Analytical-form model observers for decompressed images

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B188 J. Opt. Soc. Am. A/Vol. 24, No. 12 /December 2007 D. Li and M. Loew

Analytical-form model observers fordecompressed images

Dunling Li1,* and Murray Loew2

1Texas Instruments Inc., 20450 Century Boulevard, MS4041, Germantown, Maryland 20874, USA2Department of Electrical and Computer Engineering, George Washington University, Washington, D.C. 20052, USA

*Corresponding author: [email protected]

Received March 5, 2007; revised August 7, 2007; accepted August 28, 2007;posted September 10, 2007 (Doc. ID 80594); published November 12, 2007

We report a method for evaluating the performance of model observers for decompressed images in analyticalform using compression noise statistics. It derives test statistics and detectabilities for the ideal observer, thenonprewhitening observer, the Hotelling observer, and the channelized Hotelling observer (CHO) on decom-pressed images. The derived CHO performance is validated using the Joint Photographic Experts Group(JPEG) compression algorithm. The validation results show that the derived CHO receiver operating charac-teristics (ROCs) and areas under ROC curves predict accurately their corresponding estimated values. Theseanalytical-form quality measures of decompressed images provide a way to optimize compression algorithmsanalytically, subject to a model-observer performance criterion. They also provide a theoretical foundation forefforts to create a model observer for decompressed images. © 2007 Optical Society of America

OCIS codes: 110.3000, 100.3010, 110.4280, 110.6980.

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. INTRODUCTIONhis paper derives model observers’ performances on de-ompressed images using analytical-form compressionoise statistics. Ideally, an analytical-form quality mea-ure for decompressed images would allow us to comparearious compression algorithms and parameter sets with-ut using extensive image samples. The measure wouldlso provide a way to find optimum compression schemesr parameter sets for clinical diagnostic tasks.

As digital imaging technology advances, the amount ofedical image data increases and the need for com-

ressed images becomes more obvious. It has become arucial research area to estimate the effect of image com-ression on the accuracy of clinical diagnosis. The mostommonly used measurements of image quality, such asean-square error (MSE) or peak signal-to-noise ratio,

re not adequate for medical images. Medical image qual-ty can be better measured by human performance in vi-ual tasks that are relevant to clinical diagnosis. Thetandard method of evaluating diagnostic methods is a re-eiver operating characteristic (ROC) study [1], which isime consuming and costly because it requires a largeumber of human observations. This is compoundedhen the set of parameters changes.Model observers are algorithms that aim to predict hu-an visual performance in noisy images; they might rep-

esent the desired metric of image quality when the diag-ostic decision involves a human observer and a visualask [2]. Among all model observers, the ideal observerets an upper bound to the performance of any observer,ncluding the human; it requires knowledge of therobability-density function (PDF) of the backgroundoise. The channelized Hotelling observer is one of theost efficient and practical algorithms for prediction of

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uman performance [3–5]; it requires the first and secondoments of the background noise.Model observers have been used successfully to esti-ate decompressed image quality experimentally [6–14]

n the past. This paper first presents the compressionoise statistics of transform coding, then derives the testtatistics and performance of the ideal observer, the non-rewhitening (NPW) observer, the Hotelling observer, andhe CHO for decompressed images. These observers areurther simplified by approximating the decompressedmage statistics using background images and compres-ion noise statistics. Finally, the derived CHO and NPWbservers’ performances are validated using the Jointhotographic Experts Group (JPEG) compression algo-ithm.

. Transform Coding and the JPEG Compressionlgorithmransform coding reduces redundancy by decomposinghe source input into components whose individual char-cteristics allow more efficient representation of eachomponent than the original input [15]. To reduce theomputational complexity and memory consumption,ransform coding usually first divides the original imagesnto nonoverlapping small blocks. Then, for each block, ainear reversible transform is used to decompose theource input into a set of transform coefficients. Theransform coefficients are further quantized and coded forransmission or storage.

A typical transform image compression process ishown in Fig. 1. In transform coding, a linear reversibleransform is used to convert the source input into a set ofransform coefficients. The Karhunen–Loeve transformKLT), the discrete Fourier transform, the discrete cosine

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D. Li and M. Loew Vol. 24, No. 12 /December 2007 /J. Opt. Soc. Am. A B189

ransform (DCT), the Walsh–Hadamard transform, andther transforms have been studied extensively. Mostransforms A are orthonormal, that is, they satisfy AAT

I. The KLT, a data-dependent transform, consists of theigenvectors of the autocorrelation or covariance matrixf the data, and minimizes the mean of the variance of theransform coefficients. It can be shown that the KLT pro-ides the largest transform coding gain relative to anyther transform. The KLT, however, usually is not practi-al due to its computational complexity. The DCT is theost popular transform in image-compression applica-

ions because of its implementation simplicity and com-action ability, which is very close to that of the KLT forarkov sources with a high correlation coefficient. TheCT has been chosen for many international standards,

uch as JPEG, MPEG, and H.261.The JPEG still-image data-compression standard in-

ludes both lossless and lossy compression methods16,17]. A predictive algorithm is used in the losslessPEG compression while a DCT-based transform codinglgorithm is used for the lossy compression. In the en-oder, JPEG first partitions an original image into 8�8locks and applies the forward DCT to each block sequen-ially. After the DCT coefficients are calculated, they areuantized using uniform scalar quantizers and the quan-ization step sizes are defined in the quantization table,hich is not part of JPEG, but supplied by the user asart of the compressed image. Based on the statisticalharacteristics of the quantized DCT coefficients, entropyncoding achieves additional compression losslessly byncoding them more compactly [16,17]. Huffman codingnd arithmetic coding algorithms are specified in thePEG standard. In a JPEG decoder, the corresponding en-ropy decoding process decodes compressed data into theuantization index. Then the dequantization process con-erts the index into its corresponding quantized DCT co-fficient value. After that, the inverse DCT transformshe quantized DCT coefficients into the pixel values of theecompressed blocks. The last process is to combine theecompressed blocks into a decompressed image.

Fig. 1. Image transform coding process.

Fig. 2. Exampl

. Model Observersodel observers are task-performance-based, image-

uality assessment techniques. The objective of the modelbserver is to accurately predict human performance be-ause the ultimate observer will be a human. The modelbservers can be used for system evaluation and optimi-ation with some assurance that the best system for theodels is also the best for a human.The output of a model observer is a test statistic for de-

ection tasks. For two-alternative forced choice (2AFC)ests, model observers simulate human behavior thatould apply with the simultaneous, side-by-side presen-

ation of the two alternative image fields as shown in Fig.[1]. One contains noise only and the other contains sig-al plus noise. The observer is required to make an un-quivocal decision about which image contains the signal;he signal itself is assumed known. Obviously 2AFC testsave binary decisions and their outcomes are shown inable 1. True positive (TP) and true negative (TN) are theorrect outcomes, while false positive (FP) and false nega-ive (FN) are the incorrect decisions. The observer’s per-ormance can be defined by the TP fraction (TPF) and FPraction (FPF). TPF is the probability of a true positiveecision, while FPF is the probability of a false-positiveecision in 2AFC tests. The relationship between FPFnd TPF can be portrayed as a ROC curve, which is de-ned as the plot of FPF versus TPF.The performance of a model observer can be evaluated

y figures of merit for the ROC curve. Commonly used fig-res of merit for the model observer performance are areander the curve (AUC) and detectability dA, which are de-ned as

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We can interpret 2AFC as two-class hypothesis testing:ne hypothesis is an image without signal, i.e., g� =n� ; thether is an image with signal, i.e., g� =n� +s�, where n� , s�,nd g� are the noise image, signal image, and observed im-ge, respectively. The test statistic of a model observer is=T�g� �, where T�g� � is the observer’s discriminant func-ion. Model observers can be categorized as optimal oruboptimal. The ideal observer is optimal and its test sta-istic is the likelihood ratio. Most commonly used modelbservers use linear discriminant functions and can be

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escribed as �=WTg� . For instance, the test statistic of theonprewhitening observer is �NPW�g� �= �� 2−�� 1�Tg� . Earlyfforts on model observers concentrated on the ideal ob-erver. Later, other observer models, such as NPW ob-erver with eye filter and Hotelling observer, were alsoroposed and investigated. Among them, the CHO pro-ides a practical and effective way to predict human ob-erver performance. Therefore, this paper presents theerformance of the CHO on decompressed images.Model observers have been used successfully to evalu-

te decompressed image quality experimentally in theast. The statistics of decompressed images are, however,equired to study model observers analytically. For ex-mple, the ideal observer requires knowledge of the PDFf the decompressed images, while linear model observersequire the first and second moments of the decompressedmages. Section 2 presents the statistics of decompressedmages of transform coding.

. COMPRESSION NOISE STATISTICS OFRANSFORM CODINGtypical image transform coding process is shown in Fig.

. It is a block-based image compression algorithm.ransform coding first divides the original images intoonoverlapping small blocks and performs the same lin-ar transform for each block to produce a set of transformoefficients. The transform coefficients are further quan-ized and coded for transmission or storage. In Fig. 1,ince unpacking/entropy decoding generates exactly theame quantization index Iq as the input to entropy-oding/packing processes, the distortion from lossy com-ression comes from quantization and dequantizationlone. The amount of compression—the compressionatio—depends, however, on both the quantization andhe entropy-coding processes. Though the various imple-entations of forward and inverse transformation may

ntroduce distortion, it is insignificant compared withuantization noise. Therefore, the noise contributed fromhe finite precision in the transform calculation is negli-ible. From the compression distortion point of view, Fig.can be simplified as Fig. 3, in which compression noise

Table 1. Decision Outc

utcome Sign

ignal detected �D1� Truignal not detected �D2� Fals

ig. 3. (Color online) Image transform coding with additiveuantization noise.

distortion) R is a linear transform of quantization noise. Since model observers consider a given image as a one-imensional vector and transform coding is usually a two-imensional block-based process, the block-based imageransform coding is presented as a one-dimensional ex-ression in this paper [10].If the M�N original image is divided into Nb blocks,

hen the original and transform coefficients can be de-ned as X� = �X� 1

T,X� 2T, . . . ,X� Nb

T �T and Y� = �Y� 1T,Y� 2

T, . . . ,Y� Nb

T �T,espectively, where X� i and Y� i are vectors for original im-ge and transform coefficients at the ith block. The pro-ess of transform coding can be expressed as

Y� = AX� �3�

nd

X� r = ATY� q, �4�

here A is the block-based transform matrix [10–12,14],� and X� r are the original and decompressed images, re-pectively; while Y� and Y� q are the original and quantizedransform coefficients; respectively. If the quantizationoise Q� = �Q� 1

T ,Q� 2T , . . . ,Q� Nb

T �T and compressed noise R�

�R� 1T ,R� 2

T , . . . ,R� Nb

T �Tare defined as

Q� = Y� q − Y� �5�

nd

R� = X� r − X� , �6�

here Q� i and R� i are ith block quantization and compres-ion noise vectors, respectively, then the mean vector�R� � and covariance matrix Cov�R� � of compression noisere

E�R� � = ATm� Q� �7�

nd

Cov�R� � = ATCov�Q� �A, �8�

espectively, where m� Q� and Cov�Q� � are the mean vectornd covariance matrix of the quantization noise [10,14].The decompressed image X� r is the sum of the original

mage X� and compression noise R� . As compression noises introduced by quantization, we assume that X� and R�re independent; thus the mean vector and covarianceatrix of decompressed images are E�X� r�=E�X� �+E�R� �

nd Cov�X� r�=Cov�X� �+Cov�R� � respectively. The joint PDFf compression noise is

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PR� �R� � = �n� �Dn�pY� �AR� + Tn� �/�A�, �9�

here pY� is the joint PDF of the transform coefficients;� = �n� 1

T ,n� 2T , . . . ,n� Nb

T �T; and Dn� is defined as Dn� = �n� T :nij�−� , . . . , � , i= �1,2, . . . ,Nb , j= �1,2, . . . ,B2, where B islock size and Nb is the number of blocks in transformoding. By assuming that most of the quantization noiseomponents in one block are independent, then from theyapunov central limit theorem we conclude that thearginal PDFs of compression noise are distributed nor-ally. To simplify the computation for the joint PDF of

ompression noise, we assume that it has a jointly Gauss-an distribution, i.e.,

pR� �R� � = exp� − ��R� − E�R� ��TCov�R� �

��R� − E�R� ��/2�/�2��NbB2�Cov�R� ��, �10�

here E�R� � and Cov�R� � are the mean vector and covari-nce matrix that are defined in Eqs. (7) and (8), respec-ively. The above statistics of decompressed images will bepplied to derive the analytical-form CHO for decom-ressed images.

. MODEL OBSERVER FOR DECOMPRESSEDMAGESf the corresponding vectors of original image, signal im-ge, and background image are defined as X� , S� 0, and N� ,hen

X� =� N� signal absent

S� 0 + N� signal present. �11�

f R� is compression noise, then the decompressed image� r can be expressed as

X� r =� N� + R� 1 signal absent

S� + N� + R� 2 signal present, �12�

here S� is the detection signal, and R� 1 and R� 2 are com-ression noise with signal present and absent, respec-ively. The mean vectors and covariance matrices of de-ompressed images are

E�X� r� =� E�N� � + E�R� 1� signal absent

S� + E�N� � + E�R� 2� signal present�13�

nd

Cov�X� r� =�Cov�N� + R� 1� signal absent

Cov�N� + R� 2� signal present, �14�

espectively.Theoretically, compression noise R� i is a function of the

riginal image N� and the quantization matrix. As com-ression noise is a linear transform of quantizationoise—which for fine quantization step sizes may not de-end on the quantizer input—we assume that compres-ion noise is independent of the original images to sim-

lify the statistics of the decompressed images. Therefore,ov�X� r �Ti�=Cov�N� �+Cov�R� i�, where T1 and T2 representignal absent and present, respectively. The PDFs of theecompressed images are p�X� r �T1�=�PN�N� �PR1�X� r

N� �dN� and p�X� r �T2�=�PN�N� +S� �PR2�X� r−N� �dN� for T1 and2, respectively.If the background image N� is a lumpy-background im-

ge, i.e., has a spatially correlated Gaussian distribution,hen the PDFs of decompressed images are Gaussian, ashe compression noise has a Gaussian distribution.herefore, the joint PDF of decompressed lumpy-ackground images is

p�X� r�Ti� = exp�− �X� r − m� i�Cov�X� r�Ti�−1

��X� r − m� i�/2�/�2��L�Cov�X� r�Ti��, �15�

here i= �1,2, L is the number of pixels in the originalmages, m� 1=E�N� �+E�R� 1�, and m� 2=S� +E�N� �+E�R� 2�.

To perform 2AFC tests on decompressed images, theignal S� can be the original signal image S� 0 or its decom-ressed signal image S� r. The first case would apply whenhe observer does not have any experience with a signalistorted from compression, while the latter case repre-ents the situation in which the observer has full knowl-dge of signal distortion caused by compression. The fol-owing sections will use S� to derive the test statistics andetectability of commonly used model observers for theecompressed images. The top and bottom rows in Fig. 4how the 2AFC tests for the original and decompressedumpy-background images with circle disk signal, respec-ively. Figures 4(A) and 4(C) are the original lumpy-ackground images with signal present and absent; Figs.(D) and 4(F) are decompressed JPEG images with signalresent and absent; Figs. 4(B) and 4(E) are the originalnd decompressed signals, in each case respectively.

. Ideal Observer2AFC test can be interpreted as two-class hypothesis

esting: one hypothesis is an image without signal, thether an image with signal. The ideal observer uses theaximum a posteriori probability decision criterion toinimize decision error. The test statistic of the ideal ob-

ig. 4. 2AFC tests for original and decompressed lumpy-ackground images.

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erver on decompressed images can be calculated as theollowing log-likelihood ratio:

�ideal�X� � = log�P�X� r�T2�/P�X� r�T1��. �16�

If the original images are lumpy-background images,hen we can show that the test statistic of the decom-ressed images is

�ideal�X� r� = �S� 0 + E�R� 2� − E�R� 1��T�Cov�N� �

+ �Cov�R� 1� + Cov�R� 2��/2−1X� r, �17�

here E�R� i� and Cov�R� i� are the mean vector and covari-nce matrix, respectively, of the decompressed images fori, i= �1,2. The detectability of the decompressed imagean be derived as

dideal2 = �S� 0 + E�R� 2� − E�R� 1��T�Cov�N� � + �Cov�R� 1�

+ Cov�R� 2��/2−1�S� 0 + E�R� 2� − E�R� 1��. �18�

Usually the mean vectors of compression noise are ze-os because a quantizer was designed to minimize aver-ge distortion in transform space. Therefore, the test sta-istic and detectability of an ideal observer onecompressed images are

�ideal�X� r� = S� 0T�Cov�N� � + �Cov�R� 1� + Cov�R� 2��/2−1X� r

�19�

nd

dideal2 = S� 0

T�Cov�N� � + �Cov�R� 1� + Cov�R� 2��/2−1S� 0.

�20�

. Hotelling Observern 1931 Hotelling proposed the T2 measure to test theull hypothesis that two samples of random vectors wererawn from populations with the same mean vector. Itas used as a feature extraction method in linear dis-

rimination. If S1 and S2 are the interclass scatter matrixnd intraclass scatter matrix respectively—i.e., S1P1P2�� 2−�� 1�� 2−�� 1�T and S2=P1K1+P2K2, where Pi,

� i, and Ki are the prior probability, mean vector, and co-ariance matrix, respectively, of class i= �1,2—then theotelling trace is defined by J=tr�S2

−1S1�, where tr is therace of the matrix. The Hotelling observer computes S1nd S2 using sample means and covariance matrices in-tead of the ensemble values. A key difference is that thensemble S2 matrix usually is invertible, but the samplene usually is not. The test statistic and detectability forhe Hotelling observer are given by �hot= �� 2−�� 1�TS2

−1X�

nd dhot2 = �� 2−�� 1�TS2

−1�� 2−�� 1�, where X� is an observedmage. We can show that the test statistic and detectabil-ty of the Hotelling observer on decompressed images are

�hot = S� T�Cov�N� � + �Cov�R� 1� + Cov�R� 2��/2−1X� r �21�

nd

dhot2 = S� T�Cov�N� � + �Cov�R� 1� + Cov�R� 2��/2−1S� , �22�

espectively, where S� is the detected signal in 2AFC tests.hen S� is the original signal S� 0, the Hotelling observer

ecomes the ideal observer for decompressed lumpy-ackground images.

. NPW Observerhe test statistic of the NPW observer depends only on

he mean difference between two hypotheses. The testtatistic and detectability of the NPW observer on decom-ressed images are

�NPW�X� r� = S� TX� r �23�

nd

dNPW2 = 2�S� TS� �2/�S� T�Cov�N� � + �Cov�R� 1� + Cov�R� 2��/2S� �,

�24�

espectively, where X� r is the decompressed image, S� is theetected signal in 2AFC tests, Cov�N� � is the covarianceatrix of the original images, and Cov�R� 1� and Cov�R� 2�

re covariance matrices of the compression noise for sig-al absent and present, respectively.

. Channelized Hotelling Observerhe CHO can be considered a Hotelling observer thatakes its detection on the channel responses instead of

he images. The channel response of the original imagess defined as �� =TTX� , where T is a channel matrix inhich each column is the spectral profile of a channel, i.e.,

he spatial weights of the channel (see Subsection 4.B). Till be an L�J matrix if J channels are applied to the�1 original image X� . The channel response of the de-

ompressed images can be calculated by �� r=TTX� r=��

R� ch, where R� ch is the channel response of compressionoise and R� ch=TTR� . The CHO test statistic and detect-bility evaluated on decompressed images are

�chot�X� r� = S� T„TT�Cov�N� � + �Cov�R� 1�

+ Cov�R� 2��/2T…

−1TTX� r �25�

nd

dchot2 = S� TT„TT�Cov�N� � + �Cov�R� 1� + Cov�R� 2��/2T…

−1TTS� ,

�26�

espectively, where N� are original images with signal ab-ent, X� r are the decompressed images, S� are the signals inAFC tests, and R� 1 and R� 2 are the compression noise forignal absent and present images, respectively.

. Approximationo calculate the model observer performance on decom-ressed images, the covariance matrices of compressionoise both with signal absent and present are required. Inetection tasks of medical applications, for a given cat-gory of images, the statistics of background images aresually stable, but the detected signal varies. That is be-

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ause background images represent normal images, whilehe detected signal represents tumors or lesions. To sim-lify the computation of model observers for decom-ressed images, we use the covariance matrix of compres-ion noise for background images with signal absent topproximate the one with signal present. Thus, the cova-iance matrix term in model observers’ test statistics andetectabilities becomes

�Cov�N� � + �Cov�R� 1� + Cov�R� 2��/2 Cov�N� � + Cov�R� 1�,

here N� and R� 1 are the original background image andts compression noise, respectively. This approximationemoves the requirements of detected-signal-related com-ression noise statistics; it allows us to calculate the the-retical model observers’ performances for various de-ected signals using the same background-imagestatistics. It is a reasonable assumption because modelbservers usually perform under low detected-signal lev-ls and that would introduce only small amounts ofhange in the compression-noise statistics.

For lumpy-background images described in Section 4,able 2 shows the relative MSE differences of covarianceatrices between signal-absent and -present decom-

ressed images. The relative MSE between covarianceatrices C1 and C2 is defined as

Drmse = �i=1

L

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L

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L

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here L�L are the dimensions of C1 and C2. The de-ected signals in Table 2 are circle disks S1 (radius 10 andmplitude 10), S2 (radius 10 and amplitude 5) and S3 (ra-ius 5 and amplitude 5). In Table 2, Cn is the derived co-ariance matrix of signal-absent decompressed images,hile Cs1, Cs2, and Cs3 are the derived covariance matri-

es of decompressed images with signals S1, S2, and S3resent, respectively. The relative MSE differences be-ween signal-absent and -present decompressed imagesre 1.18%, 2.33%, and 0.43% for signals S1, S2, and S3,espectively. One can see that the difference decreaseshen the signal amplitude or radius decreases. The cor-

esponding differences among signal-present covarianceatrices are 3.49%, 1.18%, and 1.23%. Table 2 also lists

he relative MSE of variances between two sets of 102402�32 WGN images as reference. It is found to be 1.95%n MATLAB simulations.

Figure 5 shows the comparison of the above covarianceatrices in a 2D plot. Figure 5(A) is the covariance ma-

rix Cn of decompressed images, while Figs. 5(B)–5(D)how the differences between Cn and Cs1, Cs2, and Cs3,espectively. One can see that the differences are rela-

Table 2. Relative MSE (Percent) between CovarImages with Diff

GNa �Cn,Cs1� �Cn,Cs2� �Cn

.95% 3.501 1.1798 0.4

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ively small; therefore, the covariance matrices of signal-resent decompressed images can be approximated by theignal-absent one.

. VALIDATION TESTShis section validates the derived CHO’s performances.he following three types of statistically known imagesre used in the validation tests: lumpy-background im-ges, uncorrelated uniform noise images, and uncorre-ated Gaussian noise (WGN) images. The circle disk andaussian disk signals chosen in the validation tests are

hown in Fig. 6. Figures 6(A) and 6(B) are the originalnd decompressed circle disks while Figs. 6(C) and 6(D)re original and decompressed Gaussian disks. The circleisk signal can be represented as S�R ,A�, where R and Are the radius and amplitude, respectively. The Gaussianisk can be defined as S�� ,A�, where �2 is the variance ofhe Gaussian distribution and A is the signal amplitude.he size of images is 32�32. To simplify the simulation,

he statistics of each background image are fixed, but thentensity of the signal is changed and the radius of theircle disk or the variance of the Gaussian disk mayhange as well.

The lumpy-background images are generated accordingo the method of Eq. (28) below. The amplitude p is cho-en as 255. The white Gaussian and uniform noise imagesre generated using a modified version of Marsaglia’subtract-with-borrow algorithm in MATLAB. Their am-litudes are chosen as 128 and 12812. The dynamicange of AUCs is chosen roughly between 0.9 and 1. Theovariance matrices are fixed for each type of background,nd various AUCs are obtained by adjusting the signalevels and sizes.

. Statistics of Lumpy-Background Imageshis section will use several JPEG, decompressed, lumpy-ackground images to compare the derived compressionoise statistics and CHO performance on decompressed

mages. Lumpy-background images are mathematicalodels to synthesize medical images, such as mammo-

raphic images [17–20]; they were created to evaluate aarticular imaging modality and anatomy using modelbservers. The lumpy-background images are generatedy the low-pass filtering of uncorrelated Gaussian noise.he original images used in this paper are generated by

he following steps: i. generate uncorrelated Gaussian im-ges with zero mean; ii. calculate the 2D Fourier trans-orm of the generated images; iii. multiply the Fourier co-fficients by low-pass filter coefficients pixel-by-pixel; iv.alculate the 2D inverse Fourier transform of the filteredourier coefficients and take the real parts of the outputss lumpy-background images.The lumpy background can be expressed as

Matrices of Lumpy-Background DecompressedPresent Signals

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X� = p Re�F−1WFN� �, �28�

here p is the power level of uncorrelated Gaussian noise,and F−1 are the forward and inverse Fourier transformatrices represented in one dimension, and W is a diag-

nal matrix whose diagonal elements are filter coeffi-ients. Uncorrelated Gaussian noise images with zeroeans and unit covariance matrices are given by N� .he lumpy-background images that have a jointlyaussian distribution are represented by X� . Its mean

s zero, and its covariance matrix is Cov�X� �p Re�F−1WF�Re�F−1WF�T.The transform coefficients Y� of lumpy-background im-

ges are Y� =AX� =pA Re�F−1WFN� �, where A is the trans-orm matrix of block-based DCT in the JPEG compressionlgorithm. The mean vector and covariance matrix of theransform coefficients are m� y=Am� x=0� and Cov�Y� �pA Re�F−1WF� Re�F−1WF�TAT, respectively. The jointDF of the transform coefficients is

PY� �Y� � = exp�− Y� TCov�Y� �−1Y� /2�/�2��L�Cov�Y� ��, �29�

here L is the number of pixels in one lumpy-backgroundmage. The marginal PDF of the ith transform coefficients

PY� �yi� = exp�− yi2/2�yi

2 �/2��yi

2 , �30�

here �yi

2 =Covi,i�Y� �. Its pairwise joint PDF is

Fig. 5. (Color online) Comparison of covariance

Fig. 6. Original and d

PY� �yi,yj� = exp�− �yi2/�yi

2 − 2�i,jyiyj/��yi�yj

� + yj2/�yj

2 �/

2�1 − �i,j2 �/�2��yi

�yj1 − �i,j

2 �, �31�

here �i,j2 = �Cov�Y� ��i , j��2 / �Cov�Y� ��i , i�Cov�Y� ��j , j��.

To reduce the computational complexity of the modelbserver performance on decompressed images, the vali-ation tests use an image of size 32�32. The decom-ressed images are generated using a floating-pointCT/IDCT transform and the JPEG recommended quan-

ization table. Because entropy coding for quantization in-ices does not introduce any distortion, we eliminate en-ropy coding and decoding in the simulation. To find theompression ratio, however, requires implementing thentropy-coding process. Figures 7(A) and 7(B) show theerived and estimated covariance matrices of JPEG com-ression noise for lumpy-background images, respec-ively. Their differences, shown in Fig. 7(C), are smallompared with those of Figs. 7(A) or 7(B). It is clear thathe derived covariance matrix can well predict the actualovariance matrix.

. Laguerre–Gauss Channelshe CHO consists of a set of spatial frequency channels

hat are used to model the human visual system in detec-ion tasks. Its detection process operates on the channelutputs whose dimensionality has been greatly reducedy the channelization process. Therefore, the CHO is veryomputationally efficient. Several channel profiles haveeen used to predict human performance [19,20]. Since

ces of lumpy-background decompressed images.

ressed signal images.

matri

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he signal used in the simulation is symmetrical and theackground images are smooth, Laguerre–Gauss func-ions are chosen to define the channels in the simulationf the CHO. The nth-order Laguerre–Gauss functionsre defined as LGn�r�=2� /a2 exp�−�r2 /a2�Ln�2�r2 /a2�,here a is width and Ln is the nth-order Laguerre poly-omial, defined as

Ln�x� = �m=0

n

�− 1�m� n

m� xm

m ! .

Figure 8 shows the first five Laguerre–Gauss functionshat are used in the simulation. From left to right in Fig.

are the first–fifth-order 32�32 Laguerre–Gauss tem-lates. The channel matrix T is constructed by combininghe five templates in column vectors together. Therefore,

is a 1024�5 matrix.

. Performance of CHO on Decompressed JPEG Imageshe performance of the CHO in 2AFC tests can be mea-ured by the ROC curve and the AUC. Both derived andstimated ROC curves are computed using the test statis-ic of Eq. (25) by varying the decision thresholds. The de-ived ROC, however, uses the derived covariance matriceshile the estimated ROC uses the covariance matrices

hat are estimated from actual decompressed images. Theerived AUC value can be calculated from the CHO de-ectability of Eq. (26) on decompressed images using Eq.7). The estimated AUC can be calculated using the esti-ated ROC curve.

Fig. 8. Laguerre

Fig. 7. (Color online) Derived and estimated covariance m

The JPEG compression algorithm with default quanti-ation table is used to compress the original images.aguerre–Gauss channels in Fig. 8 are chosen in the cal-ulation of CHO performance. The validation tests useircle and Gaussian disks as the detected signals. EachAFC test chooses 4096 images with signal and 4096 im-ges without signal for lumpy-background images, uni-ormly distributed random images, and Gaussian-istributed random images.Table 3 lists AUCs of the CHO under various condi-

ions. The first column lists the background images. Theecond column lists the original signals: RxAy representscircle disk with radius x and amplitude y, while �xAy

epresents a Gaussian disk with standard deviation x andmplitude y. The third and fourth columns list the de-ived and estimated AUCs of CHO on the original images.he fifth, sixth, and seventh columns list the derived, es-

imated, and approximated AUCs on decompressed im-ges using the original signal in 2AFC tests, while theighth, ninth, and tenth columns are derived, estimated,nd approximated AUCs using the decompressed signaln 2AFC tests. All of the test cases show that the derivedHO AUCs accurately predict the corresponding esti-ated ones on decompressed images, and the approxi-ated AUCs agree closely with their corresponding actualUCs.Comparing various test cases, one can conclude that

he CHO AUCs increase monotonically as the amplitudencreases for a given signal size. For example, the AUCsf circle disk R10A10 are larger than those of R10A5; the

s channel profile.

of compression noise for JPEG lumpy-background images.

–Gaus

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UCs of Gaussian disk �10A20 are larger than those of10A10. CHO AUCs do not, however, increase monotoni-ally with the signal extent for a given amplitude. For ex-mple, the CHO AUCs of R5 circle disk signal are theargest among the other radii (R10, R12, R3) signals, andHO AUCs of Gaussian disk �3A20 are larger than thosef �5A20. Comparing with the AUCs on original imagesn the same case, one can conclude that the compressionistortion reduces the AUCs on both original and decom-ressed signal for circle disks. For example, in the case ofircle disk R10A10, the derived AUC on original images is.9665, and the AUCs on decompressed images are 0.9478nd 0.9582 for original and decompressed signals, respec-ively.

Compression does not, however, always degrade theUCs for Gaussian signals. For example, in the case ofaussian disk �5A20, the derived AUC on original im-ges is 0.9180 while the AUCs on decompressed imagesre 0.9280 and 0.9301 for the original and decompressedignal, respectively. Also, using the decompressed signaloes not always reduce ROCs. For example, the AUCs ofecompressed images for circle disk R10A10 on originalignal are smaller than those of the decompressed signal,ut for R10A5 the reverse is seen.Figure 9 presents the CHO ROCs of lumpy-background

mages with circle disk signal R10A5. The solid curve andhat with star symbols (asterisks) are the derived and es-imated ROCs on the original images, respectively. The

Table 3. AUCsaof CHO

ackground/Signal Original Images

ackgroundmages Signal Original Signal

Originalsignal Derived Estimated Derived

umpy R10A10 0.9665 0.9663 0.9478R10A5 0.9305 0.9320 0.9181R12A5 0.8631 0.8632 0.8497R5A3 0.9955 0.9954 0.9724R3A5 0.8950 0.8963 0.8639

R3A10 0.9939 0.9942 0.9860�5A20 0.9180 0.9162 0.9300�3A20 0.9665 0.9663 0.9478

�10A20 0.9897 0.9890 0.9903�10A10 0.8764 0.8739 0.8788

aussian-distributed

R5A30 0.9214 0.9207 0.9214R5A40 0.9703 0.9701 0.9704R10A30 0.9978 0.9976 0.9978R5A10 0.6813 0.6810 0.6814�5A40 0.9748 0.9744 0.9748�5A30 0.9289 0.9282 0.9289

�10A30 0.9955 0.9956 0.9955�5A10 0.6876 0.6871 0.6877

niformlydistributed

R5A30 0.9214 0.9207 0.9214R5A40 0.9703 0.9699 0.9704�5A30 0.9289 0.9276 0.9289�5A40 0.9748 0.9741 0.9748

aNormalized units.

ashed curve and those with plus symbols and dashes–ots are, respectively, the derived, estimated, and ap-roximated ROCs on decompressed images containing theriginal signal. The dotted curve and those with circlesymbols and dots are the derived, estimated, and approxi-ated ROCs on decompressed images containing the de-

ompressed signal. All the curves in Fig. 9 show that theerived ROCs predict their corresponding estimatedOCs and that the approximated ROCs are very close to

heir derived ROCs.

. CONCLUSIONased on the compression noise statistics of transformoding, this paper derives in analytical form the perfor-ance of the ideal observer, the NPW observer, the Ho-

elling observer, and the CHO for decompressed images.t also mathematically approximates those model observ-rs using the compression noise statistics of backgroundmages while making the decompressed image statisticsndependent of signal changes.

The validation tests show that (1) the derived CHO per-ormance matches the simulated performance, and (2) thepproximated analytical model observers show close pre-ictions of the actual performance in the simulation. Theenefit of such analytical-form image quality measures ishat they allow users to calculate the performance of im-ge compression algorithms without going through com-

ecompressed Images

Decompressed JPEG Images

al Signal Decompressed Signal

ted Approximated Derived Estimated Approximated

8 0.9536 0.9582 0.9584 0.94909 0.9047 0.7792 0.7784 0.77466 0.8280 0.8006 0.7989 0.79764 0.9625 0.7001 0.6998 0.70116 0.8801 0.8746 0.8755 0.82176 0.9906 0.8525 0.8532 0.85630 0.9178 0.9301 0.9288 0.92998 0.9536 0.9582 0.9584 0.94905 0.9896 0.9878 0.9871 0.98772 0.8760 0.8701 0.8677 0.8657

7 0.9203 0.9320 0.9314 0.93100 0.9697 0.9718 0.9715 0.97116 0.9977 0.9980 0.9978 0.99809 0.6805 0.6988 0.6980 0.69804 0.9747 0.9758 0.9753 0.97561 0.9286 0.9216 0.9210 0.92135 0.9955 0.9954 0.9954 0.99530 0.6874 0.6527 0.6515 0.6524

5 0.9203 0.9320 0.9312 0.93108 0.9697 0.9718 0.9713 0.97114 0.9286 0.9216 0.9201 0.92130 0.9747 0.9758 0.9750 0.9756

for D

Origin

Estima

0.9450.9160.8490.9680.8640.9860.9280.9450.9890.876

0.9200.9700.9970.6800.9740.9280.9950.687

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ression and decompression processes on extensive imageamples and various parameter sets. It also is a theoreti-al approach that enables the user to choose compressionarameters that predictably satisfy the tradeoff betweenompression ratio (or the size of the lossy-compressed im-ge) and the preservation of diagnostic information.It is important to note that the compression ratio in the

PEG compression algorithm is determined by both theuantization table and the entropy-coding scheme, butompression distortion depends on the quantization tablelone (ignoring round-off errors). It is for that reason thathe compression ratio is not calculated here: We are ex-mining only the compression-induced distortion and itsffects on detection; entropy coding has no effect on thatistortion, and hence none on the detection measures.hese closed-form quality measures of decompressed im-ges provide a way to optimize compression algorithmsubject to a model-observer performance criterion. Theylso provide a theoretical foundation for efforts to create aodel observer for decompressed images. We thus have

hown the theoretical underpinnings, demonstrated aracticable implementation, and made clear the clinicalalue of this unified approach.

EFERENCES1. H. Barrett and K. Myers, Foundations of Image Science

(Wiley, 2004).2. M. Eckstein, C. Abbey, and F. Bochud, “A practical guide to

model observers for visual detection in synthetic andnatural noisy images,” in Handbook of Medical Imaging, J.Beutel, H. Kundel, and R. van Metter, eds. (SPIE Press,2000), Vol. 1, Chap. 10, pp. 593–628.

3. H. Barrett, J. Yao, J. Rolland, and K. Myers, “Modelobservers for assessment of image quality,” Proc. Natl.Acad. Sci. U.S.A. 90, 9758–9765 (1993).

4. A. Burgess, X. Li, and C. Abbey, “Visual signal detectabilitywith two noise components: anomalous masking effects,” J.Opt. Soc. Am. A 14, 2420–2442 (1997).

5. C. Abbey and H. Barrett, “Human- and model-observerperformance in ramp-spectrum noise: effects ofregularization and object variability,” J. Opt. Soc. Am. A18, 473–488 (2001).

Fig. 9. (Color online) ROC curves for CHO o

6. M. Eckstein, C. Abbey, F. Bochud, J. Bartroff, and J.Whiting, “The effect of image compression in model andhuman performance,” Proc. SPIE 3663, 284–295 (1999).

7. C. A. Morioka, M. P. Eckstein, J. L. Bartroff, J. Hansleiter,G. Aharanov, and J. S. Whiting, “Observer performance forJPEG versus wavelet image compression of x-ray coronaryangiograms,” Opt. Express 5, 8–19 (1999).

8. B. Schmanske and M. Loew, “Bit-plane-channelizedHotelling observer for predicting task performance usinglossy-compressed images,” Proc. SPIE 5034, 77–88 (2003).

9. B. Schmanske, “Task-based assessment of quality appliedto compressed radiological images,” Ph.D. dissertation(George Washington University, 2003).

0. D. Li, “Statistics of transform coding and assessment ofdecompressed image quality,” Ph.D. dissertation (GeorgeWashington University, 2007).

1. D. Li and M. Loew, “Closed-form quality measures forcompressed medical images: Statistical preliminaries fortransform coding,” 25th Annual International Conference ofIEEE Engineering in Medicine and Biology Society (IEEE,2003).

2. D. Li and M. Loew, “Closed-form quality measures forcompressed medical images: compression noise statistics oftransform coding,” Proc. SPIE 5372, 218–229 (2004).

3. D. Li and M. Loew, “Model-observer based qualitymeasures for decompressed medical images,” IEEEInternational Symposium on Biomedical Imaging: Nano toMacro (IEEE, 2004), pp. 832–835, Vol. 1.

4. D. Li and M. Loew, “Closed-form compression noise inimages with known statistics,” Proc. SPIE 5749, 211–222(2005).

5. K. Sayood, Introduction to Data Compression, 2nd ed.(Morgan Kaufmann, 2000).

6. A. Papoulis, Probability, Random Variables, and StochasticProcesses, 3rd ed. (McGraw-Hill, 1991).

7. G. Wallace, “The JPEG still picture compression standard,”Commun. ACM 34, 30–44 (1991).

8. W. B. Pennebaker and J. L. Mitchell, JPEG: Still ImageData Compression Standard (Van Nostrand Reinhold,1993).

9. B. D. Gallas, “Signal detection in lumpy noise,” Ph.D.dissertation (University of Arizona, 2001).

0. C. K. Abbey, M. P. Eckstein, S. S. Shimozaki, A. H.Baydush, D. M. Catarious, and C. E. Floyd, “Human-observer templates for detection of a simulated lesion inmammographic images,” Proc. SPIE 4686, 25–36 (2002).

y-background images with circle disk signal.
n lump

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Analytical-form model observers for decompressed images

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