NON-STATIONARY TEXTURE SYNTHESIS FROM RANDOM FIELD MODELING Abdourrahmane M.

Atto1, Zhangyun TAN1, Olivier ALATA2, Maxime MOREAUD3

2

France France

LISTIC, EA 3703, University of Savoy, Polytech Annecy-chambery Lab. Hubert Curien, CNRS UMR 5516 - Jean Monnet University of Saint-Etienne 3 IFP Energies Nouvelles, BP 3, 69360 Solaize 1

France

ABSTRACT

This paper presents a generalized non-stationary and fractional model for texture synthesis. The model is based on convolution and modulation operations of fractional Brownian elds and its associated spectral representation contains many poles with unit norm. Synthesized textures generated from this model can exhibit several non-trivial fringes which can be visualized in natural textures such those involved in high resolution transmission electron microscopy. Index Terms— Texture Synthesis ; Random elds ; Fractional Brownian elds ; Non-Stationary textures ; Transmission Electron Microscopy 1. INTRODUCTION

One-Dimensional (1D) fractional Brownian motions that are non-stationary random processes, have shown relevancy in modeling uid particle interactions and chemical cell structures, etc. Their 2D generalizations to model spatial random processes (or 2D random elds) have shown eciency in texture analysis [1, 2, 3, 4] and texture synthesis [5, 6, 7]. This justi es their interest in this paper. The class of multifractional Brownian motions derived in [8], [9] is an extension of fractional Brownian motions associated with an evolving covariance structure in time [10]. However, the parameterization of this class is too intricate for making possible an intuitive spectral characterization. Thus, we address hereafter an extension of fractional Brownian motions with respect to speci cations in the spectral domain. The model generalization of fractional Brownian elds proposed in this paper has the following properties:  an arbitrary number of spectral singularity points,  an arbitrary Hurst parameter, associated to each spectral pole, which characterizes the exponential decay of the spectrum in its neighborhood. To this aim, a fractional-type random eld is obtained from convolution and modulation operators over a sequence This work has been supported by a research grant from the ARC6 of region Rh^one-Alpes, France.

of fractional Brownian random elds associated with an arbitrary sequence (being either random or deterministic) of Hurst parameters. By doing this way, we speci cally describe isotropic random eld models involving several spectral poles with unit norm and dierent power decays at these poles, whereas the standard isotropic fractional Brownian eld admits a spectral representation with only one pole located at zero frequency. The consequence is a model with rich spatial structure (or textural) information. Such a spectral content can be found in many textures issued from materials. In particular, we will explore the case of High Resolution (HR) Transmission Electron Microscopy (TEM) for visualizing soot nanostructures. The paper is organized as follows: Section 2 provides, in a discrete 2D framework, some preliminary results on isotropic fractional Brownian elds. Section 3 presents the generalized fractional random eld model proposed in the paper. It addresses their spectral characterization. Section 3 also highlights the capability of the generalized fractional random eld model in generating non-trivial stochastic textures. Section 4 discusses the contribution of the generalized model, in a context of HR-TEM images and concludes the paper. 2. ISOTROPIC FRACTIONAL BROWNIAN RANDOM FIELDS 2.1. Spectral properties of fractional Brownian fields

Let us consider a zero-mean real valued isotropic fractional Brownian eld FHq (k, `) with Hurst (long-term memory) parameter Hq , 0 < Hq < 1, where k, ` ∈ Z. A spectrum can be associated to FHq by considering transforms having stationarization properties [1]. This spectrum, de ned by association1 , can be written in the form γFHq (u, v) = ξ(Hq )

1 (u2 + v2 )Hq +1

(1)

1 The spectrum associated with F Hq is not directly the Fourier transform of its lag-based autocorrelation function since this autocorrelation function, RFHq (k, `, m, n) used in Eq. (4), cannot be written as a function of the lags (k − m) and (` − n).

We consider a sequence of Hurst parametersNHQ = ) of Q

where ξ(Hq ) is given by ξ(Hq ) =

2−(2Hq +1) π2 σ2 . sin(πHq )Γ 2 (1 + Hq )

(2)

Equation 1 shows that The spectrum of FHq exhibits a singular frequency point located at the frequency grid origin (zero frequency point). The following provides generalized isotropic fractional elds which can admit many spectral singularities. 2.2. Spectrum-Pole-Shifted non-stationary isotropic fractional Brownian fields

We now address the generation of a non-stationary fractional random eld involving a spectral pole located at an arbitrary frequency point in the grid [0, π] × [0, π]. In this respect, the non-stationary random eld FHq is modulated by a complex exponential `wave' associated with a frequency point (uq , vq ) ∈ [0, π] × [0, π]. Let us de ne a random eld GHq by setting: GHq (k, `) = eiuq k eivq ` FHq (k, `).

(3)

The random eld GHq is centered and its autocorrelation i h function RGHq (k, `, m, n) = E GHq (k, `)GHq (m, n) is: RGHq (k, `, m, n) = RFHq (k, `, m, n)eiuq (k−m) eivq (`−n) . (4)

This autocorrelation function involves a stationary separable exponential term multiplying RFHq . Note, from the expansion of RGHq given in Eq. (4), that GHq is non-stationary: its autocorrelation function involves contributions that cannot reduce to lag terms (k − m, ` − n). Similarly to FHq , one can associate the spectrum γGHq to GHq , where γGHq (u, v) = γFHq (u − uq , v − vq ), thus: 1 . γGHq (u, v) = ξ(Hq ) ((u − uq )2 + (v − vq )2 )Hq +1

(5)

From Eq. 5, the spectrum γGHq is unbounded near frequency point (uq , vq ) (spectral pole with unit norm). This spectral pole is a shifted version of the pole of γFHq . In the neighborhood of (uq , vq ), the spectrum γGHq has the same exponential decay as γFHq around zero. 3. Q-FACTOR GENERALIZED FRACTIONAL BROWNIAN FIELDS (GFBF) 3.1. Generalized fractional Brownian fields with Q spectral poles

The fractional Brownian eld FHq admits one spectral pole located at zero frequency point. Its modulated version, GHq , admits one spectral pole located at frequency (uq , vq ). In the following, a method for constructing generalized fractional elds is provided. The generalization concerns the number of poles associated with the spectral density of the fractional Brownian eld.

{H1 , H2 , . . . , HQ } and de ne a convolution (notation non-stationary elds G = {GHq , q = 1, 2, . . . , Qg: EHQ =

Q O

GH q

(6)

q=1

where sequence G is assumed to be composed with independent random elds. Random eld EHQ , hereafter referred as Q-factor Generalized Fractional Brownian Field (GFBF) is non-stationary. From some straightforward calculus involving some properties of the convolution operator, we derive the spectral density of EHQ as: γEHQ (u, v) ∝

Q Y q=1

1 [(u − uq )2 + (v − vq )2 ]Hq +1

(7)

Q

with proportionality factor equal to Q q=1 ξ(Hq ). As it can be seen from Eq. (7), the GFBF is associated with a spectrum admitting several poles and these poles are not necessarily located at the zero of the frequency grid. 3.2. Random field Q-factor GFBF based texture synthesis

The present paper addresses texture synthesis from random eld modeling (stochastic textures). The main diculty for the synthesis of such textures is linked to the capability of the random eld considered in  encompassing intricate spatial dependencies and non-

stationarities,

 having a simple parameterization that eases character-

ization and synthesis from random number generators.

Table 1. Overview of the algorithm for Q-factor GFBF syn-

thesis. Select, a sequence HQ = {H1 , . . . , HQ } of Hurst parameters; Select, a sequence WQ = {(u1 , v1 ), . . . , (uQ , vQ )}of poles. Do, for k ∈ {1, 2, 3, . . . , Q}: (E0 is a dirac) Ek ← Generate[Field, Hk ] Ek ← Modulate[Ek , (uk , vk )] Ek ← Convolve[Ek−1 , Ek ]

Table 1 provides the procedure used for generating Qfactor GFBF from Eq. (6). In this table, the \Generate" function provides a realization of a standard fractional Brownian eld (GFBF synthesis) [5]. The \Modulate" function applies Eq. (3) over a discrete frequency grid. The \Convolve" function applies the standard discrete convolution operation. Dierent Q-factor GFBF realizations are provided in Figure 1. These realizations exhibit several non-trivial structures (that do not reduce in line-wise or column-wise delineations).

Gaussian poles and uniform Hurst parameters EH2

EH3

EH4

Gamma random poles and uniform Hurst parameters EH2

EH3

EH4

EH8

EH10

EH16

EH19

EH21

EH22

Uniform poles and uniform Hurst parameters EH6

EH7

EH8

Uniform poles and constant of Hurst sequence EH6

EH6

EH6

Fig. 1. Q-factor GFBFs EHQ for dierent values of Q. Poles and Hurst parameters are generated from random variables distributed as Gaussian, Uniformly and Gamma respectively. Poles are distributed in [0, π/2] × [0, π/2] and Hurst parameters are distributed in ]0, 0.5[.

Gaussian poles and uniform Hurst parameters SW EH2

SW EH3

SW EH4

Uniform poles and uniform Hurst parameters SW EH6

SW EH7

SW EH8

Uniform poles and constant of Hurst sequence SW EH6

SW EH6

SW EH6

Gamma random poles and uniform Hurst parameters SW EH2

SW EH3

SW EH4

SW EH8

SW EH10

SW EH16

SW EH19

SW EH21

SW EH22

Fig. 2. Wavelet spectra SW EHQ of Q-factor GFBFs EHQ texture images given in Figure 1. Spectra are given in [0, π/2] × [0, π/2]. When poles are very close, they fused in one 'big' pole. Poles associated with small Hurst parameters tend to be

dominated (display setup) by other poles. Spectra are estimated from images in order to highlight the impacts of sampling and estimation.

Wavelet Spectrum SW ITEM of ITEM on [0, π] × [0, π]

HR-TEM image, ITEM

I1 = Z(ITEM ,1)

SW I1

I2 = Z(ITEM ,2)

SW I2

I3 = Z(ITEM ,3)

SW I3

Fig. 3. HR-TEM image ITEM (top-left) and its wavelet spectrum SW ITEM (top-right). Bottom: some Zooms (Z) on regions of image ITEM for texture visualization and the corresponding spectral information. The frequency grid is [0, π] × [0, π] for SW ITEM and [0, π2 ]×[0, π2 ] for SW I1 , SW I2 , SW I3 (most textural information is concentrated on the latter grid, see for instance SW ITEM ). 4. DISCUSSIONS AND CONCLUSION

Figure 2 provides spectra of GFBF textures given in Figure 1. The spectra have been computed from the wavelet packet method given in [1]. Figure 2 highlights the spectral peaks and the exponential decay in the neighborhood of these peaks, of the synthesized textures from GFBF EQ . Many natural textures have spectra which possess such characteristics. In [11], the spectral peaks in locally calculated Fourier spectrum, are used to detect crystallized ceria nanoparticles in silica mesoporous supports. For instance, soot nanostructure visualized with HR-TEM (see Fig. 3) exhibits such spectral characteristics as it can be seen in the spectra given in Figure 3. These spectra have also been obtained with the wavelet packet spectrum estimator given in [1]. One can note that the SW ITEM and SW I3 spectra exhibit many non-uniformly distributed peaks (see Fig. 3). Modeling these textures by a Q-factor GFBF thus requires the estimation of dierent parameters: the location of frequencies associated with poles and their number; the local exponential decays near those frequencies relating the Hurst parameters. These estimations are challenging and will be addressed in a future work (synthesis based on real data spectral characteristics). Estimation of the poles can be performed by using algorithms such as matching pursuit (poles are rare and with large amplitudes). Estimation of the Hurst parameters can be addressed by using log-regression methods.

We plan to investigate Q-factor GFBF parameter estimation in future work. A visual analysis highlights that Q-factor GFBFs E21 and E22 of Figure 1 present textural structures similar with some fringes observed on HR-TEM image given by Figure 3 (see the zoomed versions). The spectra SW EH21 and SW EH22 given by Figure 2 con rm this remark and show the relevancy of considering generalized fractional models for texture description. This assumption will be carefully checked, in a future work, by addressing parameter estimation of GFBF models on HR-TEM databases. We thus conclude this paper by enhancing the speci city of the Q-factor GFBF: these models are non-stationary and associated with several spectral poles, which makes them powerful tools for random eld synthesis. Indeed, textures that are synthesized from Q-factor GFBF have shown appealing structural content and similarities with some fringe structures encountered in HR-TEM textures. 5. REFERENCES

[1] A. Atto, Y. Berthoumieu, and P. Bolon, \2-d wavelet packet spectrum for texture analysis," Image Processing, IEEE Transactions on, vol. 22, no. 6, pp. 2495{ 2500, 2013. [2] L. M. Kaplan, \Extended fractal analysis for texture

[3]

[4]

[5] [6]

[7]

[8] [9]

[10]

classi cation and segmentation," IEEE Transactions on Image Processing, vol. 8, no. 11, pp. 1572{1585, Nov. 1999. B. Pesquet-Popescu, \Wavelet packet decompositions for the analysis of 2-d elds with stationary fractional increments," IEEE Transactions on Information Theory, vol. 45, no. 3, pp. 1033{1039, Apr. 1999. L. M. Kaplan and C. C. J. Kuo, \An improved method for 2-d self-similar image synthesis," IEEE Transactions on Image Processing, vol. 5, no. 5, pp. 754{761, May 1996. D. P. Kroese and Z. I. Botev, \Spatial process generation," arXiv eprint available at http://arxiv.org/abs/1308.0399, 2013. Y.-F. Pu, J.-L. Zhou, and X. Yuan, \Fractional differential mask: A fractional dierential-based approach for multiscale texture enhancement," Image Processing, IEEE Transactions on, vol. 19, no. 2, pp. 491{511, 2010. B. Pesquet-Popescu and J.-C. Pesquet, \Synthesis of bidimensional α-stable models with long-range dependence," Signal Processing, Elsevier, vol. 82, no. 12, pp. 1927{1940, 2002. D. R. A. Benassi, S. Jaard, \Gaussian processes and pseudodierential elliptic operators," Rev. Mat. Iberoamericana, vol. 13, no. 1, pp. 19{89, 1997. R.-F. Peltier and J. Levy Vehel, \Multifractional Brownian Motion : De nition and Preliminary Results," INRIA, Rapport de recherche RR2645, 1995, projet FRACTALES. [Online]. Available: http://hal.inria.fr/inria-00074045 A. Ayache, S. Cohen, and J. Vehel, \The covariance structure of multifractional brownian motion, with application to long range dependence," in Acoustics, Speech,

and Signal Processing,

2000. ICASSP '00.

, vol. 6, 2000, pp. 3810{3813 vol.6. [11] M. Moreaud, D. Jeulin, A. Thorel, and J.-Y. ChaneChing, \A quantitative morphological analysis of nanostructured ceria{silica composite catalysts," Journal of microscopy, vol. 232, no. 2, pp. 293{305, 2008. Proceedings. 2000 IEEE International Conference on