image representation using non-canonical discrete … · image representation using non-canonical...

IMAGE REPRESENTATION USING NON-CANONICAL DISCRETE MULTI WINDOWGABOR FRAMES

Nagesh K. Subbanna and Yehoshua Y Zeevinagesh @tx.technion.ac.il zeevi @ee.technion.ac.il

Department of Biomedical Engineering,Technion - Israel Institute of Technology,

Haifa, Israel.

ABSTRACT

We extend non-canonical multiwindow Gabor representa-tions to quincunx sampling and discuss the advantages ofrepresentation of textures using non-canonical discrete mul-tiwindow Gabor expansions, with the time-frequency planesampled in a quincunx fashion. We show that the quin-cunx sampling is more efficient in terms of placement of thetiling zones in the time-frequency and achieves better time-frequency localisation when compared to the usual rectan-gular sampling. To emphasise the advantage of quincunxsampling, we compare images reconstructed using multi-window Gabor frames using the canonical dual, a non-canonical dual, and finally a non-canonical dual under aquincunx sampling lattice and discuss our results. We showthat the use of non-canonical duals permits a great deal offlexibility in the choice of analysis functions (and conse-quent generation of Gabor coefficients) and that duals canbe optimised to the situation at hand.

1. INTRODUCTION

Gabor expansions are a popular choice of representation ofsignals in the time-frequency space and have been used inseveral practical applications. Gabor representations, withthe Gaussian window function, achieve optimal localisationof signals in the time-frequency plane and this criterion isof importance in texture segmentation [8] where the authorshave attempted to utilise the localisation of frequencies inthe combined space to segregate textures. In [61, it wasestablished that single windows are insufficient for properlocalisation of frequencies and therefore, we choose multi-window Gabor expansions as suggested in [15].

The multiwindow Gabor expansion of a rectangularlysampled, discrete time, finite energy signal f [k] C CL, oflength L signal, is given by

R-1 T)- U-~1

f [k] -7 ~ ~~~~~[k] (1)r= mOm n=C

where Yt,m,rL[k] = yr[k -na]ei 2 7,mbk/L_ is the r-th windowfunction of the expansion frame, Rt is the number of win-dows used, a and b are the shifts along the time and the fre-quency axes, -a and T are the number of shifts along the timeand frequency axes respectively and c-,m,fl are the rectan-gularly sampled multiwindow Gabor coefficients given by

k-O

(2)

where gr,m,n = g7 4k - na]e 27rbk/L are the analysis win-

dow functions. It is obvious that given the set of expansionvectors (set of frame vectors) -yr,m,,[kl, the task is to findthe coefficients C.m,,,, and consequently, the analysis win-dow functions gT,r,,,, [k], so that good localisation of fre-quencies is achieved by the set of coefficients.

It has been established that both undersampling and crit-ical sampling are generally unsatisfactory [16]. [2] as theformer is an incomplete representation and the latter suffersfrom lack of stability in reconstruction (shown by the BalianLow theorem). Further, the Gabor vectors are not orthog-onal to each other, which necessitates the computation ofduals to be able to reconstruct the signals. As oversamplingis the only remaining alternative in Gabor expansions, wechoose oversampling in this paper. Oversampling impliesthat there are more vectors than necessary and these vectorsare linearly dependent, and therefore the dual is not unique.Any right inverse of the set of frame vectors constitutes adual.

The traditional Gabor function based image reconstruc-tion techniques fall into one of the two categories - the firsttechnique is to use nearly orthogonal vectors so that recon-struction of the image can be made directly from the coeffi-cients [5], [7]. The second is to utilise canonical duals [11I]to find the biorthogonal functions to reconstruct the signal.The first technique suffers from a sharply restricted choiceof analysis functions, since the near orthogonality impliesthat the 'areas of significance' have to be so placed as tohave the least overlap [8], [5]. The limitation of analysis

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functions is evident in certain examples in [7]. The latterchoice of using the canonical dual [11], [9] which involvescomputation of the pseudoinverse of the set of expansionvectors, has two limitations. First, given an arbitrary choiceof window functions, the computational difficulties of cal-culating the canonical may present several difficulties [12].Second, the canonical dual provides the least square solu-tion of the problem, which may not be desirable in casesof image reconstruction [ 11. The human eye does not seemto work on the least square distance [7] and texture seg-mentation based on the square distance may not be accu-rate. Therefore, in order to alleviate these problems, wepropose to use non-canonical duals to localise frequenciesaccurately. Further, we propose to utilise the denser andmore uniform tiling of the quincunx lattice to alleviate theproblems associated with the rectangular lattice.

The paper is organised as follows: In section 2, we dis-cuss briefly non-canonical duals for rectangular sampling.In section 3, we extend the non-canonical duals to quincunxsampling and suggest techniques to obtain the dual easily.In section 4, we pick two natural images and two texturesand reconstruct them using the most prominent coefficientsobtained in the three methods and discuss them.

2. NON CANONICAL DUALS

Given a signal f [k] C CL, the Gabor expansion of the signalcan be written in vector form as

f =rc. (3)

where f is the signal f [k] of length L, c is the vector of co-efficients ~rmnof length R.Cb and r is the set of expansion(or reconstruction) vectors given by[ NY,oo[O] ... _ý,1ý10

'Y0,0, 11] ... _YRZ,0~ 1]r = (4)

- 1].O L. - 1]j

Given the expansion frame, we need to compute the coeffi-cients. The traditional way of generating coefficients is touse the canonical dual, which involves computing the pseu-doinverse of the matrix r, i.e.,

C = (f, (F)I) (5)

where r~, is the pseudoinverse of P. However, the pseu-doinverse has some disadvantages as stated in the previoussection. Recognising that r is a rectangular matrix and hasinfinite possible right inverses, we introduce a dual of theform [4],

-----------

LFig. 1. An illustration of the non-zero elements in the ma-trix J7G* and the block circulant structure of the matrix. Ininteger oversampling, the block circulant matrices end upbeing diagonal or zero blocks.

where G is another matrix of the same form and dimensionsas r but with a different set of window functions g,- Thenecessary and sufficient condition for the existence of dualsof form (6) is that the matrix rc* be invertible. The condi-tions for the existence of the inverse have been investigatedin detail in [ 13].

Briefly, we may state that the matrix rG is a block cir-culant matrix and the inversion of block circulant matricescan be handled in several ways [3]. There are several waysof inverting the matrix rG - primarily by using block dis-crete fourier transforms 1 13] and using Zak transforms [ 15].However, the kind of oversampling (integer or rational) de-termines the techniques that can be used.

It was proved in [13] that, under the conditions of inte-ger oversampling, if the set of window functions -y, are pos-itive definite' sequences and g, are of all of the same sign 2 ,

then the matrix rG* will always be invertible. Since theFourier transforms of Gaussian window functions are posi-tive definite and Gaussian window functions give optimumresolution in the combined space and are used most oftenin practice, we shall choose both -(r and g, to be Gaussianfunctions, and the time-frequency space oversampled by aninteger factor.

The inversion of the matrix rG* can be accomplishedin two ways in the case of integer oversampling. We havea simple technique [13], where the inversion of the matrixcan be solved by the computation of Fourier transforms andinversion of a diagonal matrix. Another method proposedin 110] would have similar computational complexity. De-

l Positive definite sequences are those sequences whose DPI' is real andpositive at al points.

2 Windows are positive or negative throughout the sequence.

483

al -------


Fig. 2. An illustration of rectangular and quincunx samplingwith similar rates of oversampling.

pending on the circumstances, either may be better for theactual case. Similar results can be achieved using Zak trans-forms [ 15], which involves summation of a set of diagonalmatrices and inverting them.

For the case of rational oversampling, we can still en-sure that the complexity of inversion of the matrix is limitedusing Zak transforms [ 15], [13] or by using a combinationof perfect shuffle matrices and block discrete Fourier trans-forms [13].

3. QUINCUNX SAMPLING

Quincunx sampling has been a popular alternative to therectangular sampling for a number of reasons [14]. Briefly,two good reasons for using the quincunx sampling wouldbe more uniform tiling of the time-frequency space and thedenser packing, as can be seen in Fig. 2. The Gaussian func-tion has a circular/elliptical tiling on the combined space,and it is obvious from the shape of the tiling that havinga hexagonal (quincunx) shaped lattice would leave fewerIgaps' in combined space than the rectangular sampling withsimilar rate of oversampling, as can be seen in Fig. 2. Thishas the added advantage of leading to a tighter frame [ 14].

3.1. Gabor vectors under Quincunx sampling

In rectangular sampling, the time and frequency shifts areindependent of each other. In the quincunx sampling, weno longer have the independence of the time and frequencyshift operators. In odd rows, we have the shift increasedby a for every shift used and the actual shift is 2a betweentwo adjacent time-shifted Gabor vectors. However, there isa simple way to reduce the quincunx sampling to a patternsimilar to rectangular matrix. A close observation of thepatterns shown in 3 indicates that the 'rows' of the quin-cunx matrix differs from the rows of the rectangular sam-pling only by a modulation factor. This can be achieved bymodulating the rectangular sampling translation matrix T

1 0 0

0 0

Fig. 3. An illustration of conversion of rectangular to quin-cunx sampling with similar rates of oversampling using asimple modulation operator.

Gabor vectors by a modulation matrix Q where Q is givenby

1 0o ej 27rb/L

o 0 .. ej27 rb/L Iand

(7)

(8)T

Given the easy conversion from the quincunx samplingto the rectangular sampling, we can use the non-canonicaldual, only both the matrices G and r will be accompaniedby the modulation matrix M. Apart from this difference, thetwo matrices can be manipulated in the same way the rect-angular sampling matrices were handled and inverted usingthe same techniques. A full discussion of the quincunx lat-tice and methods of handling the resultant matrices can befound in [ 14]. Briefly, we can say that both BDFT tech-niques and Zak transform techniques can be applied with-out any loss of generalisation to the quincunx lattice. Theother way to handle the quincunx. lattice is to reduce it intoa rectangular lattice as shown in Fig. 3. The technique hasbeen elaborated upon in [9].

4. SIMULATIONS AND DISCUSSIONS

We choose images of two types - natural images and tex-ture images. In the former category, we choose two images,viz, Lena, and Nails. In the latter category, we choose two

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images as well, viz, Honeycomb and Stripe. We desire toreconstruct each of them using the least number of signifi-cant coefficients, with the expansion vectors -y, already be-ing set and the oversampling rate fixed. We define the rateof oversampling as the number of samples for reconstruc-tion divided by the length of the signal, i.e., Ra-b/L andin our case, the rate of oversampling is 4. In our work, wechoose the absolute value of the coefficients I Cr,m,n Ias themeasure of the significance of the coefficients. We choose10% of the most significant coefficients as the ones neces-sary for construction. An alternate possibility would be touse the 'energy' of the coefficients as the measure for the

choice of the significant coefficients [5). In this instance.we have chosen four windows (all four windows are Gaus-sians, whose spreads 3 are in arithmetic progression) for therepresentation of the image using the multiwindow Gabor

coefficients. Here, we are aiming not at perfect reconstruc-tion, but to measure the localisation of frequencies usingnon-canonical duals, with different sampling.

As may be seen in Figs. 5 and 6, the canonical dualtends to 'smear' the coefficients across frequencies and com-promise the exact localisation of frequencies. The indicatedareas in Fig. 5 show how the lines in the hat (region A),and the reflection in the mirror (region C), are both blurredwhile the exact texture in region B is lost in the canonical

dual, and is less distinct in the non-canonical with rectangu-lar sampling case. The observed results are in consonancewith the observations of Chen et. al. [1] as shown in Fig.4. Using non-canonical duals with rectangular sampling,we obtain better frequency localisation, but the best resultsare seen in the case of non-canonical duals with quincunxsampling. The result is fairly intuitive and bears out theobservation that the uniform placement of areas of uniformspread and the better 'packing' by the quincunx samplingachieves better frequency localisation.

In the examples, we have limited ourselves to 10% ofthe coefficients. When we choose 50 % of the coefficients,the reconstruction in both canonical and non-canonical tech-niques is nearly perfect. Further, for certain cases of win-dows, both non-canonical and canonical windows give goodresults. However, it is when the windows are arbitrary andthe frequencies in the images diverse that the non-canonicalsolution really shines.

However, in the case of the Fig. 7, it is obvious that thereis not too much change between the three outputs, especiallyin the honeycomb pattern itself. The lack of difference be-tween the outputs is even more pronounced in the stripe im-age in Fig. 8. It must be remarked that the honeycomb andthe stripe images have very little in the way of frequencyvariations - both of them are regular textures. The presentscheme seems to indicate that the technique of using non-canonical duals is particularly well suited to a cadre of im-

3The spread is the standard deviation of the Gaussian function

Tim,

Fig. 4. The reconstruction of the signal 'Carbon' (top left)and its synthesis phase plane (top right). The spread of co-efficients due to the pseudoinverse (bottom left) and finallythe values of the coefficients [1 ]

ages where the expansion frame is fixed and there is a ratherlarge variation in the set of frequencies in the image. Thenon-canonical dual performs better since it is better able to' adapt' to the the frequencies, by choosing duals other thanthe one which minimises the least square distance. It is ofimportance to mention that the choice of the norm is left to

the user. We used the Li norm here, but any conceivablenorm could be applied to the set of non-canonical duals. Itaffords the user a far greater freedom without inhibiting the

ease of computational complexity to any great degree. Theflexibility of the non-canonical duals is one of the most im-portant features of the non-canonical Gabor scheme.

The adaptability of the non-canonical dual is enhancedby the choice of quincunx sampling, which improves uponthe coefficients (both in terms of the uniformity and in terms

of the greater density of packing of the tiling areas) whencompared to the rectangular sampling. The optimal place-ment of the tiling areas in the case of the quincunx samplingalleviates the problems of frequencies falling in the 'blindspots' of the rectangular sampling.

4.1. Computational Complexity

The computational complexity of the localisation of fre-quencies using non-canonical duals is made up of two parts.The first part is the creation and inversion of the matrixFG*'. The latter part consists of choosing the best coeffi-

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Fig. 5. The original Lena image (top left) and the image re-constructed using 10% of the largest coefficients using thecanonical dual (top right), the non-canonical dual with rect-angular sampling (bottom left) and using the non-canonicaldual with quincunx sampling (bottom right)

cients for the reconstruction of the image.

As discussed in [13], the creation of the matrix FG* is,in the best case, 2Rab multiplications and its inversion isof order O(2Llog(b)). The part about reconstruction is justchoosing the set of coefficients to reconstruct from and thiscan be achieved by a good sorting algorithm. Reconstruc-tion of the signal consists of a multiplication of a matrix ofsize R~b- x L with a vector of length Rubý.

5. CONCLUSION

From the examples, we can see that choosing a non-canonicaldual can lead to more efficient representation in the time-frequency plane and helps in better localisation of frequen-cies. It is also apparent that non-canonical representationof signals can lead to better texture classification, with thechoice of the metric remaining with the user, adding to theflexibility of the system. Other possible applications includespeech processing and DNA sequence analysis. It is perhapsof interest to mention that the technique has already beenused with considerable success in the latter application.

Fig. 6. The nails image from Brodatz: album (top left)and the image reconstructed using 10% of the largest co-efficients using the canonical dual (top right), the non-canonical dual with rectangular sampling (bottom left) andusing the non-canonical dual with quincunx sampling (bot-tom right)

Fig. 7. The original honeycomb image (top left) and theimage reconstructed using 10% of the largest coefficientsusing the canonical dual (top right), the non-canonical dualwith rectangular sampling (bottom left) and using the non-canonical dual with quincunx sampling (bottom right)

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Fig. 8. The original Stripe image (top left) and the image re-

constructed using 10% of the largest coefficients using thecanonical dual (top right), the non-canonical dual with rect-angular sampling (bottom left) and using the non-canonicaldual with quincunx sampling (bottom right)

6. ACKNOWLEDGEMENT

The authors are grateful to Prof. Y. V. Venkatesh for his sug-gestions. This research program has been supported in part

by the Ollendorf Minerva Research Center, by the HASSIPResearch Network Program HPRN-CT-2002-00285, spon-sored by the European Commission, and by the Fund forPromotion of Research at the Technion.

7. REFERENCES

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[3] P. J. Davis, "Circulant Matrices", New York, Wiley,c 1979.

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[5] A. K. Jain, and F. Farrokhnia, "Unsupervised TextureSegmentation Using Gabor filters", Pattern Recogni-tion, Vol. 24(12), pp. 1]167-1186, 199 1.

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[7] B. S. Manjunath, and W. Y. Ma, "Texture features forbrowsing and retrieval of image data"', IEEE Trans-action on Pattern Analysis and Machine Intelligence,Vol. 18, pp. 837-842, August 1996.

[8] M. Porat, and Y. Y. Zeevi, "Localized texture process-ing in Vision: Analysis and synthesis in the Czahorianspace", IEEE Transactions on Biomedical Engineer-ing, Vol. 36(1), pp. 115-129, 1989.

[9] P. Prinz, "Calculating the dual Gabor Window forGeneral sampling sets", IEEE Transactions on SignalProcessing, Vol. 44(8), pp 2078-2082, August 1996.

[10] S. Qiu, "Block Circulant Matrix Structure and Dis-crete Gabor Transforms", Optical Engineering, pp.2872-2878, October 1995.

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[13] N. K. Subbanna. Y. C. Eldar, and Y. Y. Zeevi,"Oversampling of the Generalised Multiwindow Ga-bor Scheme", in Proceedings of the InternationalWorkshop on Sampling Theory and Applications 2005(SampTA 05).

[14] A. van Leest, "Non-Separable Gabor schemes: TheirDesign and Implementation", PhD thesis, Universityof Eindhoven, January 2001.

[15] M. Zibulski, and Y. Y. Zeevi, "Discrete MultiwindowGabor type transforms", IEEE Transactions on SignalProcessing, Vol. 45(6), pp. 1428-1442, June 1997.

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image representation using non-canonical discrete … · image representation using non-canonical...

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