Signal decimation aimed at optimal spectral packing has
a variety of applications in areas ranging from array pro
cessing to image processing In this article we propose
and discuss a new method for determining decimation
grid and prelter that best t the spectral extension of
any D signal dened on an arbitrary sampling lattice
The method has been implemented and tested on digital
images in order to evaluate quality degradation due to
optimal spectral truncatio

A. Sarti

F. Pedersini

S. Tubaro

SARTI, AUGUSTO

TUBARO, STEFANO

Archivio istituzionale della ricerca - Politecnico di Milano

Selection of Sampling Grid and Prelter for ImageDecimation Based on Spectral Extension AnalysisFederico Pedersini Augusto Sarti Stefano TubaroDipartimento di Elettronica e Informazione  Politecnico di MilanoPiazza L Da Vinci   Milano ItalyTel 	 Fax 	pedersinsartitubaroeletpolimiitABSTRACTSignal decimation aimed at optimal spectral packing hasa variety of applications in areas ranging from array processing to image processing In this article we proposeand discuss a new method for determining decimationgrid and prelter that best t the spectral extension ofany D signal dened on an arbitrary sampling latticeThe method has been implemented and tested on digitalimages in order to evaluate quality degradation due tooptimal spectral truncation IntroductionSampling multidimensional analog signals causes theirspectrum to replicate over a regular point structurewhose density is inversely proportional to the samplingdensity Minimizing the gap among spectral replicas iswellknown to reduce information redundancy  Thisfact could be useful in a variety of applications thatrange from from array processing  to Image Processing  	 Spectral packing through decimation is not an easytask as it consists not just of a rational selection of datasamples but it also needs a careful spectral truncationfor avoiding aliasing In order to perform antialiasingpreltering in fact knowing the area of the spectral extension spectral occupancy is not sucient we alsoneed to consider its shape The spectral energy of nonsynthetic images however usually occupies regions withquite irregular and complex shape  which makes theoptimal design of a prelter very dicult On the otherhand the spectral extension may be dened and estimated according to the class of prelter we adopt fordecimation purposes More specically if we restrictthe class of prelters we are interested in to those havingan arbitrary compact and convex passband all we needabout spectral extension shape is the direction aroundwhich the spectral energy is maximally concentratedprincipal axis and a measure of the energy dispersionabout this axis This way of quantifying the anisotropyof the spectral distribution corresponds to approximating the spectral extension with an ellipse whose shapeis decided by the ratio between the inertia moments ofthe power spectrum while its size is chosen accordingto the severity of the spectral truncation we are willingto applyIn this paper we propose and test a method basedon the above convexity assumption for jointly and automatically determining decimation grid and prelterfor twodimensional discrete signals dened on arbitrarylattices The method has been implemented in a fullyautomatic computer procedure and tested on several images Mathematical preliminariesGiven a nonsingular matrix A  RM the M dimensional lattice  generated by A is dened as  LAT A x  RMx  An n  ZMwhich is the set of all possible linear combinations withinteger coecients of the M linearly independent vectors basis of the lattice that are given by the columnsof AGiven a basisA it is possible to derive any other basisAof   LAT A as A AUU being a unimodularmatrix j detAj  A fundamental cell S of an M dimensional lattice is a closed region of RMsuch that the collectionS fS  a a  g of all shifted version of S on thepoints of  tiles RMwithout overlapping Notice thatthere exist innite fundamental cells for a single latticebut their hypervolume is always the same and is givenby d  j detAj a measure of the lattice density isthus d Notice also that there exist no general geometric classications of all possible fundamental cellsof a given lattice The only results that are availablein the literature concern convex cells  and are particularly simple in the D case In fact the only convexregions that tile Rare parallelograms and hexagonswith central symmetry  LAT B is a sublattice of   LAT A ie asubset of  with lattice structure if and only if there exists a nonsingular integer matrixH such that B  AHThe integer number j detHj  j detBjj detAj corresponds to the decimation ratio All kth order sublattices of   LAT A can be obtained as iLAT Wi where Wi AHi Hi HMk where HMkis the set of integer matrices in Hermite normal formwhose determinant is equal to k The Fourier transform U f  of a discrete signal uxdened on the lattice  is periodic and its periodicity centers are specied by  LATAT which iscalled reciprocal lattice of  	 The Fourier transform U f  is thus completely specied by its valuesin any fundamental cell P of  Decimating u from  LAT A to   LAT B  being an nth ordersublattice of  returns the signal vx  ux x  and the relationship between Fourier transforms 	 results asV f  nXaIU f  a I being any period of  In order to be able toperfectly reconstruct a signal u dened on  from itsdecimated version v on  it is thus necessary for thesupport of U f  to be conned inside some fundamentalcell P of  In this case the reconstruction can bedone by using an interpolating lter from  to  havingfrequency response Hf   n in P and zero elsewhere Decimation approachThe rst step of the procedure we propose consists ofestimating the spectral extension of the signal Fromsuch information we determine an upper bound for theindex of the sublattices to choose among by simply computing the spectral occupancy of the estimated spectralextension The maximum index of decimation krepresents the order from which to start looking for suitabledecimation gridsOnce the maximum decimation ratio kis available we generate all distinct kth order sublatticesand among them we discard all those that are noncompatible with the spectral extension If no compatible subgrids of order kcan be found we decrease theorder and repeat the search until some compatible onesare found For each one of them we generate the fundamental cell that best ts the elliptical spectral extensionby using a geometrical approach A selection of the bestsublatticecell pair can nally be performed among theremaining candidates according to some specic criterionAs already mentioned in the Introduction the classof prelters we are considering has a convex passbandregion As the only convex regions that tile Rare parallelograms and hexagons with central symmetry theonly shape information on the spectral extension weneed is given by the direction around which the spectral energy is maximally concentrated and a measureof the energy dispersion about that axis Quantifyingthe spectral anisotropy through the secondorder energydistribution corresponds to approximating the spectralextension with an ellipse whose shape is decided by theratio between the inertia moments of the power spectrum while its size is chosen according to the severityof the spectral truncation we are willing to apply Spectral extension estimation as a consequence consists ofnding the inertia ellipse of the power spectrumd d r whose coordinates d d are referred to the principalaxes of the power spectral distribution while and are the radii of gyration which in turn are a functionof the inertia moments The parameter r can be chosenin such a way for the ellipse to enclose a specied portionof the signal energyThe area A  of the spectral ellipse can beused as an estimate of the spectral occupancy of thesignal As a consequence the largest integer kbelowareaPA P being a fundamental cell of the samplinggrid can be used as an upper bound for the index of thesublattices that could be used for decimationOnce the maximum decimation ratio kis availablewe can generate all kth order sublattices with k  kby determining all matrices in Hermite normal formwith determinant k The spectral extension can now be used for decidingwhether the sublattice  of the original signal support is suitable for decimation More precisely we needto check its compatibility by verifying that the ellipticalspectral replicas generated by decimation does notoverlap In order to do so we rst need to determine thek points of the reciprocal lattice that fall inside oneperiod of Rcenters of replication of the originalspectrum We then should test whether overlappingsbetween any two replicas of the spectral extension occurwhich can be done by checking for overlappings betweenthe replica about the origin and the othersThe fact that the spectral extension model is ellipticalmakes the compatibility check particularly simple Infact the limitregion for the replication centers beyondwhich no overlapping occurs is itself an ellipse thresholdellipse whose radii of gyration are twice the spectralellipses radii as shown in Fig A sublattice  is compatible with the signal spectralextension if none of the replicas overlap with the originalextension ie if all points of lie outside the threshold ellipse All compatible subgrids are good candidatesfor decimation therefore we need a criterion for deciding among them As the choice of decimation grid isstrongly inuenced by the shape of the passband regionof the prelter fundamental cell of the reciprocal of thesublattice in order to be able to decide among the compatible subgrids we need a fast method for generatinga fundamental cell that well ts the spectral extensionfor each compatible sublatticeIn general there exists a nonnumerable multitude offundamental cells for a sublattice and the degrees ofd2d1ρ1th ρ2thOΞ∗f1f2Figure  Threshold ellipse for checking the compatibility of a sublattice   freedom in their shape are enough to make the searchextremely dicult Restricting the class of fundamentalcells to the convex ones however greatly simplies thesituation as according to the results of Section  allconvex fundamental cells of a twodimensional latticeare hexagons with central symmetryA method for determining a hexagonal fundamentalcell of a given compatible sublattice which entirely encircles the elliptical spectral extension is described inFig  The method consists of determining the  pointsof that are closest to the threshold ellipse buildingtwo triangles by using two triplets of alternate pointsand determining the hexagonal cell as the intersectionof the two triangles Notice that when all six pointsused for constructing the hexagon lie on the thresholdellipse the above method generates the tangent hexagonof Fig  The fundamental cell we obtain can be used forconstructing both the antialiasing and the reconstruction lters In fact both of them will have a passbandregion that resembles the fundamental cell determinedaboveLast step of the decimation procedure consists ofchoosing the best decimation setup among the compatible subgridprelter pairs Notice that all the availablecandidates are acceptably good therefore the choicemust be made according to some criterion of optimalitythat takes into account some measure of the tness between fundamental cell and spectral extension In practice we select the sublattice whose prelter has minimum impact on the principal axis of the power spectrald2d1P1P2P3P4P5P6OFigure  Construction of the prelter the hexagonis given by the intersection between the two trianglesfP P Pg and fP P Pg built on the six closestpoints of the latticedistribution More precisely we can choose the prelterthat minimizes the angle between the principal axis ofthe nonpreltered spectrum and that of the prelteredspectrum Examples of applicationThe method proposed in this article has been implemented into a completely automatic computer procedure and tested over a series of real imagesAn example of application is shown in in Fig 	aThe spectrum of this image Fig 	b exhibits a certainanisotropy The principal axes of the elliptical spectralextension have been chosen to be 	 times the inertiaaxes of the power spectrum samples The maximumorder of decimation in which some compatible sublattices can be found is k   Among the hexagonalfundamental cells that are associated to all compatibleth order sublattices the one whose principal axes areclosest to those of the elliptical spectral extension is chosen to dene the passband region of the image preltersee Fig 	b while the relative sublattice is the corresponding decimation grid At this point the image canbe preltered through spectral windowing by using asmoothened version in order to prevent ringing fromoccurring of the ideal preltered obtained above Decimating the image over the selected subgrid causes thetruncated power spectrum to replicate like in Fig 	cwhere it is quite apparent how the elongation of thepower spectrum due to the prevalence of some edgesalong a specic direction causes subgrid and prelterto preserve the spectrum in that directionThe same lter used for avoiding aliasing is now usedfor reconstructing the original image from the decimatedone The reconstruction results of the test image are reported in Fig 	d where a comparison is made betweencorresponding zoomedin details of original left and reconstructed right images As we can see the blurringdue to the lowpass antialiasing ltering is still acceptable despite a reduction of  times in the amount ofsamples that are actually being used for describing theimage itself This result would not be possible with moretraditional decimation methods ConclusionsIn this paper we presented a new technique for decimating discrete D signals which is capable of considerablyreducing the spectral redundancy while suppressing theleast amount of spectral energy Spectral characteristicsof the signal are taken into account for determining bothdecimation grid and antialias lterWe have implemented our decimation technique intoa fullyautomatic computer procedure and tested it overa series of images The quality of the reconstruction results have proven the eectiveness of the method showing that decimation factors between  and  can bereached with acceptable blurring in spite of the factthat the resulting sampling grids are uniformReferences KF Cheung and RJ Marks II Image samplingbelow the Nyquist density without aliasing Journal of the Opt Soc of America  A  Jan  DE Dudgeon and RM Mersereau Multidimensional Digital Signal Processing PrenticeHall Englewood Clis 	 E Dubois The sampling and reconstruction oftimevarying imagery with applications in videosystems Proceedings of the IEEE 	Apr  GM Cortelazzo and R Manduchi On the determination of all the sublattices of preassigned index and its application to multidimensional subsampling IEEE Trans on Circ and Syst forVideo Technology 	 P MacMullen Convex bodies which tile space bytranslations Mathematika 	 affb cd eFigure 	 Decimation of the image Lenna with thethorder decimation basis v  T v  	T aoriginal image b power spectrum of the original imagelog scale elliptical extension and optimal prelter cspectrum of the decimated image zoomedin details ofthe original d and the reconstructed e images

Selection of Sampling Grid and Prefilter for Image Decimation Based on Spectral Extension Analysis

AIR Universita degli studi di Milano

Publication in the conference proceedings of EUSIPCO, Trieste, Italy, 199

Pedersini, F.

Sarti, A.

Tubaro, S.

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

SELECTION OF SAMPLING GRID AND PREFILTER FOR IMAGE DECIMATION BASED ON SPECTRAL EXTENSION ANALYSIS

http://www.eurasip.org/Proceedings/Eusipco/1996/paper/mdsp_5.pdf

Selection of Sampling Grid and Prefilter for Image Decimation Based on Spectral Extension Analysis

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Archivio istituzionale della ricerca - Politecnico di Milano

AIR Universita degli studi di Milano

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY