Herramientas para la enseñanza y el aprendizaje

lation problem) plays an essential role in signal and image processing as it constitutes a bridge between the discrete and continuous worlds. Two reconstruction paradigms over the real line have been widely developed in the literature: variational and statistical. In the variational

Chapter 2. Review of Parametric Snakes

approach, the reconstructed signal is solution of an optimization problem that establishes a tradeoff between fidelity to the data and smoothness conditions via a regularization term [85]. In the statistical approach, the signal is modeled as a random process defined from a Gaussian white noise and is optimally reconstructed using estimation theory [86]. Two fundamental results are that 1) these two frameworks are deeply connected [87]; and 2) the solution of either problem can be expressed as a spline function in relation with a differential operator involved in regularization (variational approach) or whitening (statistical approach) [88, 89]. In the context of this thesis, we are interesting in the reconstruction of a continuous closed curve from its samples. This implies to reconstruct periodic continuous-domain functions that are the coordinate functions of the closed curve. This motivated us to develop the theory of the variational and statistical approaches in a periodic setting and in a very general context,

i.e., for a broad class of differential operators and for general measurements that include

sampling. This theory is fully presented in Appendix B and is subject to our publication [76]. In this section, we summarize some of our results showing that periodic exponential splines naturally appear when we optimally reconstruct a closed curve from its samples.

We consider the following problem. Let {r(tn)}n∈{1,...,N }, tn ∈ T = [0, 1), be N samples of a

continuous closed curve r(t ) = (r1(t ), r2(t )), t ∈ T, where the coordinate functions r1and r2

are 1-periodic. We look for the optimal closed curve roptthat best connects its N (possibly

noisy) observed data yn= (y1,n, y2,n) ≈ r(tn), for n = 1,..., N .

Variational Approach

We consider the variational problem

ropt= Ã r1,opt r2,opt ! = arg min (r1,r2) Ã _N X n=1 ³ ¡ y1,n− r1(tn)¢2+¡ y2,n− r2(tn)¢2 ´ + λ³kLαr1k2L2+ kLαr2k 2 L2 ´ ! , (2.3.6) where Lαis the differential operator given by 2.3.3 and the parameterλ quantifies the tradeoff between the fidelity to the data and the regularization constraint. The solution roptof (2.3.6) is

unique and its coordinate functions ri ,opt, i = 1,2, are periodic exponential splines associated

to the operator (L∗_αLα) (see Appendix B.3). Statistical Approach

We change perspective and consider that the coordinates functions ri, i = 1,2, of r are real

periodic Gaussian processes with zero-mean and are related to L_α. We are looking for the optimal estimator ropt= ˜rMMSEoverT of r, in the sense that each of its coordinate functions

˜

ri ,MMSE, i = 1,2, satisfies

˜

ri ,MMSE= arg min

˜ ri Ehkri− ˜ri(·|{yi ,n}n∈{1,...,N })k2L2 i (2.3.7) 20

2.3. Splines

(a) L_α= D + I and N = 40. (b) L_α= D2+ 4π2I and N = 15.

(c) L_α= D and N = 40. (d) L_α= D2and N = 40.

Figure 2.8 – Reconstruction of stochastic closed curves for different operators Lα. Solid blue line: unknown stochastic curve r; dashed red line: estimator ˜rMMSEof r; Dots: samples.

among the estimators ˜ri(·|{yi ,n}n∈{1,...,N }) of ri such that ˜ri(tn|{yi ,n}n∈{1,...,N }) for n = 1,..., N .

The solution of (2.3.7) is unique and is a periodic exponential spline associated to (L∗_αLα) (see Appendix B.4). It means that the unique optimal closed curve ˜rMMSEhas coordinate functions

that are periodic splines. In Figure 2.8, we optimally reconstruct (in the sense of (2.3.7)) stochastic closed curves from their observed data {yn}n∈{1,...,N }for different operators Lα.

3

Parametrization with Local Refine-

ment

The geometric representation of active contours/surfaces determines their ability to approxi- mate the shape of interest as well as the speed of convergence of related optimization algo- rithms. It is thus of great interest that one can allocate additional degrees of freedom to the curve/surface only where an increase in local detail is required.

A crucial aspect in the development of local refinement algorithms is to refine specific re- gions while keeping the rest of the curve/surface unchanged. This local refinement is not inherent to standard methods as Non-Uniform Rational Basis Splines (NURBS) or classical parametrizations of curves/surfaces. Existing methods to insert points at specific locations were developed in [90–94].

In this chapter, we present a new parametrization for curves and tensor-product surfaces where the degrees of freedom (i.e., control points) can be locally increased without altering the shape of the curve/surface. In a segmentation context, these additional control points then allow to locally deform the shape with better accuracy. We locally improve the level of detail of the parametric model by inserting basis functions at specific locations. Our approach is generic and relies on refinable and scaling functions that are related to wavelets [95, 96]. Among all scaling and refinable functions, throughout this chapter we focus on the one of compact support, as it is a desirable property in practice.

This chapter is based on our publications [68, 69], in collaboration with D. Schmitter and M. Unser. The chapter is organized as follows: In Section 3.1 we fix the notations. We review refinable functions in Section 3.2. Finally, the main contribution is described in Section 3.3, where we propose a novel and generic parametrization of closed curves, as well as tensor- product surfaces, that are locally refinable.

Chapter 3. Parametrization with Local Refinement

Figure 3.1 – Refinement of a quadratic B-spline with a refinement factorρ equal to 2 (solid blue line). It can be expressed as a linear combination of four contracted versions (green dashed lines) of integer-shifted quadratic B-splines.

3.1 Notations

We denote by t a continuous parameter inR. We defineα = (α1,α2, . . . ,αL) where theαncan be

non-distinct, and denote by L_αn the multiplicity of the elementαn∈ α, for n = 1, . . . , L. We de-

note by ϕ_α a function that reproduces exponential polynomials in

span©eαnt_{, . . . , t}Lαn−1_eαntª

n∈{1,...,L}, i.e., for all i ∈ {0,...,Lαn−1} there exists a sequence {c[m]}m∈Z

such as

tieαnt

= X

m∈Z

c[m]ϕ_α(t − m). (3.1.1)

Exponential B-splines (see Section 2.3.1) are examples of such functions [81].