Note on the Existence of Minimizers for Variational Geometric Active Contours

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Note on the Existence of Minimizers for

Variational Geometric Active Contours

El Hadji S. Diop∗Val´erie Burdin†V. B. Surya Prasath‡

Abstract

We propose here a proof of existence of a minimizer of a segmentation

functional based on a priori information on target shapes, and formulated

with level sets. The existence of a minimizer is very important, because

it guarantees the convergence of any numerical methods (either gradient

descents techniques and variants, or PDE resolutions) used to solve the

segmentation model. This work can also be used in many other segmen-

tation models to prove the existence of a minimizer.

Keywords: Image Segmentation, Energy Minimization, Bounded Varia-

tion, Variational Model, Level sets, Shape priors.

1 Introduction

Image segmentation is still subjects of intensive researches due to emerging

acquisition techniques and capacity of image processing and computer vision

models in modeling and solving real life problems. In many applications, e.g.

medical imaging, it is essential by just a ﬁrst step towards the true applica-

tion (e.g. results analysis, therapeutic evaluation, . . . ). Accuracy is then very

important, and due to that fact, we developed a segmentation model [10] for

X-rays medical images that suﬀer from a poor contrast between objects of in-

terests (femur or tibia for instance) and other structures (soft tissues, pelvis,

hip, ﬁbula, tarsus, metatarsus, . . . ), problems of edge salience, occlusions phe-

nomena in bones joints, . . . . Prior shapes were incorporated in a level set-based

variational formulation of the segmentation problem, and then, as common,

associated Euler-Lagrange equations were derived constituting our segmenta-

tion model. A proof of existence of a minimizer of a segmentation energy is

∗Department of Mathematics, University of Thies, Thies BP 967, Senegal. Email: ehs-

diop@hotmail.com

†Image and Information Department, Institut Mines Telecom - Telecom Bretagne,

Technopole Brest-Iroise CS 83818, 29238 Brest Cedex 3, France. Email: valerie.burdin@imt-

atlantique.fr

‡Division of Biomedical Informatics, Cincinnati Children’s Hospital Medical Center,

Cincinnati, OH 45229 USA. Also with the Departments of Biomedical Informatics, Pedi-

atrics, Electrical Engineering and Computer Science, University of Cincinnati, OH, USA.

Email: surya.iit@gmail.com, prasatsa@uc.edu

arXiv:2210.12773v1 [math.CA] 23 Oct 2022

not always provided, the segmentation model is solved numerically (gradient

descent, partial diﬀerential equation numerical resolutions, . . . ), rather. The

main reason is the complexity of the designed segmentation functionals, and

then, many criteria (e.g. convexity) are not usually satisﬁed in order to apply

classical optimization results. Many techniques have been developed e.g. con-

vex relaxation techniques [21,6], which are weak formulation of the original

problem. Although the existence of a minimizer was not proven, the eﬃciency

and robustness of the segmentation model [10] were showed on various image

types; namely, on synthetic images, digitally reconstructed images, and real

radiographic images. Also, quantitative evaluations of obtained segmentation

results were provided. Here, we propose a proof of existence of a minimizer for

the variational segmentation functional [10]. Diﬀerent works [8,5,9,16] and

more recently [17,24,22,19,20] pursued the same goal, even if the proposed

segmentation functionals were diﬀerent.

The article is organized as follows. The segmentation functional is presented

in Section 2. The proof of the existence of a minimizer is proposed in Section 3.

We conclude with some perspectives in Section 4.

2 Variational Image Segmentation Model

Let I: Ω →R, I ={I(x), x ∈Ω}be a continuous image, Ω ⊂R2its domain

assumed to be open, bounded and with a Lipschitz boundary denoted by ∂Ω.

2.1 Prior Shapes Design

We consider a set {Cn}1≤n≤Nof Naligned contours embedded as signed dis-

tance functions denoted by φn,∀n≥0. Let {φn}1≤n≤Nbe the training dataset,

and denote ¯

φ=1

n=1

φnits mean. In order to learn the local deformations of

the dataset, we follow [15] and perform a P CA on {φn}1≤n≤N. Advantages in

doing that on (φn)ninstead of on (Cn)nwere already discussed [10]. So, this

means looking for the best orthonormal basis {uk}1≤k≤Ks.t. the projection of

φnon (uk)khas a minimal distance in L2sense. Let Ube composed with p

ﬁrst columns of Vholding the orthogonal modes of variations, new shapes can

be built then by: φ=¯

φ+Uλ, where U= [ui]1≤i≤pis the eigenvectors matrix

and λ= [λi]t

1≤i≤pthe shape parameters vector.

2.2 Segmentation Functional and Variational Formulation

The segmentation functional is deﬁned as follows [10]:

F(ϕ, λ, V, Iin, Iout) = α

2F1(ϕ) + F2(ϕ, λ, V ) + βF3(ϕ) + νF4(λ, V, Iin, Iout),(1)

where α, β, ν > 0 for counterbalancing diﬀerent energy terms deﬁned as follows:

F1(ϕ) = ZΩ

(|∇ϕ| − 1)2dx, (2)

F2(ϕ, λ, V ) = ZΩ

[ξg +γ

2φ2(λ, h(x))]δ(ϕ)|∇ϕ|dx, (3)

F3(ϕ) = ZΩ

g H(−ϕ)dx, and (4)

F4(λ, V, Iin, Iout) = ZΩ

|I−Iin|2+µ|∇Iin|2H(φ(λ, h(x))) dx

+ZΩ

|I−Iout|2+µ|∇Iout|2(1 −H(φ(λ, h(x)))) dx+ζZΩ

dH1(C(λ, h(x)));

(5)

with ϕbeing the level set function; ξand γare positive parameters; δ(·) is the

unidimensional Dirac distribution; gis a positive and strictly non increasing

function, typically: g=1

1 + η|∇Gσ? I|2,?standing for the convolution oper-

ator, η > 0, Gσis a Gaussian kernel with a variance σ.Iin and Iout designate

the Mumford-Shah functional terms; νis a positive parameter. H(·) is the

Heaviside function, i.e. H:R−→ {0,1}, s.t. H= 1 in R+,H= 0 in R−

?, and

δ=H0in the sense of distributions. φis the prior shape; λis the shape pa-

rameters vector; h: Ω −→ Ω; x7−→ h(x) = τRθx+Taccounts the spatial rigid

transformations achieved through a vector Vholding the scale parameter factor

τ, the rotation matrix Rθof angle θand the translation vector T= [Tx, Ty];

C(λ, h(x)) = {x∈Ω s.t. φ(λ, h(x)) = 0};H1is the one dimensional Hausdorﬀ

measure.

•F1(2) guarantees the signed distance function property, and avoids the re-

initialization process. In fact, when keeping |∇ϕ|bounded, one ensures the

correct computations of ϕderivatives. The most common way is to apply

the re-initialization procedure by periodically solving the PDE [23,25,7]:

(∂ϕ

∂t = sign(ϕ0)(1 − |∇ϕ|)

ϕ(x, 0) = ϕ0(x),

(6)

where ϕ0refers to the level set to be re-initialized.

•F2(3) makes the active contour evolve towards high gradients areas, and

also towards similar regions of the prior shape. Minimizing F2increases

the similarity between the active contour and the shape prior.

•F3(4) could be interpreted as a weighted area term of the target object of

interests. Notice that if gequal to 1, then F3is the area of the region of

the object of interests; i.e. {x∈Ω,s.t. ϕ(x)<0}. Minimizing it provides

another force pushing the active contour quickly towards the edges.

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NoteontheExistenceofMinimizersforVariationalGeometricActiveContoursElHadjiS.Diop*ValerieBurdinV.B.SuryaPrasathAbstractWeproposehereaproofofexistenceofaminimizerofasegmentationfunctionalbasedonaprioriinformationontargetshapes,andformulatedwithlevelsets.Theexistenceofaminimizerisveryimportant,becau...

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