Fernando Merchan
email: fernando.merchan@laps.ims-bordeaux.fr

Flavius Turcu
email: flavius.turcu@laps.ims-bordeaux.fr

Mohamed Najim
email: mohamed.najim@laps.ims-bordeaux.fr

OUTER FACTOR 2-D MA MODELS FOR PURELY INDETERMINISTIC FIELDS AND WOLD-TYPE TEXTURE DECOMPOSITIONS

In this paper, we propose a new method to compute the parameters of finite approximations of 2-D MA infinite models associated with purely indeterministic stationary fields, by using spectral factorizations. This method is developed in the framework of model based approaches for texture analysis and synthesis. The proposed parameter estimation method is then exploited to derive a scheme to separate the indeterministic and the deterministic components of a texture in Wold decomposition based models. Unlike existing approaches, the resulting decomposition scheme does not require any support detection technique in the spectral domain. exp

INTRODUCTION

Nowadays, a great deal of interest has been paid to texture analysis and synthesis, especially in the fields of image compression, image resolution enhancement, etc. [START_REF] Ghouti | Image compression using modelling[END_REF][START_REF] Komodakis | Image completion using global optimization[END_REF]. In that case, when looking at model based approaches, the parameter estimation issue must be addressed.

Among the approaches of image modeling that use random processes, one consists in using texture models based on the 2-D Wold decomposition [START_REF] Helson | Prediction theory and Fourier series in several variables, I,II[END_REF][START_REF] Francos | A Unified Texture Model Based on a 2-D Wold Like Decomposition[END_REF]. This result extends the original 1-D result [START_REF] Wold | A study in the analysis of stationary time series[END_REF] which was initially used for time series. In this 2-D approach, the texture is represented as a regular stationary field that can be decomposed in orthogonal components:

A purely indeterministic field characterized by a continuous spectral density, a deterministic field which is itself decomposed in two orthogonal components: an harmonic field corresponding to the 2-D harmonics components (i.e. sharp peaks in the frequency domain) and an evanescent field which represents periodic fields characterized by straight lines in the frequency domain.

In [START_REF] Francos | A Unified Texture Model Based on a 2-D Wold Like Decomposition[END_REF], Francos et al. present a generic model for each Wold component. It is known that the most general model for the purely indeterministic component is the moving average (MA) representation [START_REF] Helson | Prediction theory and Fourier series in several variables, I,II[END_REF]. Then, under some assumptions 1 , an autoregressive (AR) representation is suggested in [START_REF] Francos | A Unified Texture Model Based on a 2-D Wold Like Decomposition[END_REF].

In order to carry out the parameter estimation, the Wold components are usually separated. Thus, Francos [START_REF] Francos | A Unified Texture Model Based on a 2-D Wold Like Decomposition[END_REF] proposes an iterative procedure to detect the support of the deterministic field in the texture periodogram. It consists in searching the sharpest peaks. For this purpose, a threshold is introduced and initially set to the maximal value of the periodogram. Then, it is gradually lowered as long as only sharp peaks are detected. The purely indeterministic field is obtained by subtracting the estimated deterministic field from the original texture. However, it is difficult to define a stopping rule. In addition, this approach cannot be used for a large variety of textures.

A more robust method for Wold decomposition is presented by Liu and Picard in [START_REF] Liu | A Spectral 2-D Wold Algorithm for Homogeneous Random Fields[END_REF]. Their algorithm is based on the intrinsic fundamental-harmonic relationships to identify harmonic frequencies and on the Hough transform to detect spectral evanescent components. This method has the drawback of searching of the support of each deterministic component individually.

In this paper, two aspects of the Wold-based models for texture analysis are studied: modeling and parameter estimation of the purely indeterministic field, separation of the deterministic components and the purely indeterministic ones.

Our contribution is twofold. Firstly, we propose to represent the purely indeterministic field as a finite approximation of an infinite 2-D MA expansion. The MA parameter estimation is then based on the Taylor representation of a two-variable analytic function, called outer function, whose square absolute value on the bi-torus of this function equals almost everywhere the density of the spectral measure of the purely indeterministic field. Thus, the proposed MA representation differs from the limited order 2-D MA models whose parameters are obtained from the truncated correlation function by algebraic methods (i.e. as proposed in [START_REF] Francos | Parameter Estimation of Two-Dimensional Moving Average Random Fields[END_REF]).

Secondly, we present a new decomposition scheme. Unlike existing approaches, the proposed scheme does not require any support detection techniques in the spectral domain. This scheme is based on the proposed MA parameter estimation method applied to the mixed texture. The estimated parameters are then exploited in a filtering and separation procedure.

The remainder of the paper is organized as follows: section 2 deals with the theoretical background and the proposed MA parameter estimation scheme. In section 3, the implementation procedure is presented. In section 4, we derive the decomposition scheme to separate the deterministic and the purely indeterministic sub-fields. An example is provided to illustrate the algorithm.

THEORETICAL BACKGROUND

Let us first recall how infinite MA representations can be theoretically associated to purely indeterministic fields. Given the Helson-Lowdenslager extension [START_REF] Helson | Prediction theory and Fourier series in several variables, I,II[END_REF] of the Wold decomposition theorem, the field ) , ( n m y is uniquely decomposable as the orthogonal sum:

) , ( ) , ( ) , ( n m v n m w n m y + = (1)
where { }

∫ = = Ω T dm i L ϕ σ . ( 2
)
The purely indeterministic field { }

∑ - - = ) , ( ) 0 , 0 ( ) , ( ) , ( ) , ( l k l n k m i l k a n m w ≺ , (3)
driven by the white noise { }

) , ( n m i
, where ) , ( l k a are some square summable coefficients. Thus, the transfer function f associated to the model: , one can make any particular choice of MA parameters provided they satisfy [START_REF] Rudin | Function theory in polydiscs[END_REF], and this choice affects only the driving white noise.

∑ ≥ = ) 0 , 0 ( ) , ( 2 1 2 1 ) , ( ) , ( l k l k z z l k a z z f (4)
For practical purposes (see also [START_REF] Gavarecki | Proper Moving Average Representations and Outer Functions in Two Variables[END_REF], [START_REF] Korezlioglu | Spectral factorization of wide Sense stationary processes on Z2[END_REF]), one can assume that the representation (3) has all the coefficients in the first quadrant of 2 Z . The function f may thus be considered analytic, and hence belongs to the Hardy space ) ( 2 2 D H . With this additional assumption, it is well known [START_REF] Rudin | Function theory in polydiscs[END_REF] that all the solutions of the equation ( 5) are factors of the form

) , ( ) , ( ) , ( 2 1 2 1 2 1 z z F z z u z z f = (6)
where u is any inner function (i.e. u has modulus 1 a.e. on 2 T ) and F is the unique outer function (see [START_REF] Rudin | Function theory in polydiscs[END_REF]) such that A special solution for the equation ( 5) is therefore the outer function F itself (i.e. 1 ≡ u in ( 6)), which has two advantages: the first is that F is bounded and has no zeroes in the unit bidisk, meaning that the corresponding MA filter (3) and its inverse are both BIBO-stable. The second advantage is that F is uniquely determined by the density ϕ , through the formula (see [START_REF] Rudin | Function theory in polydiscs[END_REF]):

        - + - + = ∫ ∫ π π ϕ π

PROPOSED METHOD

The proposed method uses the numerical computation of the outer factor F described above, through the following approach: consider an image representing a texture, regarded as a realization of a stationary random field. Under a standard ergodicity assumption, the density of the spectral measure of the underlying random field can be assimilated to the power spectral density (PSD) of the texture. Using the PSD, one can perform a discrete computation of a finite set of coefficients of the outer function F associated to this density, and hence obtain a finite 2-D MA approximation of the MA model (3).

The proposed estimation method is resumed in the following three steps:

∑ = n m n m n m z z a z z F , 2 1 , 2 1 ) , ( , ( 1 0 ,
< < = i j i i r e r z i θ ) (9)
by using the Cauchy formula:

In practice, the PSD is estimated via the periodogram: The Taylor coefficients are then obtained as follows: for some chosen 1 0 < < r .

In figure 1 an example of reconstruction is presented. The original texture (a) is synthesized using a 4 6× 2-D MA filter driven by a white noise field. The reconstruction texture (b) is generated using a 4 6× 2-D MA model as described above and an arbitrary white noise 2 .

  field of zero-mean random variables with finite variance, in the Hilbert space ) ( 2 Ω = L H , for some probability space Ω , and denote by y µ the spectral measure of the field y on the bi-torus

  indeterministic and deterministic fields, with their spectral measures denoted w µ and v µ .It is known that, if { } so-called innovations of the field with respect to some fixed total group order ≤ on 2Z . More precisely, consider the field { }

  linear combination of innovation samples. Therefore it admits an infinite 2-D MA representation:

  the MA model (3), i.e. the Taylor coefficients of F , are then easily obtained by the classical Cauchy formula.

3 .

 3 Estimation of the Taylor coefficients of the outer function

The purely indeterministic field has an AR representation if its spectral density is positive in the unit bicircle and analytic in some neighborhood of it.

A NEW DECOMPOSITION SCHEME

In the following, we derive a decomposition approach to separate the deterministic and purely indeterministic components of a texture. Equation (1) can be written as follows:

) , ( the MA parameters of the purely indeterministic field w. Taking the z transform 3 of equation ( 14) yields:

where F(z 1 ,z 2 ) is the outer factor defined as follows:

Now, by filtering the original image with ) , (

i.e. the sum of the z-transform of the white noise u and of some deterministic field. Thus, to compute the deterministic field v in the original texture, one has to filter out the white noise u from (17), and then to restore v by filtering again the result with F. See figure 2.

As mentioned in the theoretical section, the fact that F is an outer function (and hence has no zeroes in the bi-disk) insures that the inverse filtering (17) is stable. This motivates the particular choice of the outer factor

Step 1: Filter the FFT of the original image with the transfer outer function Step 2: Estimate the variance of the white noise u

Step 3: Filter out the white noise. To this end, a binary mask is generated, by using the threshold . In practice we choose α between 4 and 5.

The resulting mask is dilated by using a standard 4-connected structuring element. The dilated mask is used as a filter in the spectral domain (see figure 4).

Step 4: Filter the result of Step 3 with the outer factor F, computed as indicated in the previous section, and obtain an estimation of the deterministic part of the texture.

An example is presented in figure 3, illustrating the decomposition of a texture into its deterministic and purely indeterministic parts. In figure 3 The dilated binary mask is used as a filter in the frequency domain for recovering the deterministic component (i.e. an evanescent field).

CONCLUSIONS

In this paper, we propose to use the outer factor of the spectral power density of a texture to compute a 2-D MA model for the purely indeterministic part of the texture. The method is then used to derive a Wold decomposition scheme to separate the purely indeterministic and the deterministic parts of a texture. Unlike existing approaches, our decomposition scheme does not require any support detection techniques in the spectral domain.