A DISTANCE MEASURE FOR PERFORMANCE EVALUATION OF SHAPE ADAPTIVE TRANSFORMS

In a recent work, a mathematical formulation for shape adaptive (SA) 2D transforms was developed. Based on this formulation, we propose in this paper a distance measure that theoretically assesses the performance of a generic SA transform. This measure is based on the minimization of the mean square difference between the coefficients of a given SA transform and those of the optimum SA transform. Experimental results show that the proposed measure is an efficient tool for performance evaluation purposes and may be of major interest for object­ based video coding applications.


INTRODUCTION
In object-based video compression, an image sequence is encoded in such a way that objects can be separately decoded and easily manipulated at the receiving end, allowing increased interactivity and compression efficiency.Shape adaptive 2D-transforms represent important tools to this area.In a recent work reported in the literature [1], a mathematical formulation was developed for these transforms.It was shown that with a simple mapping from R~ to R, we can represent any type of SA transform by means of a concatenation of three linear operators, which characterize the ID vertical transforms, the alignment of :,j vertical coefficients and the ID horizontal transforms.The DCT used in the SA-DCT described in [2] is an example of the 1D vertical and hOlizontal transforms.Two different strategies of vertical coefficients alignment are described in [2] and [3].
It is certainly of interest to have useful distance measures to assess the peIformance of a given SA transform with respect to an optimum transform, such as the Karhunen Loeve transform (KLT).The literature presents solutions to the case of regular (rectangular) transforms.An example is the development reported in [4, p. 176] to the family of sinusoidal transforms.However, such solutions are not applied to the support region of the image segments used in object-based video coding.In this case, the algebraic formulation described in this paper allows the derivation of an appropriate distance measure.Another approach usually employed in the literature is the transform coding gain GTe over PCM [5]- [7] for each transform individually.The diference to our proposal is that we derive an analytical expression for a distance measure f.l between a given SA transform and the optimum transform in terms of a mathematical formulation which is specific for shape adaptive transforms [I].The proposed measure is, therefore, useful for formal manipulations and evaluations of SA transforms based on this mathematical formulation.
Section II of this paper contains the algebraic formulation required for the derivation of the SA-KLT matrix.It is based on the autocorrelation and cross-correlation matrices of the vectors representing the object columns.Then, an expression is obtained for the autocorrelation matrix of a specific dass of signals: the wide sense stationary 20 I>l_ order Markov processes, with statistically independent horizontal and vertical coefficients.Section III is devoted to the proposal of a distance measure between SA transforms, which allows their comparison and the selection of the more efficient ones as compared to the SA-KLT.This resource eliminates a gap in the theory of SA transfoml coding.Because this tool was not available, previous works evaluate the performance of SA transforms either on experimental basis or in terms of the transform gain over PCM.The simulation results presented in Section IV confirm the usefulness of the proposed distance measure.Section V highlights some important aspects, which are specific of the proposed metric, and summarizes the main contributions of the paper.

THE OPTIMUM SA TRANSFORM
Due to its well known coefficients decorrelation and energy compaction properties [4], the KLT is taken as the optimum transform for comparative purposes.For this reason, it is of interest to derive an expression for the discrete SA-KLT <1>*1.Since the KLT is a unitary transform, its inverse is the conjugate transpose, i.e., .y=<I>*I!~ !=<I > .Y (I ) where x is an N-dimensional random vector and y is its KLT.The rows of the KLT matrix are the complex conjugate vectors of the normalized eigenvectors of Rx, the autocorrelation matrix of x.Therefore, «1>" the k-th column of cI», I :s:: k :s:: N, is the k-th normalized eigenvector of R, and, consequently, R 1 = E{ .y.y*'}= A = diag(A" A2, •• , A;, .. , AN) (2) where A" I :s:: i :s:: N, are the eigenvalues of R~.
These concepts can be used to obtain the optimum SA transform.Before, however, we will represent the 2-D real intensities of an arbitrarily shaped object as a vector.This representation can be obtained from the mathematical formulation reported in [1].Firstly, the original object is represented as a rectangular one with dimensions MxM by the matrix where the vector foj, j = 1,... ,M, contains the Mpixels of the j-th object column, !!;j is an Ilij-dimensional null vector representing the initial elements of the j-th column that do not belong to the object, and !!u is an Ilo-dimensional null vector representing the final elements of the j-th column that do not belong to the object.Note that Ilij + Nj + IlJJ = M.
R N From F we can define a vector fo in , where N=N,+ ... +NM , obtained from the non-null vectors of the columns of F. It is given by The discrete SA-KLT matrix can be obtained from the autocorrelation matrix RIO of this vector, which is defined by Rill = E{ fo fo'} Substituting (4) into (5), and defining Rfoifoj = E{foi ro j }, i,j = I ,...,M, as the correlation matrices of the vectors fOj, j = I ,...,M, we then get ( 6) In this expression, Skj = Ilij -Ilik is the relative shift between the k-th and j-th columns, where Ilik and Ilij are, respectively, the number of pixels that do not belong to the object in the beginning of these columns.Note that the cross-correlation matrix of the random vectors fOk and fOj is, in fact, a function of this relative shift and, for this reason, it will be denoted by Rr f (Skj).

S( -01".•91 "
A I-D 1 -order Markov process WIth correlatIon coefficient equal to a is an autoregressive stationary process x (ll) that can be represented by (7) where v(n) is a white noise.Successively applying this equation rtimes, we get k=O Since x(ll) is a stationary process, the correlation between x(ll) and x(l1-n does not depend on n.It can be expressed by (9) is zero because x (ll) and v(n) are uncorrelated processes.Now consider a model which is oftenly used for image analysis: a wide sense stationary and separable (x(n,,1l1) = -'(Ill) X(1l1» 1s'-order bidimensional Markov process, with horizontal and vertical correlation coefficients (i.e., between columns and between rows) equal to /3 and a, respectively.
For simplicity, we assume 2D separability and stationarity as in references [6], [9] and [10].A more realistic 2D non separable, noncausal, and non-stationary model has been considered already for rectangular-shaped fields [11]- [ 13], but it has not been done yet for arbitrarily-shaped fields.
Using an algebraic derivation similar to the one described for the I-D case, we obtain R(tl''tZ)= E{t(IlI,IlZ) X(1l1-tl,IlZ -tl)}= where -0,_0, Replacing this expression into (6) we obtain the autocorrelation matrix Rfo of fo.The eigenvectors of this matrix are the rows of the SA-KLT matrix.

A DISTANCE MEASURE FOR EVALUATION OF SA TRANSFORMS
The distance measure proposed in this paper to evaluate a given unitary transform T i is given by the mean square difference between the transform coefficients of the original object represented by fo and those of the SA-KLT cI»*' applied to the same vector fo.It is defined by E{d 1 (T i fo -cI»*' fo)} (12) where the power of the coefficients of T; is normalized by the ones of the SA-KLT.For real vectors and transforms, we have (13) which can be written as Since the temlS in ( 14) are scalars, they can be written as the trace Tr(.) (sum of the elements of the main diagonal of a matrix) of the Ixl matrix defined by the corresponding scalar.This means that Now, we can use the fact since the traces of inner and !outer products are identical, Tr(fol fo)= Tr(fo fol).In addition, .) Tr(A')=Tr(A).After trivial manipulations it results that Applying the expectation operator to this expression we get which can be expressed by where Ak are the eigenvalues of the autocorrelation matrix of fa.To derive (18) we have used the fact that Tr ( E{ fa fa' ) ) = L,A.due to the energy conservation property of unitary transforms.For comparative purposes we can eliminate the constant 2 in (18).We can also normalize the metric to values between 0 and 1 by dividing the right-hand side of (18) by the sum of the eigenvalues.The final expression is given by

SIMULATION RESULTS
In this section we will provide numerical results, obtained from two different SA transforms, to determine how the proposed distance measure behaves as a performance criterium to evaluate them.In our experiments, we have employed the six synthetic segments shown in Fig. 1.These segments are generated from a wide sense 1 51 stationary and separable order bidimensional Markov process, with both horizontal and vertical correlation equal to 0.95.
The two different SA transforms to be compared use the I-D DCT for the vertical and horizontal transforms and two different vertical coefficients alignment (see U] for the mathematical formulation of generic SA transforms).One of these alignment procedures was recently proposed in [3] (Alignment by Phase -AP) and the other is described in [2] (Equal Indices -EI).These two schemes were compared in [3] using as performance criterion the cumulative energy (CE) curve.It was concluded that the method AP is definitely better than the EI approach for images that approximate a separable first order 2-D Markov process with high correlation coefficients.This is the case of the synthetic segments depicted in Fig. I.I shows the results obtained from the theoretical evaluation with the metric proposed in this paper.For comparative purposes we also show in Fig. 2 the CE curves for the segments employed in the experiments.As can be seen from Table I, the nearest distances to the SA-KLT is for the method that employs the AP scheme.This is in accordance with the results presented in Fig. 2, where the CE curves obtained for the AP scheme is always larger than or equal to the ones provided by the EI approach.We can also verify that the segment (8x7 (s=O» for which the difference in CE (between the AP and the EI schemes) is the smallest one is also the same segment for which the difference between the distance measure proposed in this paper is the smallest one.
The results presented in Table I corroborate the ones obtained with the CE criterion for the two SA transforms taken as examples in this paper.These results show that the proposed metric is an efficient distance measure between the SA-KLT and a given SA transform.For this reason, it can be used as an effective measure to assess the potential of a given SA transform in an object-based coding (or compression) scheme.Other interesting and important features concerning the proposed metric can be remarked.It may also be a profitable tool to choose the best operator that composes an SA transform.Based on the mathematical formulation described in [1], it was shown that with a simple mapping from R 2 to R, we can represent any type of shape adaptive transform by means of a concatenation, T = T, T2 T I, of three linear operators.The first one (T I) describes the one dimensional transforms of the object columns (the vertical transforms).The second operator (T2) represents the alignment of the vertical transform coefficients before the application of the horizontal transforms.Finally, the third operator (T ,) corresponds to the computation of the horizontal transforms.The simulation results presented in this paper arl' concerned with the performance evaluation of two possibilities for the T 2 transformation, while T 1 and T~ are fixed.E.\j1ressing the distance measure given by ( 19) in terms of these 0pl'rators we get 1l=1- LA" " Note that thi, l'\lwession can be minimized in terms of any of the tlnl'l' 011L'rators.Although this is not an easy task, (21) cenainl~ rl'prescnts a mathematical tool that can be of major interl'S\ ,Jnd usdulness for further research in the area of SA tranSIOl"l1h

DISCUSSION AND CONCLUSIONS
In this P'Ij1l'l.\\ l' have presented a theoretical method to evaluate the' l'crlorlllanCe of shape adaptive transforms.The mathematic,li lormulation of SA transforms presented in [1] was taken as .I hasis for the theoretical development reported in lhi, papl'!".The performance analysis of the SA OCT and otllL'I SImilar transforms [2], [5]-[10] described in the literature' arc' l'jther based on experimental results or in terms of traIhlorlll gains over PCM.They often miss the relationship 10 Ihl' panicular mathematical structure being examined III this sense, the approach introduced in this paper ma: hc' c"ollsidered as a valuable resource to fill this gap.
A metric" \\ ,IS proposed to assess the distance between a given SA transform and the optimum SA transform.An algebraic dl'l l'lopment was presented, and a simple expression \\ .Is obtained for this metric.Simulation results show the \alidity of this metric as compared to experimental result" previously reported in the literature [3].
Similarl~ to other performance criteria, such as the cumulative cnL'l"gy (CE) and the transform gain over PCM (GTcl, thc proposed metric allows the selection of the best transform among a set of several ones.Nevertheless, a major point is that this is not its sole use.Another important application i" that the proposed metric may also be a profitable tool to choose the best operator that composes an SA transform. R(O,O) = E{X(IlJ, 111) X(Il" Ill)}' Applying this model to the sub-matrices in (6) we get R r .r.(slJOO)=R(O.O)/3l i -;1.

Figure 1 :
Figure 1: Synthetic segments employed in the experiments TableIshows the results obtained from the theoretical evaluation with the metric proposed in this paper.For comparative purposes we also show in Fig.2the CE curves for the segments employed in the experiments.As can be seen from TableI, the nearest distances to the SA-KLT is for the method that employs the AP scheme.This is in accordance with the results presented in Fig.2, where the CE curves obtained for the AP scheme is always larger than or equal to the ones provided by the EI approach.We can also verify that the segment (8x7 (s=O» for which the difference in CE (between the AP and the EI schemes) is the smallest one is also the same segment for which the difference between the distance measure proposed in this paper is the smallest one.The results presented in TableIcorroborate the ones obtained with the CE criterion for the two SA transforms taken as examples in this paper.These results show that the proposed metric is an efficient distance measure between the SA-KLT and a given SA transform.For this reason, it can be used as an effective measure to assess the potential of a given SA transform in an object-based coding (or compression) scheme.

Table I :
Results (jI values) obtained from the theoretical evaluation