Nonparametric Image Segmentation Using Rényi’s Statistical Dependence Measure

Abstract

In this paper, we present a novel nonparametric active region model for image segmentation. This model partitions an image by maximizing the similarity between the image and a label image, which is generated by setting different constants as the intensities of partitioned subregions. The intensities of these two images can not be compared directly as they are of different modalities. In this work we use Rényi’s statistical dependence measure, maximum cross correlation, as a criterion to measure their similarity. By using this measure, the proposed model deals directly with independent samples and does not need to estimate the continuous joint probability density function. Moreover, the computation is further simplified by using the theory of reproducing kernel Hilbert spaces. Experimental results based on medical and synthetic images are provided to demonstrate the effectiveness of the proposed method.

Appendix

In this section, we give the basic theories of RKHS we have used in this paper.

Let E be an arbitrary set and H be a Hilbert space of real functions on E. We say that H is a RKHS if the linear map F _x :f→f(x) is a bounded functional for any x∈E.

By this definition, F _x ∈H ^⋆, which is the dual space of H. Therefore, Reisz representation theorem shows that there exists a unique K _x ∈H, such that

$$f(x) = \langle F_x, f\rangle = \langle K_x, f\rangle,\quad \forall f\in H.$$

Define K:E×E→ℝ as K(x,y)=〈K _x (⋅),K _y (⋅)〉. It is easy to see that K has the following properties:

• 1. K is symmetric: K(x,y)=K(y,x).

• 2. Reproducing property: f(x)=〈K(x,⋅),f(⋅)〉.

• 3. K is positive definite: $\sum_{i,j}^{n}a_{i}a_{j}K(x_{i},x_{j})\geq 0$ holds for all x ₁,x ₂,…,x _n ∈E, a ₁,a ₂,…,a _n ∈ℝ and the equality holds if and only if a _i =0, i=1,2,…,n.

We call such an K the reproducing kernel for the Hilbert space H.

On the other hand, suppose K:E×E→ℝ is symmetric and positive definite, then according to the Moore-Aronszajn theorem [26], there is a unique Hilbert space of functions on E for which K is a reproducing kernel. In fact, let H ₀(E) be the linear span of the functions {K(x,⋅)|x∈E} and define the inner product in H ₀(E) to be

$$\Biggl\langle \sum_{i=1}^na_i K(x_i, \cdot), \sum_{j=1}^mb_j K(y_i, \cdot)\Biggr\rangle = \sum _{i=1}^n\sum_{j=1}^ma_i b_j K(x_i,x_j).$$

Let H(E) be the completion of H ₀(E) with respect to this inner product. It is not difficult to check that H(E) is the unique RKHS with reproducing kernel K.

For the particular case E=ℝ. Let C ₀(ℝ) be the space of real valued continuous functions vanishing at infinity with the supremum norm. Then we have the following result

$$\sup_{f,g\in V(\mathbb{R})} CC\bigl(f(X),g(Y)\bigr)=\sup_{f,g\in H_0(\mathbb{R})} CC\bigl(f(X),g(Y)\bigr),$$

(51)

where V(ℝ) is the space of all real Borel measurable functions with finite positive variance. This is the main result we have used in this paper and (17), (18) follows directly from this result.

The proof of this result can be obtained through the following three steps (We omit the details here).

• 1. H ₀(ℝ) is dense in C ₀(ℝ).

• 2. Let V(B) be the space of all real bounded Boreal measurable functions, then

  $$\sup_{f,g\in V(B)} CC(f(X),g(Y))=\sup_{f,g\in C_0(\mathbb{R})} CC(f(X),g(Y)).$$
• 3.
  $$\sup_{f,g\in V(\mathbb{R})} CC(f(X),g(Y))=\sup_{f,g\in V(B)} CC(f(X),g(Y)).$$