

image. In black in the last image, p oints where this laplacian image is large. This exp erimentsimulates the rst step of

the laplacian pyramid. The laplacian image is, for most digital images, a sparse representation, therefore well adapted to

compression.

proposed \edge detection" as a main to ol : it is assumed that the apparentcontours of the ob jects and

also the b oundaries of the facets of ob jects, result in step discontinuities in the image, while, inside those

boundaries, the image oscillates mildly. The apparentcontour p oints, or \edges p oints" will b e computed

as points where the gradientisissomesenselargest. Twoways to do so : Hildreth and Marr proposed the

points where 

crosses zero. A signicantimprovementwas done byCanny, who proposed to compute

the points where

is maximal on the gradient lines. Such points satisfy

(

Du; Du

) = 0. Figure

1.10 displays what happ ens when we smooth the image with the heat equation and compute the p oints

where

(

Du; Du

)=0and

is large enough. At rst, everything in the image is boundary : the

image, being a very oscillatory function, has inexion points everywhere ! After some evolution of the heat

equation, we can see what happ ens : we are able to extract some structure.

Figure 1.10:

Heat equation and Canny's edge detector. Boundaries, or \edges" of the image can b e dened as p oints where

the gradient attains a maximal and large value along the gradient lines. This amounts to say that edge points are p oints

where

(

Du; Du

) crosses zero and

is large. Canny's edge detector computes those points. On the left, the original

image, followed by the edge points found. They makeavery dense set, b ecause of the oscillatory character of the image.

Next, the image blurred by the Gauss kernel (heat equation) and the Canny edges found. The heat equation has removed

the \irrelevant" edges (Chapter 3)

If the heat equation is, under sound invariance requirements, the only go o d linear smo other, there are

instead many nonlinear ways to smo oth an image. The rst one was prop osed byPerona and Malik in

87. The idea is roughly to smo oth out what has to b e smo othed, the irrelevant, homogeneous, regions and

to enhance instead the b oundaries. Thus, the diusion should look likethe heat equation when

small and an inverse heat equation should instead b e applied when

is large. Here is the equation in

1.1. THE WHEN, WHY AND HOW OF P.D.E. MODELS IN IMAGE PROCESSING

divergence form.

div

(

))

;

where

(

s

decreases when

increases. It is easily checked that wehave a diusion equation when

j



and an inverse diusion equation when

>

. In order to do so, we rewrite the equation in

the following way. We consider the second derivativeof

in the direction of



(

;

)

and the second derivative in the orthogonal direction,



(

;

)

;

where

) and



)

The laplacian can be rewritten in the intrinsic coordinates

(

; 

)as





The Perona-Malik equation rewrites









)



(1 +



)

So the rst term always appears as a one-dimensional heat equation in the direction orthogonal to the

Figure 1.11:

Perona-Malik equation and edge detection : same exp erimentasinFigure

, but the heat equation has b een

replaced bythePerona-Malik equation. Notice that the edge map lo oks slightly b etter localized as with the heat equation.

gradient, tuned by the size of the gradient though. The second term can be a directional heat equation, or

reverse heat equation in the direction of the gradient. So we indeed mix in this model the heat equation

and the reverse heat equation ! We compare in Figure 1.11 the Perona-Malik with the classical heat equa-

tion in terms of accuracy on the boundaries obtained byCanny's edge detector : at a comparable scale of

smoothing, we clearly gain some accuracy in the boundaries and get rid of more \spurious" b oundaries.

The representation is b oth more sparse and more accurate.

Now, this ambitious mo del attempts to put in a single op erator two very dierent goals which we

already mentionned, namely restoration and analysis. This has a cost : the mo del contains a \constrast

threshold" which can only b e xed manually. Mathematical existence and uniqueness are not guaranteed,

despite some attempts byKichenassamyand Weickert. Let us summarize the involved parameters : we

need to x both



and the smo othing scale(s)

and the threshold on the gradient in Canny's detector as

F. Guichard, J-M. Morel,

Image Analysis and PDE's 17

CHAPTER 1. INTRODUCTION

well. Weobviously must take the same gradient threshold



in Canny's detector and in the Perona-Malik

equation. All the same, wehaveatwo parameters game : how will this b e dealt with in automatic image

analysis ? This question seems to have no general answer for the time being. An interesting attempt based

on statistical arguments is made, however, in Black, Sapiro.

If any nonlinear diusion can b e an image analysis mo del, why not trying them all ? This is exactly

what has happ ened in the past ten years. We can claim that almost all possible nonlinear parab olic

equations have b een proposed. The logic in this proliferation of mo dels is this : each attempt xes one

intrinsic diusion direction and tunes the diusion by the size of the gradientorby the size of a nonlinear

estimate of the gradient. Sometimes, the prop osed mo dels are even systems of PDE's, but in order not

to blow up this preface, we shall focus on the simplest prop osed examples. We can start with Rudin-

Osher-Fatemi's mo del, which consists, for the smoothing term, of minimizing the total variation of

. The

gradient descentfor

writes

div

(



Written in that way, the metho d app ears as diusion in the direction orthogonal to the gradient, tuned

by the magnitude of the gradient. Caselles, ? ? and ? ? proved that this equation is indeed well posed in

the space of b ounded variation. Avariantwas proposed indep endently by Alvarez and al.,



div

(





;

where the tuning of the gradient is nonlo cal. Kimia, Tannenbaum and Zucker proposed, endowed in a

more general shape analysis framework, the simplest equation of the list,

div

(

;



This equation had been prop osed some time b efore in another context by Sethian as a to ol for front

propagation algorithms. This equation, which we call in continuation \curvature equation", is a \pure"

diusion in the direction orthogonal to the gradient. The Weickert equation is a variant of the curvature

equation, with nonlo cal estimate of the direction orthogonal to the gradient : the diusion direction

SEigen

(



(



)) is computed as the eigenvector of the least eigenvalue of



(



): if

the convolution kernel is removed, this eigenvector simply is

. The three mentionned mo dels can b e

interpreted as diusions in a direction orthogonal to (an estimate of ) the gradient, tuned by the magnitude

of the gradient (Figure 1.14). Other diusions have been considered as well : For interpolation goals,

Caselles et al. proposed a diusion whichmaybe interpreted as the strongest p ossible image smo othing,

(

Du; Du

)

This equation is not used as the other ones as a prepro cessing of the image, but a way to interpolate

between the level lines an image with sparse level lines (Figure 1.13). Zhong and Carmona prop osed

a diusion in the direction

SEigen

(

) of the eigenvector with least eigenvalue of

(Figure

1.12). Sochen, Kimmel and Malladi prop ose instead a nondegenerate diusion, asso ciated with a minimal

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