Fast cartoon + texture image filters

Abstract:

Can images be decomposed into the sum of a geometric part and a textural part. Ina

theoretical breakthrough, Yves Meyer [28] proposed variational models that force the

geometric part into the space of functions with bounded variation, and the textural part

into a space of oscillatory distributions. Meyer’s models are simple minimization

problems extending the famous total variation model. However, their numerical solution

has proved challenging. It is the object of a literature rich in variants and numerical

attempts.

This paper starts with the linear model, which reduces to a low-pass/high-pass

filter pair. A simple conversion of the linear filter pair into a nonlinear filter pair

involving the total variation is introduced. This new-proposed nonlinear filter pair retains

both the essential features of Meyer’s models and the simplicity and rapidity of the linear

model. It depends on only one transparent parameter: the texture scale, measured in pixel

mesh. Comparative experiments show a better and faster separation of cartoon from

texture. One application is illustrated: edge detection.

Introduction

The cartoon + texture algorithm decomposes any image into the sum of a cartoon

part, where only the image contrasted shapes appear, and a textural part with the

oscillating patterns. Such a decomposition is analogous to the classical signal processing

low pass-high pass filter decomposition. However, the cartoon part of an image actually

contains strong edges, and therefore all frequencies, up to the high ones, while a texture

can also contain middle and high frequencies. Thus, a linear decomposition is never clear

cut. It blurs out edges, and leaves behind some texture in the low pass filtered part.

To overcome this difficulty, the algorithm proposed, is a nonlinear low pass-high

pass filter pair. At each point, a decision is made of whether it belongs to the cartoon part

or to the textural part. This decision is made by computing a local total variation of the

image around the point, and comparing it to the local total variation after a low pass filter

has been applied.

Edge points in an image tend to have a slowly varying local total variation when

the image is convolved by a low pass filter. Textural points instead show a strong decay

of their local total variation by convolution with a low pass filter. The cartoon+texture

algorithm is based on this simple observation.

The cartoon part keeps the original image values at points termed as non-textural

points. At points identified as texture points, the cartoon part takes the filtered value. At

points where the cartoon/texture decision is ambiguous, a weighted average of them is

given. The texture part simply is the difference between the original image and its

cartoon part.

The presented algorithm deals with color images. The cartoon-texture

decomposition is made independently on all channels.

A grey level or color image will be denoted by f : (x,y) Є Ω → IR (respectively

IR3) where is an open subset of IR2, typically a rectangle or a square. An image f is

defined on a continuous domain by interpolating a digital image defined on a finite set of

pixels. We are interested in decomposing f into two components f = u + v, such that u

represents a cartoon or geometric (piecewise-smooth) component of f, while v represents

the oscillatory or textured component of f. The oscillatory part v should contain

essentially the noise and the texture.

Scope of the Project:

The proposed algorithm can be used in Multimedia Industry for fast generation of

cartoon images and textures.

Module Description:

As indicated in the algorithm, the cartoon+texture decomposition only requires

the application on the gradient image of two low-pass filters, which are performed

directly by a discrete convolution. The gradient is computed by the simplest centered

difference scheme. The main modules are:

Apply the low pass filter Lσ to the image f.

Compute the modulus of the image gradients of f and Lσ ∗ f.

Convolve these moduli with the Gaussian Gσ to get the local total variation of f

and Lσ ∗ f.

Deduce the value of λ(x) at each point in the image.

Deduce the value of the cartoon image as a weighted average of f and Lσ ∗ f.

Compute the texture as the difference u-f.

ALGORITHMS USED

The algorithm used is based on a theory proposed by Yves Meyer in “Oscillating

patterns in image processing and nonlinear evolution equations”.

DOMAIN

DIGITAL IMAGE PROCESSING

Digital image processing is the use of computer algorithms to perform image

processing on digital images. As a subfield of digital signal processing, digital image

processing has many advantages over analog image processing; it allows a much wider

range of algorithms to be applied to the input data, and can avoid problems such as the

build-up of noise and signal distortion during processing.

SOFTWARE REQUIREMENTS

MATLAB 7.0 AND ABOVE

MATLAB

MATLAB is a high-performance language for technical computing. It integrates

computation, visualization, and programming in an easy-to-use environment where

problems and solutions are expressed in familiar mathematical notation.

Typical uses include:

Math and computation

Algorithm development

Modeling, simulation, and prototyping

Data analysis, exploration, and visualization

Scientific and engineering graphics

Application development, including Graphical User Interface building

MATLAB is an interactive system whose basic data element is an array that does not

require dimensioning. This allows you to solve many technical computing problems,

especially those with matrix and vector formulations, in a fraction of the time it would

take to write a program in a scalar non-interactive language such as C or FORTRAN.

SYSTEM DESCRIPTION:

PRIOR WORK:

The general variational framework for decomposing f into u + v is given in Meyer’s models

as an energy minimization problem

Where F1, F2 ≥ 0 are functional and X1, X2are spaces of functions or distributions such

that F1(u) < ∞ and F2(v) < ∞ if and only if(u, v) ∈ X1 × X2. The constant λ > 0 is a

Tuning parameter. A good model for (1) is given by a choice of X1 and X2 so that if u is

cartoon And if v is texture, then F1(u) << F2(u) andF1(v) >> F2(v) (such conditions

would insure a clear cartoon +texture separation; in other words ,if u is only cartoon,

without texture, then texture Components must be penalized by F1, but not by F2, and

vice-versa).

The long story of this problem can be summarized in a list of proposed choices for both

spaces X1 and X2, and both functional F1(u) andF2(v). In fact the choice for F1(u) has

quickly converged to the total variation of u, that excludes strong oscillations but permits

sharp edges. The main point under discussion has been what spaceX2 would model the

oscillatory part. Since the discussion is complex, we refer to Table I and its legend, which

present the main models. This table extends the model classification outlined in, and

adopts the same terminology.

One of the first nonlinear cartoon + texture models is the Mumford and Shah model [30],

for image segmentation, where f ∈ L2() is decomposed into u ∈ SBV () ([12], [2], [29], ,

a piecewise-smooth function with its discontinuity set Ju included in a union of curves

whose overall length is finite, and v = f − u ∈ L2() represents the noise or the texture.

The minimization problem is

Where H1 denotes the 1-dimensional Hausdorff measure (the length if Ju is

sufficiently smooth), and λ > 0 is a tuning parameter. With the above notations, X1 =

SBV () is the De Giorgi space of special functions with bounded variation. F1is

composed of the first two terms in the energy from (3), while the third term is F2(v) = R

v2,the quadratic norm. It is difficult to solve this model in practice, because of its non-

convex nature coming from F1(u).An easier decomposition can be obtained by the

Rudin, Osher, and Fatemi (ROF) total variation(TV) minimization model [37] for image

de noising. Their functional is convex and therefore more amenable to efficient

minimization.

The variational model is

Denotes the total variation of u in , also denoted by T V (u) or by |u| BV (). The

component u belongs to the space of functions of bounded variation BV () = u ∈ L1() :

R |Du| <∞. This space penalizes oscillations (such as noise or texture), but allows for

piecewise-smooth functions, made of homogeneous regions with sharp boundaries. Since

almost all level lines (or isolines) of a BV function have finite length, the BV space is

considered adequate to model images containing shapes. These shapes can actually be

extracted by edge detection or by image binarization and morphology.

The bibliography on algorithms minimizing the ROF functional and its multi-

scale variants, Convex dual numerical methods have been tested Hybrid models with

wavelets are described in Models where the L2 norm is replaced by the L1 norm are now

classical.

In strong mathematical geometric arguments are put forward in favor of the T V -

L1 model: explicit solutions can be computed for simple geometric objects. These

examples demonstrate that, based on the perimeter/area ratio, shapes are unambiguously

put either in the TV part or in the L1 part. This study connects the T V -L1 model with

the classical morphological granulometry. Accurate regularity results for the level set

boundaries of minimize of the T V −L1 model is also given, in any dimension, in [1].

Probably the most popular T V minimization algorithm is Chambolle’s projection

algorithm [14]. Recent years have, however, shown a trend to abandon the BV norm and

replace it bya so-called “non-local” norm inspired from.

Yet, as pointed out in, T V −L2 or T V −L1do not characterize the oscillatory

components. Indeed, these components do not have small norms in Lp(), p ≥ 1, . To

overcome this drawback, Y. Meyer [28] proposed in his seminal book weaker norms to

replace k · k2L2 in the ROF model, that would better model oscillatory components with

zero mean. The Meyer model is

Where ||.||*_ is the norm in one of the following spaces, denoted by G, F or E

(defined below for (Ω = IR2).

Definition 1: A distribution v belongs to G if and only if v = div(~g) for some ~g ∈ (L1)2

in the distributional sense. The endowed norm is

The introduction of the spaces G, F and Eis motivated by the fact that highly

oscillatory signals or images have small norms in G, F or E. For instance, || cos nx||G =

1n. The presence of a non-BV part in images is corroborated by the experimental-

numerical study . However, the three norms proposed by Meyer are not expressed as

integrals and are therefore difficult to compute.

It is also difficult to set up the right value of λ for real images. This problem is

addressed. The numerical experiments have shown promising results and justified further

inquiries. There has been an extensive line of papers modifying and interpreting Meyer’s

models, and proposing minimization schemes: An extensive mathematical analysis of

Meyer’s model in a bounded domain is performed .For many formal properties of the G-

norm the reader can refer. In the G-norm is replaced by the H−1 norm. This approach

using Sobolev spaces with negative exponents was extending ding. The F = div (BMO)

variant was numerically studied. There have also been extensions intending to decompose

u into three components, namely BV , texture, and a residual (e.g., noise). In the model

(where the space G = ˙W −1,1 is approximated by Gp = ˙W −1,p for large p), this is done

by solving

In the norm ||v||G of v = div~g is approximated by ||pg1 + g22||p, p ≥ 1 which is

of course far from the real problem with p = ∞. Aujol etal. Addressed the original Meyer

problem and proposed an alternate method to minimize

Where the above mentioned variants and others are summarized. Following this

paper’s terminology, the funding models that inspired this line of research are T V − L2

(ROF) and the original Meyer models T V −div (L1), T V −div (BMO) (numerically tried

in. A simpler variant is T V −H−1, since also the H−1 norm is small on oscillatory

signals. The hierarchy of the spaces used for the oscillatory part is complex: div (L1) and

div (BMO) are distributional first derivatives of vector fields in L1 and BMO

respectively .The Besov model takes the oscillatory part v into which is a space of second

derivatives of functions satisfying a Zygmund regularity condition. Since this condition is

close to assuming a Lipschitz bound on the functions, it is fair to say that the Besov

model

Defines distributions that are second derivatives of functions that have (almost)

bounded gradients.

In conclusion (as also pointed out by Y. Meyer), the four spaces G = div(L1),

H−1 =_(H1), F = div(BMO) and can be considered as variants of each other, since they

all appear as first derivatives of (bounded like)functions. Experimental evidence does not

favor one of them.

Generalizing T V −H−1, a generic T V -Hilbertmodel [11] can be defined using a

smoothing kernel K. The associated Meyer energy is

This model has also been proposed in [19]. TheL2 norm of K ∗ (f − u) can be

substituted by an Lp norm, p ≥ 1. One obtains slightly better results with p = 1 [18]. Our

numerical trials yield no significant difference between T V -Hilbert and the other

mentioned T V − X models. Because of its simplicity, we shall retain this version in the

experiments after fixing adequately the kernel K. This is precisely the object of the next

section. The main goal of the manuscript is to propose here a simpler and faster model

than the variational model (5), while better separating cartoon from texture We wish to

recall here the function spaces notations used in the next sections. H0 = L2 denotes the

space o square-integrable functions The Sobolev space H1 is defined by

We will also make use of the space defined in the Fourier domain by the set of

functions and distributions

we formulate the linear cartoon +texture H1−H−1 model inspired from Y. Meyer,

which can be easily and rapidly solved in the Fourier domain in one step. Since this

model introduces blurring in the cartoon component u, we propose in a novel nonlinear

cartoon + texture model that retains the simplicity and efficiency of the linear one, while

the cartoon component u is piecewise-smooth and with sharp edges. illustrates numerical

comparisons between the linear model, the non linear minimization model (5) and the

proposed fast non linear model; an application to edge detection is also shown, together

with a discussion on the local texture scale.

II. LINEAR VERSION OF MEYER’S MODEL

In view of the multiplicity and complexity of nonlinear models, it seems

reasonable to first fix as a reference the best linear model. Separation of scales in images

is classically obtained by applying a complementary pair of low-pass and high-pass filters

to the data f, namely u =LPF(f), and then v = f − u = HPF(f). The T V −H1 model is

easily linearized by replacing the total variation R |Du| by the Dirichlet integral R |Du|2.

Then the most natural variation linear model associated with Meyer’s ideas isH1 − H−1.

Indeed, H−1 is dual to H1, in the same way as G is dual to BV. The low pass filter f → u

is obtained by the minimization

The meaning of σ4 will be shortly explained. This model can be compared with the

classical Tikhonov quadratic H1 − L2 minimization

Which is equivalent in the Fourier domain to the low-pass filter ˆu = 11+(2πσ|ξ|)2

ˆ f. This Wiener filter is known to remove high-frequency components due to the edges of

f, and not only those due to oscillations (See Fig. 1).Using the Fourier transform in (6),

the H1semi-norm of u is R |Du|2 = R(2π|ξ|)2|ˆu(ξ)|2and the H−1 semi-norm of v is R |

ˆv(ξ)|2(2π|ξ|)2 . This implies in particular that u − f = v has zero mean, since feasible

solutions satisfy ˆv(0) =0.Minimizing this quadratic functional (6) in yields in Fourier the

unique solution ˆu = ˆLσ ˆ f, where

The meaning of the parameter σ is now easily explained: if the frequency ξ is

significantly smaller than 12πσ, then the ξ frequency is kept in u, while if ξ is

significantly larger than 12πσ ,then the frequency ξ is considered a textural frequency and

attributed to v. Thus, the solution (u, v) = (Lσ ∗ f, (Id − Lσ) ∗ f) is nothing but a pair of

complementary low pass and high pass filters. Note that as σ → 0, Lσ → Id. We will also

consider the filter Kσ, where ˆKσ(ξ) =e−(2πσ|ξ|)4 , which behaves still more like the

characteristic function of the ball centered at zero with radius 12πσ .

It is worth mentioning that related linear and nonlinear three-term decompositions

f = u+v+w based on the H1−H−1 duality were introduced the linear case is the (H1,H0,

H−1) decomposition, while the nonlinear decomposition uses piecewise (H1,H0,H−1)

(where piecewiseH1 is the SBV space for the cartoon, combined with piecewise H−1 for

the texture).

The observed efficiency of the linear pair (Lσ, Id − Lσ) (see Figure 3) leads to

consider nonlinear versions that would retain its main feature, namely the excellent

extraction of the texture by a high pass filter Id−Lσ. On the other hand, the non-

oscillatory parts of the initial image f should be kept unaltered even if they have sharp

edges. This is of course impossible with a linear filter. Thus, a local indicator must be

built to decide at each point x whether it belongs to a textural region or to a cartoon

region. The main character of a cartoon region is that its total variation does not decrease

by low pass filtering. The main character of a textured region is its high total variation

due to its oscillations. This total variation decreases very fast under low-pass filtering.

Formalizing these remarks leads to define the local total variation (LTV) at x,

(note that Lσ can be substituted by Kσ). The relative reduction rate of LTV is defined by

afunction x 7→ λσ(x).

PROPOSED METHOD:

The observed efficiency of the linear pair (Lσ, Id − Lσ) (see Figure 3) leads to

consider nonlinear versions that would retain its main feature, namely the excellent

extraction of the texture by a high pass filter Id−Lσ. On the other hand, the non-

oscillatory parts of the initial image f should be kept unaltered even if they have sharp

edges. This is of course impossible with a linear filter. Thus, a local indicator must be

built to decide at each point x whether it belongs to a textural region or to a cartoon

region. The main character of a cartoon region is that its total variation does not decrease

by low pass filtering. The main character of a textured region is its high total variation

due to its oscillations. This total variation decreases very fast under low -pass filtering.

Formalizing these remarks leads to define the local total variation (LTV) at x,

(note that Lσ can be substituted by Kσ). The relative reduction rate of LTV is defined by

a function x 7→ λσ(x), given by

which gives us the local oscillatory behavior of the function f. If λσ is close to 0,

we have

which means that there is little relative reduction of the local total variation by the

low pass filter. If instead λσ is close to 1, the reduction is important, which means that the

considered point belongs to a textured region. Thus, a fast nonlinear low pass and high

pass filter pair can be computed by weighted averages of f and Lσ ∗ f depending on the

relative reduction of LT V . We can set

where w(x) : [0, 1] → [0, 1] is an increasing function that is constant and equal to

zero near zero and constant and equal to 1 near 1. In all experiments the soft threshold

function w is defined by

fixed to 0.25 and 0.5. If λσ(x) is small, the function f is non-oscillatory around x and

therefore the function is BV (or cartoon) around x. Thus u(x) = f(x) is the right choice. If

instead λσ(x) is large, the function f is locally oscillatory around x and locally replaced

by (Lσ∗f)(x). The choice of λσ = 12 as underlying hard threshold is conservative: it

permits to keep all step edges on the cartoon side, but puts all fine structures on the

texture side, as soon as they oscillate more than once. Of course changes in the

parameters a1 or a2 would slightly modify the separation results.

Since it is desirable to have a one-parameter method, it seems advisable to fix the

threshold function w once and for all, as has been done in all experiments. In that way the

method keeps the scale σ of the texture as the only method parameter. That this last

parameter cannot be avoided is obvious: textural details become shapes when their sizes

grow, and therefore should be moved from the texture to the BV side.

Advantages:

Edges are highly preserved.

It differentiates the sharp and flat regions.

The output appears visually to a cartoon image.

Applications:

The developed project can be widely used in multimedia industry.

Literature Survey:

Various image filters used in the previous papers are reviewed. And the filters

used in those papers are listed below.

[1]L. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal

algorithms. Physica D, 60(1-4):259–268, 1992.

A constrained optimization type of numerical algorithm for removing noise from

images is presented. The total variation of the image is minimized subject to constraints

involving the statistics of the noise. The constraints are imposed using Lanrange

multipliers. The solution is obtained using the gradient-projection method. This amounts

to solving a time dependent partial differential equation on a manifold determined by the

constraints. As t → ∞ the solution converges to a steady state which is the de-noised

image. The numerical algorithm is simple and relatively fast. The results appear to be

state-of-the-art for very noisy images. The method is noninvasive, yielding sharp edges in

the image. The technique could be interpreted as a first step of moving each level set of

the image normal to itself with velocity equal to the curvature of the level set divided by

the magnitude of the gradient of the image, and a second step which projects the image

back onto the constraint set.

∫ |Du| + ∫ |v|2

[2]T.F. Chan and S. Esedoglu. Aspects of total variation regularized L1 function

approximation. SIAM J. on Appl. Maths., 65(5):1817–1837, 2005.

The total variation based image denoising model of Rudin, Osher, and Fatemi has

been generalized and modified in many ways in the literature; one of these modifications

is to use the L1 norm as the fidelity term. We study the interesting consequences of this

modification, especially from the point of view of geometric properties of its solutions. It

turns out to have interesting new implications for data driven scale selection and

multiscale image decomposition.

∫ |Du| + ∫ |v|

[1] AND [2] contains the classic BV or SBV + Noise Models.

[3]Y. Meyer. Oscillating patterns in image processing and nonlinear evolution

equations. American Mathematical Society Providence, RI, 2001, proposes the filter

This paper proposes an adaptive color feature extraction scheme by considering

the color distribution of an image. Based on the binary quaternion-moment-preserving

(BQMP) thresholding technique, the proposed extraction methods, fixed cardinality (FC)

and variable cardinality (VC), are able to extract color features by preserving the color

distribution of an image up to the third moment and to substantially reduce the distortion

incurred in the extraction process. In addition to utilizing the earth mover's distance

(EMD) as the distance measure of our color features, we also devise an efficient and

effective distance measure, comparing histograms by clustering (CHIC). Moreover, the

efficient implementation of our extraction methods is explored. With slight modification

of the BQMP algorithm, our extraction methods are equipped with the capability of

exploiting the concurrent property of hardware implementation. The experimental results

show that our hardware implementation can achieve approximately a second order of

magnitude improvement over the software implementation. It is noted that minimizing

the distortion incurred in the extraction process can enhance the accuracy of the

subsequent various image applications, and we evaluate the meaningfulness of the new

extraction methods by the application to content-based image retrieval (CBIR). Our

experimental results show that the proposed extraction methods can enhance the average

retrieval precision rate by a factor of 25% over that of a traditional color feature

extraction method.

∫ |Du| + ∫ ||v||Ẁ-1,∞

[4]S.J. Osher, A. Sol´e, and L.A. Vese. Image decomposition and restoration using

total variation minimization and the H−1 norm. Multiscale Modeling and

Simulation, 1(3):349–370, 2003.

In this paper, we propose a new model for image restoration and image

decomposition into cartoon and texture, based on the total variation minimization

ofRudin, Osher, and Fatemi [Phys. D, 60 (1992), pp. 259–268], and on oscillatory

functions, which follows results of Meyer [Oscillating Patterns in Image Processing and

Nonlinear Evolution Equations, Univ. Lecture Ser. 22, AMS, Providence, RI, 2002]. This

paper also continues the ideas introduced by the authors in a previous work on image

decomposition models into cartoon and texture [L. Vese and S. Osher, J. Sci. Comput., to

appear]. Indeed, by an alternative formulation, an initial image f is decomposed here into

a cartoon part u and a texture or noise part v. The u component is modeled by a function

of bounded variation, while the v component is modeled by an oscillatory function,

bounded in the norm dual to | · | H1 0. After some transformation, the resulting PDE is of

fourth order, envolving the Laplacian of the curvature of level lines.

∫ |Du| + ∫ ||v||Ĥ-1

[5]T.M. Le and L.A. Vese. Image decomposition using total variation and

div(BMO). SIAM Multiscale Modeling and Simulation, 4(2):390–423, 2005.

This paper is devoted to the decomposition of an image f into u + v, with u a

piecewise-smooth or “cartoon” component, and v an oscillatory component (texture or

noise), in a variational approach. Meyer [Oscillating Patterns in Image Processing and

Nonlinear Evolution Equations, Univ. Lecture Ser. 22, AMS, Providence, RI, 2001]

proposed refinements of the total variation model (Rudin, Osher, and Fatemi [Phys. D, 60

(1992), pp. 259–268]) that better represent the oscillatory part v: the spaces of

generalized functions G = div(L∞) and F = div(BMO) (this last space arises in the study

of Navier–Stokes equations; see Koch and Tataru [Adv. Math., 157 (2001), pp. 22–35])

have been proposed to model v, instead of the standard L2 space, while keeping u a

function of bounded variation. Mumford and Gidas [Quart. Appl. Math., 59 (2001), pp.

85–111] also show that natural images can be seen as samples of scale-invariant

probability distributions that are supported on distributions only and not on sets of

functions. However, there is no simple solution to obtain in practice such decompositions

f = u+v when working with G or F. In earlier works [L. Vese and S. Osher, J. Sci.

Comput., 19 (2003), pp. 553–572], [L. A. Vese and S. J. Osher, J. Math. Imaging Vision,

20 (2004), pp. 7–18], [S. Osher, A. Sol´e, and L. Vese, Multiscale Model. Simul., 1

(2003), pp. 349–370], the authors have proposed approximations to the (BV,G)

decomposition model, where the L∞ space has been substituted by Lp, 1 ≤ p < ∞. In the

present paper, we introduce energy minimization models to compute (BV,F)

decompositions, and as a by-product we also introduce a simple model to realize the

(BV,G) decomposition. In particular, we investigate several methods for the computation

of the BMO norm of a function in practice. Theoretical, experimental results and

comparisons to validate the proposed new methods are presented.

[6]J.B. Garnett, P.W. Jones, T.M. Le, and L.A. Vese. Modeling oscillatory

components with the homogeneous spaces BMO−_ and W−_,p. UCLA CAM Report

07- 21, to appear in PAMQ, 2007.

This paper is devoted to the decomposition of an image f into u + v, with u a

piecewise-smooth or “cartoon” component, and v an oscillatory component (texture or

noise), in a variation approach. The cartoon component u is modeled by a function of

bounded variation, while v, usually represented by a square integrable function, is now

being modeled by a more refined and weaker texture norm, as a distribution.

Generalizing the idea of Y. Meyer [32], where v 2 F = div (BMO) = BM˙ O −1, we

model here the texture component by the action of the Riesz potentials on v that belongs

to BMO or to Lp. In an earlier work [26], the authors proposed energy minimization

models to approximate (BV, F) decompositions explicitly expressing the texture as

divergence of vector fields in BMO. In this paper, we consider an equivalent more

isotropic norm of the space F in terms of the Riesz potentials, and study models where

the Riesz potentials of oscillatory components belong to BMO or to Lp, 1 _ p < 1 (thus

we consider oscillatory components in BM˙ O _ or in ˙W _,p, with _ < 0). Theoretical,

experimental results and comparisons to validate the proposed methods are presented.

[5] & [6] proposes the following filter equation.

∫ |Du| + ∫ ||v||BMO-1

And introduces a key new feature: The norm of the oscillatory part v decreases

when v oscillates more. This is obtained by putting a norm on v that is actually a norm on

a primitive of v. The TV-H−1, TV-DIV(BMO) AND TV-Besov Models follow the same

pattern.

[7]J.B. Garnett, T.M. Le, Y. Meyer, and L.A. Vese. Image decompositions using

bounded variation and generalized homogeneous Besov spaces. Applied and

Computational Harmonic Analysis, 23(1):25–56, 2007 (UCLA CAM Report 05-57,

2005).

This paper is devoted to the decomposition of an image f into u + v , with u a

piecewise-smooth or “cartoon” component, and v an oscillatory component (texture or

noise), in a variational approach. Y. Meyer [Y. Meyer, Oscillating Patterns in Image

Processing and Nonlinear Evolution Equations, University Lecture Series, vol. 22, Amer.

Math. Soc., Providence, RI, 2001] proposed refinements of the total variation model [L.

Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms,

Phys. D 60 (1992) 259–268] that better represent the oscillatory part v: the weaker spaces

of generalized functions G =div( L ∞ ) , F =div( BMO ) , and have been proposed to

model v, instead of the standard L 2 space, while keeping u BV, a function of bounded

variation. Such new models separate better geometric structures from oscillatory

structures, but it is difficult to realize them in practice. D. Mumford and B. Gidas [D.

Mumford, B. Gidas, Stochastic models for generic images, Quart. Appl. Math. 59 (1)

(2001) 85–111] also show that natural images can be seen as samples of scale invariant

probability distributions that are supported on distributions only, and not on sets of

functions. In this paper, we consider and generalize Meyer's ( BV , E ) model, using the

homogeneous Besov spaces , −2< α <0 , 1 p , q ∞, to represent the oscillatory part v.

Theoretical, experimental results and comparisons to validate the proposed methods are

presented.

[8]J.F. Aujol, G. Gilboa, T. Chan, and S. Osher. Structure texture image

decomposition - modeling, algorithms and parameter selection. International

Journal of Computer Vision, 67(1):111–136, 2006 (UCLA CAM Report 05- 10,

2005).

This paper explores various aspects of the image decomposition problem using

modern variational techniques. We aim at splitting an original image f into two

components u and ¿, where u holds the geometrical information and ¿ holds the textural

information. The focus of this paper is to study different energy terms and functional

spaces that suit various types of textures. Our modeling uses the total-variation energy for

extracting the structural part and one of four of the following norms for the textural part:

L2, G, L1 and a new tunable norm, suggested here for the first time, based on Gabor

functions. Apart from the broad perspective and our suggestions when each model should

be used, the paper contains three specific novelties: first we show that the correlation

graph between u and ¿ may serve as an efficient tool to select the splitting parameter,

second we propose a new fast algorithm to solve the TV ¿ L1 minimization problem, and

third we introduce the theory and design tools for the TV-Gabor model.

[7] & [8] proposes the filter equation as given below.

∫ |Du| + ∫ |K * v|2

Here, the main fact is that the model, H1 − H−1, boils down to the decomposition

into a classic low-pass and high-pass decomposition.

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