Anisotropic ChanVese segmentation Salvador Molla Vicent Pallard oJuli aab Abstract. In this paper we study a variant to ChanVese CV segmentation

2025-04-27 0 0 678.64KB 29 页 10玖币

侵权投诉

Anisotropic Chan–Vese segmentation

Salvador Molla,∗, Vicent Pallard´o–Juli`aa,b

Abstract. In this paper we study a variant to Chan–Vese (CV) segmentation

model with rectilinear anisotropy. We show existence of minimizers in the 2-

phases case and how they are related to the (anisotropic) Rudin-Osher-Fatemi

(ROF) denoising model. Our analysis shows that in the natural case of a piece-

wise constant on rectangles image (PCR function in short), there exists a min-

imizer of the CV functional which is also piecewise constant on rectangles over

the same grid that the one deﬁned by the original image. In the multiphase

case, we show that minimizers of the CV multiphase functional also share this

property in the case that the initial image is a PCR function. We also investigate

a multiphase and anisotropic version of the Truncated ROF algorithm, and we

compare the solutions given by this algorithm with minimizers of the multiphase

anisotropic CV functional.

Keywords: segmentation, image processing, anisotropy, total variation

2010 MSC: 35G60, 35Q68, 35J92, 49J10, 94A08

1 Introduction

Image segmentation consists in partitioning a given image into multiple seg-

ments in which pixels share some characteristics. One of the most relevant

models in the ﬁeld of image segmentation is the Mumford-Shah (MS) model,

introduced by the authors in [25]. This model was the seed of a very suc-

cessful approach to the problem: variational techniques with level set formu-

lations. A particular case of the Mumford-Shah model is the case in which

the objective function is piecewise constant inside some domains with ﬁnite

perimeters. This model was introduced by Chan and Vese in the 2-phases

and multiphases framework in [14] and [29], respectively. They are known as

Chan-Vese (CV) models and play a cornerstone role in diverse recent appli-

cations of image processing (e.g. see [16, 26, 30, 31]).

aDepartment d’An`alisi Matem`atica, Universitat de Val`encia, C/Dr. Moliner, 50, Bur-

jassot, Spain

bKimera Technologies, Lanzadera, C/ del Moll de la Duana, s/n, Val`encia, Spain

∗Corresponding author: j.salvador.moll@uv.es

arXiv:2210.02425v2 [math.AP] 26 Dec 2022

The starting point of this paper is the recent study [11] about the relation-

ship between the CV model in image segmentation and Rudin-Osher-Fatemi’s

(ROF) model in image denoising (see [28]). In [11], the authors show that

a thresholding in the ROF model’s solution provides a partial minimizer of

the two phases CV functional, which can be written as

CV2(Λ, c1, c2) = Per(Λ; Ω) + µZΛ

(c1−f)2dx +ZΩ\Λ

(c2−f)2dx.

A partial minimizer in the sense that one can obtain a minimizer in each

of its three variables; i.e. letting Λu:= {x:u(x)≥c1+c2

2}, with ubeing the

minimizer of the ROF functional (see [12, Proposition 2.6]), one has











Λu∈arg min

Λ:Per(Λ)<+∞

CV2(Λ, c1, c2)

1

|Λ|RΛf dx, 1

|Ω\Λ|RΩ\Λf dx∈arg min

(c1,c2)∈[0,1]2

CV2(Λ, c1, c2).

Despite this, whether Λu,1

|Λu|RΛuf dx, 1

|Ω\Λu|RΩ\Λuf dxis a true mini-

mizer of CV2remains as an open problem. On the other hand, as stated in

[11], it is of interest to understand if this relationship is still valid in some

variants of both CV and ROF models.

Our work focuses in particular anisotropic variants of these models. Our

motivation comes from some well known features of the anisotropic `1version

of the ROF model, such as sharp recovery of edges (see [15, 17]), exact com-

putability (e.g. see [10, 19, 22]) and an observed reduction of the staircasing

eﬀect detected in the isotropic version (see [15, 27]). Because of that, we

establish the anisotropic case as the `1one, i.e. we replace in the CV models

the usual total variation by the total variation with respect to |·|1, deﬁned

by |v|1:= |v1|+|v2|for v= (v1, v2)∈R2.

Our main objectives are the next ones: First, we show that there is a

global minimizer of the two phases anisotropic CV functional

ACVµ(Λ, c1, c2) := Per1(Λ; Ω) + µZΛ

(c1−f)2dx +ZΩ\Λ

(c2−f)2dx,

whose ﬁrst component is an upper level set of a minimizer to the anisotropic

ROF functional in L2(Ω), which is deﬁned as follows:

AROFλ(u) := |Du|1(Ω) + λ

2ZΩ

(u−f)2dx.

In order to show this result, we need to assume that Ω is a rectangle and

that the data considered belong to a suitable space. Namely, we will assume

that fis piecewise constant on rectangles (denoted by PCR and deﬁned in

Def. 4). We point out that this restriction is harmless from the point of

view of applications. Our strategy is as follows: First of all, we generalize

the ACV functional, which is deﬁned only for sets of ﬁnite perimeter, to an

energy functional deﬁned on L2(Ω) ×[0,1]2as follows

Gµ(u, c1, c2) := |Du|1(Ω)+µZΩ

(u(c1−f)2+(1−u)(c2−f)2)dx+ZΩ

I[0,1](u)dx,

where I[0,1] is the indicator function on [0,1]. In relation to the above, we

note that Gµ(·, c1, c2) and AROFλ(·) are solely ﬁnite on the space of bounded

variation functions, BV (Ω). On the other hand, the indicator function re-

stricts the range of Gµ(·, c1, c2) to [0,1]. Additionally, we remark that if Eis

a set with ﬁnite perimeter and u=χE,Gµ(u, c1, c2) = ACVµ(E, c1, c2).

On these functionals, we prove existence of a global minimizer of Gµ

through a direct variational method. Then, we show that a truncation of

the solution to the AROF functional (i.e. an upper level set of the solution)

yields the ﬁrst component of a minimizer to Gµ. To show that this is indeed

the case, we rely on the description of explicit solutions of AROF obtained in

[22] for any PCR datum and on the corresponding Euler-Lagrange equations

to both functionals. In doing so, we ﬁnd a global minimizer both of Gand of

ACV. All the results concerning the 2-phases case are worked out in Section

Remark 1. In the 1-dimensional case (Ω = [a, b]⊂R), it was shown in [24]

that there is a minimizer to the CV problem with ﬁrst component satisfying

the following two properties:

(a) The boundary of the set belongs to a sole level set of the datum fin a

multivalued sense.

(b) If fis piecewise constant, then the boundary of the set is contained in

the jump set of f.

The above properties are shown to be false in the anisotropic case as shown

in Example 1 as well as it is in the two–dimensional case of the standard CV

model (see [24, Remark 4.2]). However, as a by-product of our result, we ob-

tain that, in the case of a PCR datum, the ﬁrst component of a minimizer to

the ACVµfunctional is also a PCR function; i.e the minimizing set is a rec-

tilinear polygon. Moreover, its essential boundary belongs to a grid generated

by the datum itself. This last property permits to design a trivial algorithm

to compute a minimizer of the ACVµfunctional.

In second place, we deal with the n-phases anisotropic CV model. In this

case, we decide to slightly modify the original (anisotropic) functional, which

reads as

ACVn

µ(Ω,c) :=

i=1 Per1(Ωi; Ω) + µZΩi

(ci−f)2dx,(1)

with µ > 0 , Ω:= {Ωi}n

i=1 ∈ Pn(Ω), where, by Pn(Ω) we denote the set of all

non empty n-parts (disjoint) partition of Ω and c:= {ci}n

i=1 ∈[0,1]n. The

proposed modiﬁcation consists in not counting more than once the length

of the possible overlaps of the boundaries of the partition. To do this, we

propose the following energy functional:

µ(Σ,c) =

i=1 Per1(Σi−1; int(Σi)) + µZΣi\Σi−1

(ci−f)2dx,(2)

with µ > 0 and c:= {ci}n

i=1 ∈[0,1]nand Σ:= {Σi}n

i=0 ∈ P∗

n(Ω), where

P∗

n(Ω) := {{Λi}n

i=0 :∅= Λ0⊆Λi⊆Λi+1 ⊆Λn= Ω}.

We observe that deﬁning Ωi:= Σi\Σi−1, the unique diﬀerence between

both functionals is that in ACVn

µsome edges will count more than twice (in

the case that an edge belongs to the boundary of more than two diﬀerent

upper level sets) in the length term while in Gn

µthey are counted only once.

As in the 2-phases case, the existence of a minimizer (Σ∗,c∗) follows from

the direct method in calculus of variations (see Proposition 2). Moreover,

one can assume that 0 ≤c∗

i+1 ≤c∗

i≤1 for any i= 1, . . . , n −1. Next, we

observe that

Σ∗∈arg min

Λ∈P∗

n(Ω)

µ(Λ,c∗) with c∗

i=1

|Σ∗

i\Σ∗

i−1|ZΣ∗

i\Σ∗

i−1

f(x)dx.

In the case that f∈PCR(Ω), by the tools developed in [22], we can show

that Σ∗

iis a rectilinear polygon whose boundaries lie on the grid generated

by f. These results concerning the multiphase case are the core of Section 4.

Thirdly, we discuss about a possible relationship between the minimizers

of a variant of ACVn

µand the truncated AROFλfunctional, from a general

point of view. For this purpose, we deﬁne the following general ACVn

µvariant

CVn

ϕ,µ(Ω,c) :=

i=1 Perϕ(∪i

j=1Ωi; Ω) + µiZΩi

(ci−f)2dx(3)

where µ:= {µi}n

i=1 ∈[0,+∞)n,Ω:= {Ωi}n

i=1 ∈ Pn(Ω) and c:= {ci}n

i=1 ∈

[0,1]n. Similarly, the truncated ROF functional for nphases is deﬁned as

TROFn

ϕ,λ(Σ,τ) :=

n−1

i=1 Perϕ(Σi; Ω) + λZΣi

(τi−f)dx(4)

where λ > 0, Σ:= {Σi}n

i=0 ∈ P∗

n(Ω) and τ:= {τi}n−1

i=1 ∈[0,1]n−1. Here,

these generalizations depend on a positively 1-homogeneous convex function

|·|ϕ. Note that CV n

ϕ,µ=Gn

µif |·|ϕis the 1-norm and µi=µfor every i.

In Section 5 we prove that, in the 2-phases case, it is possible to obtain a

relationship between the minimizers of ACV2

µand TROF2

1,λ. However, we

prove that a similar relationship in the multiphase case cannot be true, in

general. Finally, we remark the relationship between TROFn

ϕ,λ and AROFλ

in the anisotropic case.

The paper ﬁnishes with some applications of our results. In Section 6, we

show the strength of them with some examples on 2-phases and multiphase

image segmentation, by comparing them with similar isotropic processes.

2 Preliminaries

Before starting, we prescribe some notations and we provide a basic knowl-

edge on bounded variation functions and anisotropies.

2.1 Notations

Throughout the paper, Ω ⊂RNwill denote an open bounded set with bound-

ary ∂Ω and νΩwill denote the outer unit exterior normal at a point on the

boundary, when deﬁned. We denote by Lp(Ω) (1 ≤p < +∞), L∞(Ω) and

M(Ω)Mthe set of Lebesgue p−integrable functions in Ω, the set of essen-

tially bounded measurable functions in Ω and the set of ﬁnite vector Radon

measures on Ω, respectively. In the case that M= 1, we omit the index.

For any measurable set E⊂RNwith respect to the Lebesgue measure in

RN, we denote by |E|the Lebesgue measure of E. By HN−1, we denote the

(N−1)−dimensional Haussdorf measure. We denote by BV (Ω) the Banach

space of functions of bounded variation in Ω with the norm deﬁned next:

BV (Ω) := {u∈L1(Ω) : Du ∈ M(Ω)N},

kukBV (Ω) := kukL1(Ω) +kDukM(Ω)N,

where Du is the distributional gradient of u. If u∈BV (Ω), then the measure

Du can be decomposed into its absolutely continuous part and singular with

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

AnisotropicChan{VesesegmentationSalvadorMolla;,VicentPallardo{Juliaa;bAbstract.InthispaperwestudyavarianttoChan{Vese(CV)segmentationmodelwithrectilinearanisotropy.Weshowexistenceofminimizersinthe2-phasescaseandhowtheyarerelatedtothe(anisotropic)Rudin-Osher-Fatemi(ROF)denoisingmodel.Ouranalysissho...

展开>> 收起<<

Anisotropic ChanVese segmentation Salvador Molla Vicent Pallard oJuli aab Abstract. In this paper we study a variant to ChanVese CV segmentation.pdf

共29页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Anisotropic ChanVese segmentation Salvador Molla Vicent Pallard oJuli aab Abstract. In this paper we study a variant to ChanVese CV segmentation

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: