^{1}

^{1}

^{1}

^{1}

^{1}

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.

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 [

The SID method allows decomposing any signal into a superposition of Spectral Proper Mode Functions (SPMFs) [

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

For a given signal

Equation (

So the explicit form leads to the following numerical resolution:

The operator matrix,

Iterative scheme (

Let

So, the asymptotic solution in (

The asymptotic eigenvalue matrix

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

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

The diffusivity function called stopping function

Let

Similarly,

Let

Image | Width | Height | Time for |
Time for |
---|---|---|---|---|

Figure |
300 | 300 | 0,4684466 | 0.320695 |

Figure |
400 | 266 | 0,826974833 | 0,2877035 |

Figure |
400 | 272 | 0,807968 | 0,284108 |

Figure

The effectiveness of envelope computation using

In the following,

The Spectral Intrinsic Decomposition procedure is defined as the calculus of all the SPMFs for a given signal.

This decomposition is intrinsic and depends only on the position of the characteristic points of

The SID is adaptive and depends on the position of the characteristics points of the signal. It is autoadaptive and works for nonlinear and nonstationary signal. SID can decompose an IMF and can be used to separate mixing mode [

However, the main disadvantage of the proposed SID algorithm is the computation time when the size of signal is a large. This is due to matrix inversion in the algorithm. Thus a faster algorithm is proposed in the next section.

In this section we present, in Algorithm

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

(

Let us consider an image as represented by a matrix

Each line can be recomposed to build a matrix

The elementary components of a spectral decomposition are the modulation of eigenvectors by their coefficients as can be seen in (

The range of smallest eigenvalues catches higher frequencies contents of the reconstructed signal with the smallest modulated amplitude.

So we can associate components which have similar frequency and amplitude by summing the components that have same eigenvalues; this method works well for signal in one dimension. For images, it is possible numerically to have missing eigenvalues in many lines. Hence, to avoid this drawback, elementary components which have the same eigenvalues will be associated with the same belonging to a specific range of eigenvalues.

In the following, 2D SID is applied to very known images test to demonstrate the ability of this pectoral intrinsic decomposition for images analysis, particularly in components extraction. In Figures

Decomposition (a), component extraction and representation of high frequency component (

Decomposition (a), component extraction and representation of high frequency component (

Decomposition (a), component extraction and representation of high frequency component (

In this paper, we have presented a new method for autoadaptive image representation called two-dimensional version of the Spectral Intrinsic Decomposition. A new faster diffusivity function in the computation of the mean envelope operator is also provided. In future works, we will investigate how to use the Spectral Proper Mode Functions (SPMFs) to do signals classification or to treat other aspects of image processing, like edge detection, segmentation, and so on.

The authors declare that they have no conflicts of interest.