reduced complexity modeling and reproduction of colored ... · cally distributed (i.i.d.)...

510 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 9, NO. 3, MARCH 2000

in fine detail areas even though they yield better results in uniform re-gions. The sharpening effect which can be achieved using the adaptivetechnique proposed in [7] is significantly smaller.

Comparing the results of the adaptive algorithm presented in thispaper to those obtained using the competing techniques, we can see thatthe homogeneous areas of the output of our algorithm are less noisythan similar areas in Fig. 5(a) or (b). In addition, good sharpening isalso achieved in the detail areas. Thus, the adaptive algorithm also over-comes the problems of the cubic and of the order-statistics operators.In particular, the adaptive operator is able to enhance the medium-con-trast details better than these two algorithms. The noise amplificationdue to our adaptive algorithm is lower than that caused by the other al-gorithms except the cubic unsharp masking operator; however, the lownoise yielded by the cubic operator is due to its reduced enhancementof medium-contrast (but significant) details. There are some transienteffects in the output of the adaptive processor that occur while the re-cursions in the adaptive filter are moving from a detail zone to a smootharea. These transients cause an amplification of the input noise but donot appear to produce annoying visual effects.

B. Preprocessing for Interpolation

Interpolation is widely used in multirate image processing andfinds uses in applications such as pyramidal coding and zooming.The presence of antialiasing lowpass filters in the sampling andsubsampling processors often introduces some blurring effects into theinterpolated images. The nonideality of the lowpass filters employedin such systems partially suppresses useful frequency components inthe passband, and this also contributes to the loss of contrast in theoutput image. Perceptually better results can be obtained by applyinga contrast enhancement algorithm to the image before interpolation[10]. For this experiment, we processed a block of 64 × 64 pixelsof “Lena” and zoomed it to a block of size 256 × 256 pixels usingbicubic interpolation [11] after preprocessing the low-resolution blockwith the enhancement operators. The original block of the image isshown in Fig. 6(a). A general loss of contrast can be observed in Fig.6(b), which was obtained without applying any preprocessor to theinterpolator. Fig. 7 displays the result obtained using the adaptivepreprocessor of this paper. Our objective here was to slightly enhancethe input image prior to interpolation, and therefore, we chose thethreshold values�1 and�2 to be 200 and 400, respectively, to produceFig. 7. We can see that this operator provides satisfactory contrastenhancement on abrupt edges as well as fine details. Furthermore, thenoise present in the uniform areas appears to be acceptable from aperceptual point of view. We also processed the input image with theother processors discussed in the previous subsection. Preprocessingthe images using the linear UM technique, the Type 1B algorithm andthe adaptive algorithm in [7] resulted in amplified noise in smoothareas. The results obtained with the Cubic UM and the OS-UMtechniques showed a lack of enhancement of the finer details. We donot include the output images obtained using these techniques herebecause of space limitations.

IV. CONCLUDING REMARKS

This paper presented an adaptive algorithm for image enhancement.The algorithm employs two directional filters whose coefficients areupdated using a Gauss–Newton adaptation strategy. Experimental re-sults presented in this paper demonstrate that the algorithm performswell when compared with several approaches to image enhancementthat are available in the literature.

REFERENCES

[1] S. K. Mitra and H. Li, “A new class of nonlinear filters for image en-hancement,” inProc. IEEE Int. Conf. Acoustics, Speech, Signal Pro-cessing, Toronto, Ont., Canada, May 14–17, 1991, pp. 2525–2528.

[2] G. Ramponi, N. Strobel, S. K. Mitra, and T. Yu, “Nonlinear unsharpmasking methods for image contrast enhancement,”J. Electron. Imag.,vol. 5, pp. 353–366, July 1996.

[3] T. N. Cornsweet,Visual Perception. New York: Academic, 1970.[4] G. Ramponi, “A cubic unsharp masking technique for contrast enhance-

ment,”Signal Process., vol. 67, pp. 211–222, June 1998.[5] Y. H. Lee and S. Y. Park, “A study of convex/concave edges and edge-

enhancing operators based on the Laplacian,”IEEE Trans. Circuits Syst.,vol. 37, pp. 940–946, July 1990.

[6] S. Guillon, P. Baylou, M. Najim, and N. Keskes, “Adaptive nonlinearfilters for 2-D and 3-D image enhancement,”Signal Process., vol. 67,pp. 237–254, June 1998.

[7] F. P. De Vries, “Automatic, adaptive, brightness independent contrastenhancement,”Signal Process., vol. 21, pp. 169–182, Oct. 1990.

[8] C. V. D. B. Lambrecht, “Perceptual models and architectures for videocoding applications,” Ph.D. dissertation, EPFL, Lausanne, Switzerland,1996.

[9] B. Widrow and S. D. Stearns,Adaptive Signal Processing. EnglewoodCliffs, NJ: Prentice-Hall, 1985.

[10] H. S. Hou and H. C. Andrews, “Cubic splines for image interpolationand digital filtering,” IEEE Trans. Acoust., Speech, Signal Processing,vol. ASSP-26, pp. 508–517, Dec. 1978.

[11] R. G. Keys, “Cubic convolution interpolation for digital imageprocessing,”IEEE Trans. Acoust., Speech, Signal Processing, vol.ASSP-29, pp. 1153–1160, Dec. 1981.

Reduced Complexity Modeling and Reproduction ofColored Textures

Patrizio Campisi, Alessandro Neri, and Gaetano Scarano

Abstract—An unsupervised color texture synthesis-by-analysis methodis described. The texture is reproduced to appear perceptually similar toa given prototype by copying its statistical properties up to the secondorder. The synthesized texture is obtained at the output of a Single-InputThree-Output nonlinear system driven by a realization of a white Gaussianrandom field. Significant complexity reduction is gained by exploiting therank deficiency of the Cross Power Spectral Density Matrix of the colortexture samples.

Index Terms—Image color analysis, image generation, image textureanalysis.

I. INTRODUCTION

Texture reproduction is a challenging theoretical problem as wellas an important issue in practical applications. The texture synthesishas been widely investigated since it can be applied in the simulationof textured fields to be used in the performance assessment of pat-tern detection procedures, and in simulation of image background in

Manuscript received June 9, 1998; revised June 16, 1999. The associate editorcoordinating the review of this manuscript and approving it for publication wasDr. Fabrice Heitz.

P. Campisi and A. Neri are with the Dipartimento Ingegneria Elet-tronica, Università degli Studi di Roma Tre, I-00146 Roma, Italy (e-mail:[email protected]; [email protected]).

G. Scarano is with the Dipartimento INFOCOM, Università degliStudi di Roma “La Sapienza,” I-00184 Roma, Italy (e-mail: [email protected]).

Publisher Item Identifier S 1057-7149(00)01252-5.

1057–7149/00$10.00 © 2000 IEEE

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 9, NO. 3, MARCH 2000 511

Fig. 1. Color textures synthesis model employing MIMO filters.

testing automatic target recognizers. In image coding, synthetic tex-tures, produced by parsimonious mathematical models, may replacebackgrounds in natural scenes, providing very low bit rates texture rep-resentation techniques. Moreover in the area of synthetic pictures, re-produced textures may be employed for surface rendering in three-di-mensional (3-D) computer graphics. Exact reproduction of grey scale,as well as color texture synthesis, is not a simple task, since it is verydifficult to capture with a single mathematical model the unlimited va-riety of combinations of illumination conditions and pattern structure.

However, in human-oriented applications, the goal is not to generatea perfect reply of the given prototype, but to reproduce a textureper-ceptually closeto the original image. Even though the human eye hasthe outstanding capability to discriminate, after detailed examination,between quite similar patterns, it has been conjectured by Julesz [1],that the human visual system is pre-attentively unable to distinguishtextures having the same first and second order distributions, but dif-ferent higher order distributions.

During the recent past, different approaches have been proposed forthe synthesis of grey-level textures. They range from causal and noncausal autoregressive (AR) models or moving average (MA) models tomore complex models including Markov Random Fields, fractals, andmorphological models. Generally speaking, the AR and MA modelsconsist of a simple convolutional system driven by independent identi-cally distributed (i.i.d.) two-dimensional (2-D) excitations (see for in-stance [2], [3]). The parameter identification procedure is performedusing deconvolution techniques, which can use higher order statisticalanalysis, as shown in [3], [4]. These models are quite attractive due tothe simplicity of the identification and generation stages. However, thequality of the obtained synthetic texture depends on the accuracy of theconvolutional model. In order to take into account the basic character-istics of textures such asgranularity, directionality, repetitiveness, in[5]–[7], the texture is preliminary decomposed into itsdeterministic,andunpredictablecomponents, according to the 2-D generalization ofthe Wold decomposition theorem. As a matter of fact, it is often nec-essary to add complementaryad hocexcitation components in orderto copy the structured behavior of many textures encountered in prac-tice [8]. In [9] and [10], this problem is circumvented using MarkovRandom Field, but this approach is unpractical for the complexity ofthe parameter identification step.

A more suitable solution was proposed in [11]. In essence, the re-sulting scheme was a cascade of two linear system interconnected by ahard limiter, and excited by a white Gaussian zero mean random field.The binary field, driving the second linear system, has a morpholog-ical structure closely resembling the original texture's one, since it re-tains the information on the zero-crossings of the prototype that, as

well known [12], plays a fundamental role in the image characteriza-tion. Both the identification and the synthesis procedures still main-tain the simplicity of the previous convolutional schemes. The binaryexcitation and the synthesis filter were optimized by minimizing thedifferences between the first and second order statistics of the originaltextured image and its synthesized counterpart, in accordance with theJulesz's conjecture. Synthetic binary excitations with the required spa-tial correlation were then generated by hard limiting a filtered whiteGaussian random field whose spatial correlation is directly related tothe correlation of its binarized version via thearcsin law[13].

In this work, an innovative unsupervised color texture synthesis-by-analysis method is presented. The proposed model is based on a multi-input multi-output (MIMO) convolutional model where each input isconstrained to be a binary field whose autocorrelation retains most ofthe morphological properties of the prototype. The generalization ofthearcsine lawallows then to generate those binary excitations by hardlimiting a (three component) Gaussian random field. Moreover, the ex-perimental analysis on a wide set of images has outlined that, at eachspatial frequency, the cross power spectral density matrix (CPSDM) ofthe multichannel hue, saturation, and brightness (HSB) image has twoeigenvalues significantly close to zero, i.e., it has approximately rankone. This fact implies that the multi-component synthesis-by-analysisalgorithm, presented by the authors in [14], needs to be particularizedto handle with such a peculiar case. Specifically, the MIMO systememployed in the general case to jointly copy the morphological HSBbehavior, is replaced by a single input three output (SITO) system ex-cited by a scalar, white Gaussian random field. As a consequence, adrastic simplification of both the identification and the implementationof the whole system is obtained since each channel can be indepen-dently treated.

A further simplification can be obtained performing the Karhunen-Loève (KL) expansion of the three components of the color image, andretaining only the components associated to the two principal eigen-values. This in turn implies that the MIMO filter employed in the gen-eral case, here becomes a single input two output (SITwO) system.

The model here presented can also be applied to obtain very low bitrate representation of textured color images (see for example [15], forapplication to gray-scale images).

This paper is organized as follows. In Sections II and III, respec-tively, the three channel and the two channel models employed are il-lustrated. In Section IV the analysis by synthesis technique is described,and it is applied on a representative set of textures. A comparison be-tween the results obtained using the models presented in Sections II,andIII is provided. Finally conclusive remarks are outlined inSectionV.


Fig. 2. Color textures synthesis model employing SITO filter.

Fig. 3. Color textures synthesis model employing SITwO filter.

II. THREE CHANNEL COLOR TEXTURE MODELING

Let us indicate withk1[n1; n2]; k2[n1; n2]; k3[n1; n2], the threecomponents of a color image on a generic color space representation,and letk[n1; n2] be the corresponding vector field, i.e.

k[n1; n2] = (k1[n1; n2]; k2[n1; n2]; k3[n1; n2])T: (1)

Then, as illustrated in Fig. 1, let us consider the following convolutionalmodel fork[n1; n2]:

k[n1; n2] = ts[n1; n2] + z[n1; n2] (2)

ts[n1; n2] = ��(F[n1; n2] � �bs[n1; n2]) (3)

where�� is a three component zero memory nonlinearity performinga histogram matching between each component of the prototype andthe corresponding synthetic one,�� indicates the 2-D convolution,z[n1; n2] is a random field accounting for model mismatching andF[n1; n2] is a linear MIMO system, driven by the binary vector field

bs[n1; n2] = (bs1[n1; n2]; bs2[n1; n2]; bs3[n1; n2])T: (4)

Following [16], in the identification phase we determine��;F[n1; n2];bs[n1; n2], by minimizing the difference between the first and secondorder statistics ofk[n1; n2] and ts[n1; n2]. Then, according to theJulesz's conjecture, the synthesis of a color texture perceptually closetok[n1; n2] is accomplished by feeding the system (3) with any binaryexcitation sharing the same statistical distributions ofbs[n1; n2] up tothe second order. In this way, the burden to reproduce the texture is splitbetween the “design” of the binary excitation, and of the shaping filterF[n1; n2].

The choice of a binary excitation dramatically simplifies our task,since under weak symmetry conditions usually verified in practice,the second order distribution of a binary random field is characterizedby its mean and its autocorrelation function (acf). On the other hand,since a binary field can always be thought as the hard-limited versionof a Gaussian random field (see Fig. 1), its acf can be controlled in astraightforward way. In fact the autocorrelation matrixRcc[l1; l2] =Efc[n1; n2]c

T [n1 + l1; n2 + l2]g of an ergodic, zero mean Gaussianrandom field,c[n1; n2] with variance�2c , and the autocorrelation ma-trix Rb b [l1; l2] of its binarized version,bs[n1; n2] with variance�2b , are related through the generalizedarcsin law, which states that

Rb b [l1; l2]

�2b=

2

�arcsin

Rcc[l1; l2]

�2c: (5)

Thus a binary vector random field with acfRb b [l1; l2] is obtain-able by hard limiting a vector Gaussian random field with autocorrela-tion matrix

Rcc[l1; l2] = �2

c � sin�

2

Rb b [l1; l2]

�2b(6)

that can be generated by filtering a white zero mean Gaussian randomfield with a linear MIMO filter whose transfer function matrixG(!1; !2) satisfies the constraint

Pcc(!1; !2) = GH(!1; !2)G(!1; !2) (7)

wherePcc(!1; !2) = F2DfRcc[l1; l2]g, having denoted byF2�Df�gthe 2-D Fourier transformation.

In generalG[n1; n2], andF[n1; n2] are MIMO systems, but, as il-lustrated in the following, model complexity can be effectively reduced


Fig. 4. Parameter identification procedure referring to the three channel model shown in Fig. 2.

TABLE ISYNTHESIS-BY-ANALYSIS

PROCEDURE

for a wide range of natural textures with a negligible loss in reproduc-tion accuracy.

Since each texturek[n1; n2] is considered as a realization of athree component stationary random field, its spectral representation isprovided by the integrated Fourier transform (see for instance [17]).As well known, [17], while components of the increments of theintegrated Fourier transform at different frequencies are mutuallyuncorrelated, at the same frequency the covariance matrix is given byPk(!1; !2)�!1�!2 wherePk(!1; !2) = F2DfRkk[l1; l2]g is theCPSDM ofk[n1; n2].

Now, let us consider the eigendecomposition of the CPSDMPk(!1; !2):

Pk(!1; !2) =

3

i=1

�i(!1; !2)ui(!1; !2) � uH

i (!1; !2) (8)

where �i(!1; !2) is the eigenvalue associated to the eigenvectorui(!1; !2). Then, the KL expansion of the increments is obtainedby representing them in the basis constituted by the eigenvectorsui(!1; !2); in such a basis, the variances of the principal componentsare proportional to the eigenvalues�i(!1; !2).

As enlightened by the experimental analysis, for many textures en-countered in practice, the highest eigenvalue�max(!1; !2) is signif-icantly greater than the other two, say�a(!1; !2) and �b(!1; !2),i.e., the CPSDMPk(!1; !2) is approximately of rank one. Looselyspeaking, a strong correlation between texture components exists. Sucha phenomena has been experienced by the authors, for instance, whenthe HSB color space is adopted. Examples pertaining to the HSB com-ponents are illustrated in Fig. 6, where for the textures of Fig. 5 thequantity

�(!1; !2) =�a(!1; !2) + �b(!1; !2)

2 � �max(!1; !2); �(!1; !2) 2 [0; 1) (9)


Fig. 5. Original textures.

is shown. Note that when�(!1; !2) is equal to zero,Pk(!1; !2) hasrank equal to one. In summary, for a wide set of textures, their CPSDMcan be well approximated by a rank-one matrix, i.e.

Pk(!1; !2) ' �max(!1; !2)umax(!1; !2) � uH

max(!1; !2) (10)

whereumax(!1; !2) is the eigenvector associated to the principaleigenvalue�max(!1; !2). This is tantamount equivalent at havingretained only the principal component of the KL expansion of theintegrated Fourier transform increments.

The strong mutual coherence of the HSB components, witnessed bythe rank-deficiency of the CDPSM, requires a particular care in de-signing the MIMO filterG(!1; !2) which controls the cross-correla-tions of the binary processbs[n1; n2]. With reference to Fig. 1, sincethe CDPSM of the processc[n1; n2] takes the form

Pcc(!1; !2) = G(!1; !2)Pw(!1; !2)GT (!1; !2) (11)

rank deficiency can be obtained by reducing the rank of bothPw(!1; !2) andG(!1; !2); to assure mutual coherence of the HSBcomponents which is of primary concern, a viable choice consists inassigning all the rank deficiency toPw(!1; !2), retaining a diagonalstructure forG(!1; !2), i.e., (12), shown at the bottom of the page.Experimental results showed that other choices, while leading tosufficiently accurate reproduction of each component, suffer froman inherent lack of coherence. Moreover, a dramatic complexityreduction is obtained from (12), since also the transfer function matrixF(!1; !2) turns out to be diagonal, and both the analysis and the

synthesis can be performed treating each component separately. Insummary, according to (11) and (12), the reduced model of rank one,shown in Fig. 2, reproduces a fully coherent three-component binaryprocessbs[n1; n2] by using a single white Gaussian inputw[n1; n2](of unitary variance�2w = 1) filtered by a single input three output(SITO) filter, whose components are

Gii(!1; !2) = �cmax(!1; !2) [uc

max(!1; !2)] (13)

being

Pcc(!1; !2) ' �c

max(!1; !2)uc

max(!1; !2)� [uc

max(!1; !2)]H (14)

the rank one approximation of the CPSDM of the Gaussian excitationc[n1; n2]. Each binary component is then separately filtered and equal-ized. The reproduced texture is obtained by combining the split HSBsynthetic components.

III. T WO CHANNEL COLOR TEXTURE MODELING

The model complexity reduction described above has been obtainedby retaining, for each frequency, only one principal component of theKL decomposition of the increments of the integrated Fourier trans-form. This, in turn, implies the use of three input three output filters inthe synthesis(F[n1; n2] andG[n1; n2] in Fig. 2).

However, for several textures these operators can be further approx-imated by means of simpler filters, exploiting the common part sharedby the covariance matrix of the increments of the Integrated Fourier

Pw(!1; !2) = �2

w

1 1 1

1 1 1

1 1 1

; G(!1; !2) =

G11(!1; !2) 0 0

0 G22(!1; !2) 0

0 0 G33(!1; !2)

(12)


Fig. 6. Map of the values of the parameter�, for the textures depicted in Fig. 5. White pixels indicate a value of� 2 [0; 0:15), green pixels indicate a value of� 2 [0:15;0:25), red pixels indicate a value of� 2 [0:25;1).

Fig. 7. Realization of synthetic binary excitations, employing the three channel model shown in Fig. 2, (component H).

Transform at different frequencies. The evaluation of this latter con-sists in extracting the principal components of the mean (over the fre-quency axis) of the CPSDM, i.e. Pk(!1; !2)d!1 d!2, which is, bythe way, just proportional to the correlation matrix ofk[n1; n2] at zerolagRkk[0; 0]. Thus, an alternative synthesis-by-analysis scheme canbe devised by using the pointwise (spatial) KL decomposition obtained

from the eigenanalysis ofRkk[0; 0]. Since only the zero lag correlationis removed by this KL decomposition, only the smallest eigenvalue canbe usually neglected, and the filters of the synthesis stage reduce to twoinput two output. Hence, let

v[n1; n2] = (v1[n1; n2]; v2[n1; n2])T (15)


Fig. 8. Synthetic textures obtained from the original counterpart shown in Fig. 5, obtained by using the three channel model shown in Fig. 2.

Fig. 9. Realization of synthetic binary excitations, employing the two channel model shown in Fig. 3, (component associated to the principal eigenvalue).

be the vector which retains the two principal components resulting bythe KL transform performed on the original texture. As expected, ex-perimental results have confirmed that the cross power spectral densitymatrix of v[n1; n2] has rank close to one, so that the simpler single

input two output (SITwO) filter, generating the two, spatially corre-lated, principal components, cascaded with the pointwise two inputthree output inverse KL transform (IKLT), depicted in Fig. 3, can re-place the three channel model described in Fig. 2.


Fig. 10. Synthetic textures obtained from the original counterpart shown in Fig. 5, obtained by using the two channel model shown in Fig. 3.

IV. PARAMETER IDENTIFICATION PROCEDURE

The identification procedure of both the shaping filterF[n1; n2],and of the second order statistics of the binary excitationb[n1; n2] ofthe three channel model described in Fig. 2, is performed by fitting aconvolutional factorization on the three components of the given colortexture. More in detail, referring to (2) and (3), we want to perform thefollowing factorization:

ki = �� (Fiiba ) + zi (16)

having denoted with the subscripti = 1; 2; 3 the quantities relatedto thei-th componentki[n1; n2] of the color image. Specifically, letki andzi be the 1-D array obtained by rearranging the columns ofki[n1; n2] andzi[n1; n2] in lexicographic order, respectively, and letFii = Fii(��ii) be the linear operator associated with the 2-D filterfii[n1; n2] having rational transfer functions whose coefficients are ar-ranged in the vector��ii:.

Here, we considered the generalization to the multichannel case ofa blind deconvolution algorithm already employed for seismic tracesdeconvolution [18], [19]. It consists in finding the triplet(~ba ; ~��ii; ~�i)which simultaneously maximizes the log-likelihood function, i.e.,

~ba ; ~��ii; ~�i = argmaxb ;�� ;�

ln p (ki=ba ;��ii; �i)K =B ;A ;�

: (17)

This optimization criterion leads to an iterative procedure whose flowchart is depicted in Fig. 4 (see [11] and [16] for further details).

Convergence is characterized by a proportionality relation betweenthe cross-correlation betweenq[n1; n2] and the residualsr[n1; n2]and the cross-correlation betweenq[n1; n2] andba[n1; n2]. For thisreason, this kind of deconvolution scheme is referred to as “BussgangDeconvolution,” since the invariance of correlation function undernonlinear point-wise transformations characterizes the class of“Bussgang” random fields. Even though theoretical convergence isnot assured, we have experienced that, for the class of textures underexamination, after four to five iterations of the Bussgang Deconvolu-tion algorithm convergence is achieved. In texture modeling, where

only the statistical description up to the second order is perceptuallysignificant, according to the Julesz's conjecture, the proportionalitybetween covariances is employed to seek for a convergence pointwhere the difference between the residualsr[n1; n2] and their binaryversionsba[n1; n2] is small enough to allow the use of a realization ofa binary Gaussian vector random field (BGVRF) as a morphologicalsupport in the synthesis phase. At convergence, once identifiedthe inverse shaping filterH[n1; n2], the realization ofba[n1; n2],obtained during the last iteration, can be employed to estimate thecovariance matrix functionRb b [l1; l2] of the BGVRF. Then, wecan derive the covariance matrix functionRcc[l1; l2] using theinversearcsin lawas expressed by (6). Finally, the spatial-correlated Gaussianrandom field can be obtained by filtering a white Gaussian randomfield with a SITO coloring filterG[n1; n2] whose impulse responsematrix is determined according toRcc[l1; l2].

The identification procedure here outlined can be applied to the twochannel models presented in Section II with no substantial changes,except for the preliminary extraction of the two principal componentsof the image, by means of the spatial KL transform. From this point on,the identification procedure can be applied as it is, taking into accountthat all the vector random fields are bi-component, and substituting, ineach formula,q[n1; n2], with v[n1; n2] representing the two principalcomponents of the KL transform of the given prototype.

V. EXPERIMENTAL RESULTS AND CONCLUSIVE REMARKS

The whole analysis and synthesis stages are outlined in Table I. Theaccuracy of the proposed methods has been tested on a wide range oftextures. The collection of the prototypes is displayed in Fig. 5. Thesetextures are representative of different behaviors, ranging from direc-tional to random to periodic. In Figs. 7 and 9, the synthetic binary coun-terparts, obtained using the three channel model and the two channelmodel respectively, are represented. Comparing the synthetic binaryexcitations with the original textures, it is evident that they are able tocapture the basic structure of the original image. In Figs. 8 and 10, the


TABLE IICOMPUTATIONAL COMPLEXITY OF THE MIMO, SITO, AND SITwO SCHEME.

THE QUANTITY M INDICATES THE IMAGE PIXEL'S NUMBER AND k THE

NUMBER OF THE IDENTIFICATION PROCEDUREITERATIONS. THE

COMPUTATIONAL COMPLEXITY ASSOCIATED TOEACH (I)FFT CAN BE

ROUGHLY ASSUMEDPROPORTIONAL TOM log M

synthetic textures are shown. In a preattentive stage of vision the resultsare quite impressive for the different kind of textures.

As far as the choice of the color space is concerned, it must be out-lined, however, that, in principle, this should be such to reveal the rankdeficiency (if any) of the CPSDM in order to apply the reduced com-plexity methodology. Experimental results, performed on a wide rangeof textures encountered in natural scenes, have shown that the rank de-ficiency of the CPSDM substantially holds when these textures are de-scribed in the HSB color space or similar spaces such as YUV, YIQ,and Lab.

It is worth noting that the results obtained using the two channelmodel are comparable to the ones obtained using the SITO model, al-lowing a dramatic complexity reduction with respect to the generalthree channel MIMO case. Eventually, a comparison between the com-putational complexity of the synthesis plus analysis stage of the dif-ferent synthesis methods is provided in Table II. It has been assumedthat filtering is accomplished by means of circular convolution per-formed in the frequency domain by using discrete Fourier transformsof suitable length so that to employ fast algorithms whose computa-tional cost is expressed in terms of fast Fourier transform (FFT).

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3-D Contextual Bayesian Classifiers

Rasmus Larsen

Abstract—We extend a series of multivariate Bayesian two-dimensional(2-D) contextual classifiers to three-dimensional (3-D) by specifying a si-multaneous Gaussian distribution for the feature vectors as well as a priordistribution of the class variables of a pixel and its six nearest 3-D neigh-bors.

Index Terms—Classification, contextual methods, segmentation, 3-D.

I. INTRODUCTION

In [1]–[4], algorithms for two-dimensional (2-D) images that basethe classification of a pixel on the feature vectors of the pixel itself andthose of the four nearest neighbors are introduced. In [3], it is assumedthat the classes of the nearest neighbors of a pixel are conditionally in-dependent given the class of the center pixel, whereas in [1], [2] it isassumed that the pixel size is small relative to the grains of the patternunder study. In this correspondence, we will extend these algorithms tothree-dimensional (3-D) images, and carry out a series of tests on twosimulated 3-D images, one that expresses low spatial frequency of thesignal and one with high frequency signal. Furthermore, we will illus-trate the use of the 3-D contextual classification for tissue classificationin a 3-D magnetic resonance image of a human brain.

II. M ETHODS

In this section, we develop a 3-D contextual classification rule,specify a Gaussian observation model, and specify a prior distribution

Manuscript received June 26, 1997; revised August 9, 1999. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Dr. Andrew F. Laine.

The author is with the Department of Mathematical Modelling, TechnicalUniversity of Denmark, DK-2800 Lyngby, Denmark (e-mail: [email protected]).

Publisher Item Identifier S 1057-7149(00)01507-4.

1057–7149/00$10.00 © 2000 IEEE

reduced complexity modeling and reproduction of colored ... · cally distributed (i.i.d.)...

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