Journal of Electrical and Computer Engineering

Volume 2017, Article ID 7948571, 6 pages

https://doi.org/10.1155/2017/7948571

## The 2D Spectral Intrinsic Decomposition Method Applied to Image Analysis

Laboratoire de Traitement de l’Information et Systèmes Intelligents, Ecole Polytechnique de Thiès, BP A10, Thiès, Senegal

Correspondence should be addressed to Oumar Niang; ns.dacu@gnaino

Received 5 July 2016; Revised 4 January 2017; Accepted 18 January 2017; Published 20 December 2017

Academic Editor: M. Jamal Deen

Copyright © 2017 Samba Sidibe et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose a new method for autoadaptive image decomposition and recomposition based on the two-dimensional version of the Spectral Intrinsic Decomposition (SID). We introduce a faster diffusivity function for the computation of the mean envelope operator which provides the components of the SID algorithm for any signal. The 2D version of SID algorithm is implemented and applied to some very known images test. We extracted relevant components and obtained promising results in images analysis applications.

#### 1. Introduction

The need of components extraction and reconstruction in signal and image processing in time frequency analysis is very strong for many fields of application. Notorious methods that have been proposed include Fourier technics, wavelet decomposition, and Empirical Mode Decomposition. While Fourier transform is localized in frequency, wavelets are localized in both time and frequency; EMD is autoadaptive. EMD decomposes a signal in AM-FM components called Intrinsic Mode Functions (IMF) and a residue. This nonlinear and nonstationary decomposition works on 1D signals [1] and 2D signals such as images [2, 3]. The EMD algorithm is based on a procedure called sifting process which iteratively uses the upper and lower envelopes to extract IMFs. To create a mathematical model to compute envelopes directly, an envelope operator has been proposed in [4] and from this operator a new decomposition method called Spectral Intrinsic Decomposition, SID, was proposed in [5].

The SID method allows decomposing any signal into a superposition of Spectral Proper Mode Functions (SPMFs) [5]. This method has been presented in a 1D version and depends on an operator interpolating the characteristics points of the signal to be decomposed. In this paper, the two-dimensional version of the Spectral Intrinsic Decomposition for images analysis is introduced. An algorithm for a faster spectral decomposition is proposed and illustrated with some images. We first recall the SID principle in one dimension and propose a faster method to determine the signal characteristics points in Section 2. Than an algorithm of the two-dimensional SID is presented in Section 3. Applications on grayscale images are depicted in Section 3.1.

#### 2. The Spectral Intrinsic Decomposition Method

The Spectral Intrinsic Decomposition Method decomposes any signal into a combination of eigenvectors of a Partial Differential Equation (PDE) interpolation operator as presented in [5, 6].

##### 2.1. The PDEs System Interpolator

For a given signal , the upper and lower envelope are the asymptotic solution of the following PDEs system: is the tension parameter which ranges from 0 to 1. and are diffusivity functions for upper and lower envelope which are equal to zero at characteristic points of and range from zero to one. based on Maximum Curvature Points (MCP) can be computed as follows:where denotes the sign function.

Equation (1) is resolved numerically in its discrete implicit unconditionally stable scheme as follows:where is the time step, is signal value at step , and is a matrix formed with finite difference approximation coefficients of second- and fourth-order differential operators (resp., and ), as , with being the diagonal matrix constructed with discrete version of stopping function values , exactly,

So the explicit form leads to the following numerical resolution:with being the identity matrix. Finally (1) can be decomposed into a linear system from implicit numerical scheme (4) by

is given by

The operator matrix, , has real-valued eigenvalues that are always greater than or equal to 1. Then, eigenvalues, , of are always smaller than or equal to ; see [4].

##### 2.2. On the Asymptotic Solution

Iterative scheme (5) can be rewritten in terms of initial solution asAfter convergence (see [7]), the asymptotic solution, , is given by

Let be a matrix of ’s sequence of eigenvectors and a diagonal matrix having ’s sequence of eigenvalues , at the diagonal. So we have the following decomposition It is easy to see that

So, the asymptotic solution in (7) is given by

The asymptotic eigenvalue matrix is a diagonal matrix with eigenvalues only at loci where matrix is zeroed, and , where . Finally, the asymptotic solution of the PDE interpolator system is a linear combination of fixed vector point of upper and lower envelope operators.

##### 2.3. A Faster Stopping Function for Discrete Signal

For image processing we will consider region boundaries as characteristic points. The characteristic points of the upper envelope will be the local maximums and the limits of the regions where the value of the gray level of the pixel is equal to or greater than the gray level of all the pixels in their neighborhood represented, for example, by a rectangular window.

We define the diffusion function for lower envelope to be equal to everywhere except in characteristic points of the lower envelope where it will be equal to .

Similarly the characteristic points of the lower envelope are local minimums and region boundaries where the pixel value is equal to or less than the gray level of all pixels in their neighbors. We define the diffusion function for upper envelope to be equal to everywhere except in characteristics points of the upper envelope where it will be equal to .

The diffusivity function called stopping function is calculated by using morphological dilation and erosion operations [8].

Let be a structured element; the grayscale dilation of by at is given byThe grayscale erosion of by at is given by

is equal to at local maxima and when is locally constant; is equal to at local minima and when is locally constant.

is zeroed at points which are invariant to morphological dilation and variant to morphological erosion; is equal to 1 for any other point.

Similarly, is zeroed at points which are invariant to morphological erosion and variant to morphological dilation; is equal to 1 for any other point.

LetWe have

functions are faster than to compute and give satisfying results for the computation of envelope operators for real images (Table 1).