Sensitivity analysis of multilayer perceptrons appliedto focal-plane image compression

J.G.R.C. Gomes, A. Petraglia and S.K. Mitra

Abstract: The authors previously considered the application of multilayer perceptrons (MLPs) toblock coding at the sensor level of modern imaging systems, and have proposed analogue encoderswith transistor-count complexity that is low enough to suit focal-plane implementation. In thepaper, they extend the on-sensor block coding MLP study, to include a statistical analysis of theMLP sensitivity to implementation errors occuring in standard CMOS fabrication processes.Employing simple offset models, a comparison is made of the MLP with other block encodersbased on full-search entropy-constrained vector quantisation (ECVQ) of the data, and it is verifiedthat the MLPs are less sensitive over a wide range of rate-distortion compression points. By intro-ducing a realistic linear model that takes into account sensitivity and the complexity performancesfor both systems, the authors verify that, for MLPs, the sensitivity becomes less dependent on thecomplexity as the expected quality loss is allowed to increase. Without setting a limit on theexpected quality loss, the MLPs are consistently better than the ECVQs, both in terms of sensitivityand complexity for a precision equivalent to 6 bits.

List of symbols

n index of pixel block

N total number of pixel blocks on sensor

x(n) vector of principal component data (VQ input)

x(n) reconstructed version of x(n)

D mean squared error between x(n) and x(n)

H entropy of x(n) (lower bound on bit rate)

b(n) VQ output binary vector

f (n) feature vector assigned to x(n)

J Lagrangian cost function Dþ lH

l Lagrange multiplier in J

S SQNR of vector quantiser

D0 mean squared error between x(n) and x(n) atzero rate

xR training set

xT test set

N(m, s) Gaussian distribution with mean m and standarddeviation s

sp variance of random error parameters in MonteCarlo simulations

L largest length of b(n)

ck the kth vector in VQ codebook

K number of vectors in VQ codebook

# The Institution of Engineering and Technology 2007

doi:10.1049/iet-cds:20050315

Paper first received 7th November 2005 and in revised form 26th June 2006

J.G.R.C. Gomes and A. Petraglia are with the Department of ElectricalEngineering of the Federal University of Rio de Janeiro, Brazil

S.K. Mitra is with the Department of Electrical and Computer Engineering ofthe University of California, Santa Barbara, CA, USA

E-mail: gabriel@pads.ufrj.br

IET Circuits Devices Syst., 2007, 1, (1), pp. 79–86

lk length of binary codeword used for represen-tation of ck

dk(n) distance between x(n) and ck

jk(n) cost of x(n) representation, dk(n)þ llk

ek the kth canonical vector of RL

E fabrication precision

S(E) average SQNR of nonideal vector quantiser

DS(E) SQNR degradation

P maximum acceptable SQNR degradation

EDS,P precision required for DS , P

B bit-equivalent precision of nonideal VQ

j maximum percentage tolerance for precision E

u transistor count complexity of VQ

a(P) slope of j � u characteristic

b(P) offset of j � u characteristic

1 Introduction

Research on focal-plane image processors has shown thatit is possible to synthesise simple image processingoperations in analogue mode at the pixel level, beforeanalogue-to-digital (A/D) conversion. Interesting appli-cation examples have been provided [1–4]. Focal-planeimage compression can also be considered as an applicationof the on-sensor image-processing techniques that are avail-able. The relation between the rate-distortion compressionperformance and the stringent complexity constraintsimposed by the sensor area and pixel-size specificationshas been studied by Gomes and Mitra [5], for the case offocal-plane vector quantisers (VQs) to be implementedusing low-complexity multilayer perceptrons (MLPs).

An important issue regarding the study of focal-planeblock-coding schemes, derived for analogue integratedcircuit realisation, is sensitivity analysis of compression

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performance with respect to fabrication process variations.The sensitivity analysis of MLPs to coefficient errorshas already been conducted in the case of continuous acti-vation functions, (e.g. Zeng and Yeung [6]), but an analyti-cal approach to sensitivity assessment is difficult when theactivation functions are not continuous [7]. In that case, astatistical technique based on Monte Carlo simulations isa viable alternative. The statistical approach has beenapplied in the sensitivity analysis of implementation errorsin MLP-based classifiers [8].

In this paper, we report statistical simulations to comparethe sensitivity of MLPs and entropy-constrained vectorquantisers (ECVQs) to implementation errors, in thecontext of image compression. At the encoder side, a pre-processing stage encodes the average luminance value ofevery block of 4 � 4 pixels employing differential pulse-code modulation (DPCM), and it maps the 16 zero-meanresidual values into four-component data vectors x(n).The index of each 4 � 4 pixel block is n ¼ 0, . . . , N,where N is the total number of pixel blocks in the sensorarray. We focus on the vector quantisation scheme,(either based on MLPs or ECVQs) that encodes the datavectors x(n) into a sequence with entropy H, so that adecoder can reconstruct the original data by means ofapproximations x(n) with distortion D. We note that bothtypes of systems process the same input data, and the prob-ability density function of the data does not change whenthe preprocessing stage is not ideal, and so we expect allMLPs and ECVQs to have the same error sensitivity theywould have if the preprocessing stage was ideal.Discontinuities are present both at the encoder and thedecoder sides, and complexity constraints must be enforcedfor the MLPs. Monte Carlo simulations of these systemsreveal useful trade-offs involving (H, D)-pair sensitivityand complexity for MLP and for ECVQ image compressionschemes. The insight that can be won from such investi-gation, to enable feasible designs of analogue focal planeencoders, is the aim of this paper.

2 Models for sensitivity analysis

The models presented in this Section take into account inputand output offset errors, and multiplicative coefficientimplementation errors. These errors are caused by imperfec-tions in the CMOS fabrication process that implements thefocal-plane encoder.

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2.1 Hyperbolic tangent MLP model

Fig. 1 shows the typical structure of an MLP applied inblock coding. The kernel principal component analysis(PCA) technique was proposed in [9], and later adapted tovector quantisation problems. Kernel PCA is almostalways based on nonlinear activation functions calledkernels; if linear activation functions are used as kernels,then the method simplifies to the conventional PCA. Inthe following example, hyperbolic tangent activation func-tions are used. This HT-MLP maps the four-dimensionaldata vectors x(n) into binary codewords b(n) of length8. An operation outline for the example of Fig. 1 is pre-sented in Sections 2.1.1 and 2.1.2.

2.1.1 Operations: In the HT-MLP of Fig. 1, each com-ponent of the input vector x(n) ¼ [x1(n) x2(n) x3(n)x4(n)]T is multiplied by a gain factor and a bias is addedto the result of the multiplication. The kernel PCA is per-formed by a matrix-vector multiplication applied to thehyperbolic tangent function outputs, as shown in Fig. 1. Abias vector is added to the result, to generate the featurevector f(n) ¼ [ f1(n) f2(n) f3(n) f4(n)]T.

Precisely due to the nonlinear map between vectors x(n)and f(n), we partition the input space of vectors x(n) usingarbitrary curves, by partitioning the space of vectors f(n)with a rectangular grid. For that, each component of f(n)is encoded by a low-resolution scalar quantiser. In Fig. 1,f1(n), f2(n), f3(n) and f4(n) are encoded with 3, 2, 2 and 1bits of resolution, respectively, to simulate an 8-bit vectorquantiser of the input space. Using a greedy optimisation[10] algorithm that was developed for this specificproblem, the 3, 2, 2, 1 bit allocation was decided, and thecorresponding decision levels were optimised. Each scalarquantiser is implemented by subtraction between the inputvariable and the constant thresholds that define the quantisa-tion intervals, followed by hard limiters to define ther-mometer binary codes (of sizes 7, 3, 3 and 1 in theFigure). The codes with sizes 7, 3, 3 and 1 are finallyconverted into the output codes with sizes 3, 2, 2 and 1,by means of matrix-vector multiplications involving onlylow-resolution quantised values.

2.1.2 Free parameters and cost function: During theMLP design, kernel PCA is employed to compute jointlythe following parameters: hyperbolic tangent gains, hyper-bolic tangent biases, kernel PCA matrix and kernel PCAbias vector. The scalar quantisation thresholds, on the

Fig. 1 Typical structure of an HT-MLP applied to the compression of vectors x(n)Dotted lines represent coefficients that are affected by implementation errors

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other hand, must be found by nonlinear optimisationmethods to minimise a rate-distortion cost functionJ ¼ Dþ lH, where l is a Lagrange multiplier that controlsthe trade-off between rate and distortion minimisation. Weassume that entropy coding is performed outside the pixel-block, so that the MLP binary output b(n) has fixed length.

To test the HT-MLP (MLP with tanh(x) activationfunction), we took the four-component data vectors x(n),obtained from differential pulse code modulation (DPCM)and linear transformation of pixel-block values, from a data-base with seven test images, and compressed them intobinary indexes b(n). We estimated the entropy H from thebinary index histogram of the sequence b(n). The decodergenerated the reconstructed vectors x(n) from a training-setVQ codebook. We measured the distortion D as the meansquared error between x(n) and x(n) (average over n from1 to N ).

Several MLP encoders with different rate-distortion proper-ties were designed, with l values ranging from 1024 to 1021.For example, the MLP with highest signal-to-quantisationnoise ratio (SQNR) (SQNR is a log-scale normalised versionof D and is defined as S ¼ 10 log10(D0/D), where D0 is thezero-rate distortion at H ¼ 0; for the vectors in the trainingset xR, D0 ¼ 0.013; for the vectors in the test set xT,D0 ¼ 0.034) in the test-set curves (xT ) of Fig. 2 is anHT-MLP with H ¼ 4.4 bits per vector. Its distortion D issuch that 10 log10(D0/D) ¼ 9.0 dB. Its design admittedl ¼ 1.2 � 1024, assuming no implementation errors.

MLPs based on other types of activation functions weredesigned as well, by applying minor changes to theHT-MLP design steps. Except for the addition of randomperturbations, we assumed all MLP parameters to be pre-computed and constant throughout the numerical sensitivitysimulations. In other words, the MLPs were designed onlyonce, before simulations started, and no retraining wasdone after the perturbations were applied to the MLPparameters.

2.1.3 Physical implementation of basic functions:The hyperbolic tangent is synthesised by a differentialpair of CMOS transistors [11]. The circuit was modelledas a nonideal block containing the ideal hyperbolictangent function itself, an additive input offset error, anadditive output offset error, and a gain error. These threeerror parameters are modelled as N(0, sp) random vari-ables. The variance sp is chosen according to the precisionthat can be achieved with the analogue design techniquesand the fabrication process under consideration. In this

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work, we define sp ¼ 22E, where the precision level Erepresents the combined equivalent precision, in bits, ofthe analogue circuitry and the fabrication process.

The inner product is realised by a current conveyor [12,13] that adds, in current mode, the set of multiplicationresults generated by one-transistor CMOS synapses [4] con-nected to the current conveyor input node. Lumped into theblack circles indicated in Fig. 1, the current conveyorcircuits were modelled as nonideal blocks. Each blockcomprises the ideal current conveyor itself, input-sensingoffset errors, offset error in the realisation of the biasvalue, bias-sensing offset error, output offset error, andsynapse errors. The synapse errors were divided into twotypes: (i) the error in the multiplication factor, followingthe same idea of the hyperbolic tangent gain error, and(ii) the output offset error. All these errors were selectedfrom the N(0, sp) distribution.

Note that the input and output errors are of the offset type.It is therefore appropriate to assume that they have identicaldistributions. Although gain errors may have a different dis-tribution, we nevertheless assumed offset and gain errors areidentically distributed, to make the Monte Carlo simula-tions simpler. However these simulations yield results thatare useful for the design (transistor dimensioning) ofthe synapses of an image compression system to beimplemented in CMOS technology [14]. We are currentlyworking on the improvement of these models.

The threshold operation has also been implemented usinga differential pair of CMOS transistors [11] working as hardlimiters. The hard limiter circuit was modelled as an idealsign function with a threshold offset error and an input-sensing offset error. Both errors were drawn from theN(0, sp) distribution.

2.1.4 Simulations: Inside one pixel block, all error valuesof the activation functions, inner products and threshold oper-ations were generated as independent, identically distributed(IID) N(0, sp) random variables. To simplify the MonteCarlo simulations, we assumed the same error values forevery pixel block. This assumption is supported by the factthat gradient effects cause strong correlation betweenimplementation errors in neighbouring on-chip blocks.

The Monte Carlo algorithm was applied to each MLP ofour design database, so that in each iteration the followingactions were taken:

1. Generate error values for the activation function. In theexample of this Section, which involves four hyperbolic

Fig. 2 Free loss with precision E equal to 8 bits and 6 bits

a E ¼ 8 bitsb E ¼ 6 bits

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tangent functions, we drew four gain errors, four bias errors,four tanh(x) input errors, and four tanh(x) output errors.They were applied at the points indicated by dotted lines#1, and dotted arrows #2 and #3 in Fig. 1.2. Generate error values for the inner products. In ourexample, we drew four bias errors, four bias-sensingerrors, four output errors, 16 input-sensing errors, 16 multi-plier errors, and 16 multiplication output offset errors. Theywere applied at the points indicated by dotted lines #4 anddotted arrows #5 in Fig. 1.3. Generate error values for the comparators. In ourexample, we drew 14 threshold errors and 14 input-sensingerrors. They were applied at the points indicated by dottedarrows #6 in Fig. 1.4. Keeping the perturbations constant, compute H and D.To compute D, we used the original codebook obtainedfrom the training set, without any information about theGaussian perturbations.

For each MLP, 50 iterations were run, and the MLP averagevalues of H and D were stored. The simulation results aredescribed in Section 3.

2.2 Entropy-constrained VQ model

Fig. 3 shows the typical structure of an ECVQ that can beused for block coding. ECVQ is based on an extension ofthe classical Linde–Buzo–Gray (LBG) algorithm [15]that includes, at the cost function, an entropy constraintwith information about probability of codeword assignment.It achieves a locally optimal performance in terms ofaverage rate and distortion combined [16]. Although thereis an entropy constraint, the encoder structure is uncon-strained. Typically, its excellent rate-distortion performanceis achieved at the cost of a very high complexity.Variable-length coding is considered inside the ECVQstructure, as Fig. 3 indicates, because this particular solution

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achieves a complexity that is smaller than the complexity ofan ECVQ with fixed-length output. The choice of fixed orvariable-length coding is not important for a sensitivityanalysis. The details of the layer between the WTA outputand the b(n) outputs are not commented on any further inthis text. The ECVQ in this example maps the four-dimensional data vectors x(n) into variable-length binarycodewords b(n) ¼ [b1(n) b2(n) � � � bL(n)]T, withlengths up to L ¼15 in this example. An operation outlineis developed next, in Sections 2.2.1 and 2.2.2, for theexample of Fig. 3.

2.2.1 Operations: In the ECVQ of Fig. 3, K inner pro-ducts of size 5 are computed between the input vector[jjx(n)jj2; x(n)] and constant vectors [1; 2 2ck], k ¼1, . . . , K. A precomputed constant bias jjckjj

2þ llk is

added to the result of each inner product. These operationscompute the cost jk(n) ¼ dk(n)þ llk of using the codevector ck as a representation for x(n), where dk(n) is theEuclidean distance between x(n) and ck, and lk is thelength of the binary codeword that represents ck throughthe communication channel.

The computation of jjx(n)jj2 requires four circuits for therealisation of square functions and one current conveyor. Togenerate the costs jk(n), the matrix-vector multiplicationrequires K current conveyors. Each jk(n) enters awinner-takes-all (WTA) circuit, which selects the smallestjk(n) and generates a binary vector ek(n). All componentsof ek(n) are zero, except the component at the position cor-responding to the smallest jk(n). Finally, the vector ek is pro-cessed by a matrix of unit-gain synaptic connections, togenerate a binary output vector b(n) with variable length.

This matrix, presented schematically at the right-handside of Fig. 3, is not considered in the sensitivity analysis,because the map from vectors ek(n) into vectors b(n) isessentially digital, and hence it is not affected by analogueimplementation errors.

Fig. 3 Typical structure of an ECVQ applied to the compression of vectors x(n)

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2.2.2 Free parameters and cost function: Weemployed the ECVQ design algorithm [16] to jointly opti-mise the codebook, comprising the vectors ck, and the code-word lengths lk to minimise J ¼ Dþ lH. To test the ECVQin this example, as well as all the other ECVQs in Section 3,we took the vectors x(n) that were used in Section 2.1.2 andcompressed them into the binary index vectors b(n).Similarly to what was described in Section 2.1.2, wecomputed H and D.

Several ECVQs with different H and D values weredesigned by varying l between 1024 and 1021. Forexample, as discussed later in Section 3, the ECVQ withthe second lowest SQNR in the test-set curves (xT) ofFig. 2 has H ¼ 0.9 bits per vector. Its distortion D issuch that 10 log10(D0/D) ¼ 5.9 dB. Its design admittedl ¼ 0.06, assuming no implementation errors. The code-book has size K ¼ 33 and the longest binary codeword haslength L ¼ 15.

We also assumed all ECVQ parameters to be precom-puted and constant throughout the numerical sensitivitysimulations, which means no retraining was done after theperturbations were applied to the ECVQ parameters.

2.2.3 Physical implementation of basic functions:The square function is realised by the three-transistorcircuit presented in [17]. It is modelled as a nonidealblock containing the ideal square function itself, an additiveinput offset error and an additive output offset error.

In Fig. 3, current conveyors appear in three instances.First, they add the results of the square-function blocks. Inthis case, as all gains are unitary, there is no inner productinvolved. These current conveyors were modelled by theirinput-sensing offset errors and their output offset errors.Secondly, they implement inner products in the matrix-vector multiplication that computes j1(n), j2(n), . . . , j33(n).In this case, the current conveyors and their synaptic con-nections were modelled as described in Section 2.1.3.Thirdly, current conveyors can be used to add the WTAbinary results at the variable-length coder output.However, involving data that are essentially digital, theseadditions are insensitive to fabrication errors, and hencethe third instance of current conveyors was consideredideal in the simulations.

The WTA function is realised by parallel connections oftwo-transistor current conveyors, as proposed in [18].Owing to the symmetry of this operation, the nonidealmodel only takes into account the input-sensing offseterror of each jk(n), for k ¼ 1, 2, . . . , 33. There is nooutput offset error.

All the errors in the square function, current conveyorand WTA circuits are generated from the N(0, sp)distribution.

2.2.4 Simulations: As in Section 2.1.4, the errors inside apixel block were assumed IID N(0, sp) random variables,and the same set of error values was used for every pixelblock. The Monte Carlo algorithm was applied to eachECVQ of our design database, so that in each iteration thefollowing actions were taken:

1. Generate error values for the square function. In theexample of this Section, which involves four square functions,we drew four input errors and four output errors. They wereapplied at the points labelled as #1 and #2 in Fig. 3.2. Generate error values for the sum-of-squares(norm-square) current conveyor. In our example, we drewfour input-sensing errors (applied at dotted arrows #2) andone output offset error (applied at dotted arrow #3).

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3. Generate error values for the inner products. This issimilar to Section 2.1.4, except that in this ECVQexample the number of inner products is 33, so that wedrew 33 bias errors, 33 bias-sensing errors, 33 outputerrors, 132 input-sensing errors, 132 multiplier errors and132 multiplication output offset errors. They were appliedat the points labelled as #4 and #5 in Fig. 3.4. Generate error values for the WTA. In our example, wedrew 33 input-sensing errors. They were applied at the pointlabelled as #6 in Fig. 3.5. Keeping the perturbations constant, compute H and D.To compute D, we used the original codebook, obtainedfrom the training set, without any information about theGaussian perturbations.

For each ECVQ, 50 iterations were run, and the ECVQaverage values of H and D were stored. The simulationresults are given in Section 3.

3 Simulation results and practical implications

Using the methods mentioned in Section 2, 60 MLPs and 17ECVQs were designed with different l values. Each of thesesystems was tested for sensitivity, according to the modelspresented in Section 2, which generated a large amount ofdata. To simplify the exposition and discussion of thesedata, the results are divided in three Subsections. In Section3.1, we present results for the systems that remain within afixed margin of their ideal performance, having low sensi-tivity in the sense of low performance degradation. InSection 3.2, we discuss the relationships that exist betweenlow sensitivity and low complexity, verifying that they aredifferent for MLPs and ECVQs. In Section 3.3, we presentresults for systems that have good compression performanceregardless of their performance degradation.

Besides the HT-MLP, activation functions of radial-basis,polynomial and linear types are also considered in thisSection. The building blocks of radial-basis and polynomialfunctions are not described here (details can be found in[11] and [19]).

After the design, the rate-distortion (H, D) performanceof all 77 systems was measured over the training set xR,from which the codebook was designed, and over the testset xT, using the same codebook from xR, without the appli-cation of any parameter perturbation. The training set xR iscomposed by 480 512 unsigned vectors of size 4 � 1,obtained from 21 gray-level images. The test set xT is com-posed of 730 432 unsigned vectors of size 4 � 1, obtainedfrom 6 other gray-level images. Some of the test imagesare larger than the training images, and so the number ofvectors in xT is larger. All the images are typically usedin the literature for image compression problems. Togenerate the data vectors, all images were partitioned into4 � 4 pixel blocks. The average luminance was compressedby DPCM and removed from the blocks. A linear transform[20] was applied to the 4 � 4 residual blocks, and 12transform coefficients were discarded. The sign of thefour remaining coefficients was left unencoded (raw trans-mission) and the four unsigned coefficients form thevectors of xT or xR, to be compressed by VQ.

This generated the ideal performance curves that appearas dotted lines in Figs. 2 and 4. The left-hand side plotswere obtained from the training-set data, xR, and the right-hand side plots were obtained from the test-set data xT. Theplots in Fig. 5 were obtained from test-set data.

To perform the statistical analysis of all MLPs andECVQs we followed, for each of the 77 systems, the pro-cedures outlined in Sections 2.1.4 and 2.2.4. The random

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error values were drawn from a N(0, 22E) distribution, asmentioned in Section 2.1.3. We applied E ¼ 6, 8, 10, 12,14 and 16 bits in the numerical simulations.

3.1 Fixed maximum loss

For an MLP or ECVQ encoder with implementationprecision E, we defined the expected encoder qualityloss as DS(E) ¼ S 2 S(E), where S(E) is the averagesignal-to-quantisation noise (SQNR) of the encoder over50 realisations in a Monte Carlo experiment. The idealentropy and SQNR of an encoder suffering no implemen-tation error are denoted as H and S, respectively.

Letting P be the maximum quality loss desired for a givenencoder, we can find E such that DS(E) , P for thatencoder. This particular value of E is denoted as EDS,P.We assume that an encoder/decoder pair (codec) has asensitivity that is low enough to turn it robust in analogueimplementations with B-bit equivalent precision, if itpresents EDS,P , B bits over the test set.

Fig. 4a displays the computed (H, S) pairs of MLPs(curves marked with small circles) and ECVQs (curvesmarked with small squares), considering that no pertur-bation is applied to the encoder parameters. The largercircle and square symbols designated the MLPs andECVQs for which EDS,0.3 , 8 bits. Observe that most ofthe MLPs (eight of them) can be implemented with 8-bitprecision losing no more than 0.3 dB in SQNR, whereas8-bit realisations of ECVQs suffer loss larger than 0.3 dB.Fig. 4b shows the results obtained considering P ¼ 3 dBand B ¼ 8 bits. Simulations were also conducted with

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systems having an equivalent precision of B ¼ 10 bits, asshown in Figs. 4c and 4d.

The ideal curves (with no perturbation) show that,ideally, MLPs lose to ECVQs both in terms of compressionquality and entropy: the local optimality of the ideal ECVQsis due to the much higher complexity that they possess.When implementation precision is high, say 10 bits, the per-formance degradation is relatively small for MLPs andECVQs, as Fig. 4c and 4d show, and so MLPs also losein terms of compression quality and entropy for highimplementation precision. However, as the implementationprecision is reduced, the degradation of ECVQs becomesmuch more serious than that of MLPs, to the point thatthe best MLPs have better compression quality thanECVQs, although the entropy of these MLPs is stillhigher, as shown in Figs. 4a and 4b. In this situation, anupper convex hull of (H, S) points has not only ECVQsbut MLPs as well.

As the best ideal ECVQ systems have a significant SQNRadvantage, the SQNR difference between the best nonidealMLPs and the best nonideal ECVQs can be reduced byincluding ECVQs with high SQNR and relatively highSQNR degradation in the curves, which is the point of com-paring Figs. 4a and 4b. This point is discussed further inSection 3.3.

3.2 Sensitivity against complexity

MLPs have constrained complexity, as their structure isfixed throughout the training, to keep complexity low.When they are trained, the optimisation methods yield asolution of locally minimal J inside a set of fixed-structure

Fig. 4 Sensitivity with fixed loss

a P ¼ 0.3 dB, B ¼ 8 bitsb P ¼ 3 dB, B ¼ 8 bitsc P ¼ 0.3 dB, B ¼ 10 bitsd P ¼ 3 dB, B ¼ 10 bits

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encoders. The design of an ECVQ, on the other hand, allowsfor complexity-unconstrained local minimisation of J. As aconsequence, for the same H, an ECVQ achieves an SQNRthat is at least equal to that of the MLP. This advantagecomes at the expense of a much higher complexity, whichmay be prohibitive in practical implementations. As theMLPs have less parameters to suffer implementationerrors, it is expected that they will be more robust in low-precision analogue hardware realisation. This observationis supported by the results displayed in Figs. 4a and 4d.

If we choose a maximum allowable SQNR loss P, thenwe find a number of systems (MLPs or ECVQs) thatsatisfy DS , P. Each of these systems has an implemen-tation tolerance j and a complexity u. The percentage coef-ficient tolerance j represents the bounds of the intervalwithin which 99.7% of the nonideal coefficients stay.About 99.7% of the Gaussian perturbations applied to thecoefficients assume values between 23s and 3s, where3s ¼ 3/2E. We defined j as the percentage expression of3s. For example, j ¼ 0.5 means that 99.7% of the coeffi-cient perturbation values lie within +0.5% of themaximum signal level, which is 1.0. The complexity func-tion u operates on a given image compression system andreturns the number of CMOS transistors that would berequired for its analogue hardware implementation. Theplot of j versus u values, for DS , P ¼ 0.5 dB andE , 10, is shown in Fig. 5a. This result is rather intuitive.Assuming a maximum allowable SQNR loss (P), systemsthat are more complex will require more accurate coefficientimplementation (less tolerance).

Among the MLPs, 11 systems seem to have j ¼ 4.7 (6bits), because they achieve DS , 0.5 dB with tolerance

Fig. 5 Tolerance against complexity, and least-squares fitcoefficients

a Tolerance against complexity: E � 8 (dots), 8 , E � 10 (crosses),MLPs (circles), and ECVQs (squares)b Functions a(P) and b(P)

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greater than 4.7%, although these MLPs do not haveexactly the same tolerance requirement, which could beconfirmed by a simulation with E ¼ 4. Such a simulationwas not carried out, because E ¼ 6 is enough to representthe precision of current CMOS processes.

The relationship between j and u in Fig. 5a can be clari-fied by means of the linear least-squares fit between log(j)and log(u), that is,

logðjÞ ¼ aðPÞ logðuÞ þ bðPÞ ð1Þ

For example, the straight lines labelled ‘MLP’ and ‘ECVQ’in Fig. 5a were produced by the least-squares fits of theMLP and ECVQ encoders, respectively. The distributionof j and u values has a wide spread. Nevertheless, we pro-posed a linear fit relating log(j) to log(u), because the coef-ficients of this linear model, a(P) and b(P), changesmoothly as P varies from 0.1 to 5.0 dB. This shows thatthe distribution of j and u values has well-definedparameters, in spite of the large spread.

As the maximum acceptable SQNR loss, P, was variedfrom 0.1 to 5 dB we obtained, for the logarithm fitcoefficients, the functions aMLP(P), aECVQ(P), bMLP(P)and bECVQ(P), which are shown in Fig. 5b. If we find asystem which has tolerance that is not a strong function ofits complexity (i.e. a system with small jaj), then thissystem has an interesting property: it can be improvedby having its complexity increased, without making thetolerance requirements more stringent. Although theMLPs do not possess a particularly small jaj, especiallyat low P, their tolerance becomes less dependent oncomplexity as the maximum loss is allowed to increase,and they enjoy a high b at any allowable loss, meaningthat low-complexity MLPs have high tolerance toimplementation errors. The ECVQs behave in the oppositeway for all P less than 4.0 dB.

3.3 Free maximum loss

In this experiment we considered all MLP and ECVQ enco-ders that had been simulated for an implementationprecision of 6 or 8 bits, regardless of their SQNR loss. Ifthe implementation precision is high, such as 16 bits, allsystems can achieve their full performance, that is, equalto that expected from their ideal designs. In more realisticsituations, such as in 8-bit precision implementations, wenoted from Fig. 2a that MLPs achieve a smaller PSNR,whereas their complexity remains smaller than that ofECVQs. Moreover, in 6-bit precision realisations, MLPsare consistently better in terms of SQNR sensitivity andcomplexity, as indicated in Fig. 2b.

Regarding the theoretical justification of these results, wenote that all the results presented in Sections 3.1, 3.2 and 3.3can be predicted from two observations: (i) MLPs havelower complexity, so that they have fewer parameters tobe disturbed by the implementation imperfections (Figs. 2and 4); (ii) their optimisation is constrained by the limitsthat were set on complexity, which causes the maximum Jto be less conditioned on the parameters obtained fromthe optimisation process (Fig. 5). As the derivatives of theLagrangian cost with respect to the parameters cannot becomputed simply in these problems, a theoretical treatmentof this sensitivity against complexity trade-off problemremains open for future investigations.

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Fig. 6 DPCM drift

a Encoder block-mean datab Decoder raw datac Decoder post processed data

4 Conclusions

The study advanced in this paper has added new evidence tothe previous observations that were favourable to the prac-tical realisation of MLPs for focal-plane image compressionapplications, instead of full-search structure-unconstrainedECVQ schemes, for the following reasons:

† Simple MLPs can be implemented with less SQNR lossfor a given level of precision, in comparison to morecomplex ECVQs, as shown in Fig. 5.† For very low complexity levels and a large range ofSQNR loss, MLPs have looser tolerance requirements tokeep the loss small, as shown in Fig. 5b.† Setting the precision level but not worrying about theSQNR loss, to focus on the average SQNR after implemen-tation, the best SQNR at very low precision (6 bits) isobtained by MLPs, as indicated in Fig. 2b. For higherimplementation precision, ECVQs have an SQNR advan-tage over the test set data.

We are currently extending the perturbation models, toenable rate-distortion simulations with more advanced non-ideal effects obtained from electrical simulations of theschematic diagrams of all building blocks. Emphasis isbeing placed on the multiplicative error of the single-transistor synapse and on the nonlinear effects of thecurrent-conveyor circuit.

Another future research topic concerns the preprocessingstage that is used to generate the data vectors x(n). In thatstage, DPCM is used to encode the mean luminance valueof each pixel block. The existence of a feedback loop atthe DPCM predictor will cause scalar quantisation errors,unknown to the decoder because they are due to the fabrica-tion process, to accumulate in a way that the decoder cannotpredict and, as a consequence, cannot correct entirely.Although it does not impair the operation of the VQstage, the accumulation of DPCM errors causes degradationof the image quality, because pixel block average luminancevalues are not reconstructed correctly. An example of sucheffect is illustrated in Fig. 6

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