Predicting detection task performance using a visual ...

Predicting detection task performance using a visual discrimination model Dev P. Chakraborty University of Pittsburgh, 3520 5th Ave, Pittsburgh PA 15213 ABSTRACT In the visual discrimination model (VDM) approach to measuring image quality two input images are analyzed by an algorithm, which calculates the just-noticeable-index (JND) index, which is a measure of the perceptual difference between the two images in the discrimination task. It has been proposed that if one can simulate relevant lesions and backgrounds, the same method can be used to predict target detectability. One generates pairs of images which are exactly identical except for the presence of a lesion in one of them. The JND-index measured on this image pair is thought to correlate with target detectability, such as might be measure in a receiver operating characteristic (ROC) study, and some experimental studies supporting this idea have appeared. It is pointed out in this work that this method can lead to anomalous results, namely it does not predict the qualitative effect of lesion-size on lesion detectability in mammographic backgrounds. Another anomaly is that the method appears to work on single images, whereas the ROC method needs sets of normal and abnormal images. In this work we show that by modifying the method so that comparisons of near-identical images are avoided, it is possible to predict the lesion size dependence and avoid the clash with the ROC method.

Keywords: Observer performance, detection, discrimination, ROC, image quality, model, mammography

1. INTRODUCTION The terms detection performance and image quality are sometimes used synonymously in medical imaging but while the former can be objectively measured using receiver operating characteristic (ROC) methodology [1, 2] it is more difficult to objectively quantify image quality. To measure clinical image quality, as in the MQSA program [3], one needs to ask a mammographer to rate the image(s) on standardized criteria. Note that clinical image quality can be defined for a single image. In the ROC approach to measuring detection performance one presents images which can be normal or abnormal and measures the classification accuracy of the observer, i.e., how good he is at separating the normal images from the abnormal images, as measured by the area under the curve (AUC). Using a well-known theorem [4] it can be shown that AUC is the same as the probability that a randomly picked abnormal case will appear more suspicious for abnormality than a randomly picked normal case. Note that the images in each paired comparison in this two-alternative forced choice (2AFC) task are from different patients and are uncorrelated. Note also that using the ROC method it is impossible to define detection performance for a single image. The visual discrimination model (VDM) is a widely-used method of measuring image quality [5], [6] which has been successful in several optimization applications. In this approach one inputs two images, a test image and a reference image to the VDM-algorithm. Typically the test image is a slightly modified version of the reference image – e.g., the reference image can be a battlefield scene without a tank, and the test image is the same scene with a tank. The VDM algorithm, which is based on human visual system characteristics, determines a just-noticeable-difference (JND) map for this image pair. At locations where the JND-map is large, the VDM theory predicts that it should be easy to discriminate between the two images. A single-valued measure calculated from the JND-map, termed the JND-index, is often used as an image quality metric for discrimination of the target. It has been proposed [7], [8] that the VDM can be used to measure radiological image quality. In the proposed application the test and reference images that are input to the algorithm are quasi-identical, i.e., they are identical at all

Medical Imaging 2004: Image Perception, Observer Performance, and Technology Assessment, edited by Dev P. Chakraborty, Miguel P. Eckstein, Proceedings of SPIE Vol. 5372 (SPIE, Bellingham, WA, 2004) · 1605-7422/04/$15 · doi: 10.1117/12.533270

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locations except where a lesion is present. To make this point obvious, note that if one were to mathematically subtract the two input images one would be left with the lesion only – the anatomic background would completely cancel out. The JND-index produced by the analysis is proposed as a measure of image quality for discrimination of the lesion. Note that JND-based image quality can be defined for a single image (the normal image) and the task (the lesion). Note also that unlike the 2AFC method, the images entering the paired comparison in the VDM method are quasi-identical and highly correlated. Provided that the two input images to the VDM are indistinguishable from clinical images, it seems quite reasonable to use the JND-index as a measure of clinical image quality. Lacking a precise definition that the author is aware of, clinical image quality of a normal image may be defined as the probability that a lesion will be visible, if it were present in the image at a specified location. Depending on the visibility of recognizable structures in the surround, the observer can probably form an opinion as to how likely a particular lesion would be to be visualized, if it were present at the center of the region. For example, in the hilar-region of a chest radiograph the radiologist knows, based on the difficulty of seeing structures that are known to be present (e.g., the heart), that it will be difficult to see a mass, if it were present, and therefore hilar-region image quality would be rated lower than the lung-region image quality. By comparing a lesion-containing ROI with a normal ROI of the same patient, the VDM is essentially doing a similar task. However, it has been claimed recently that the VDM can measure not just clinical image quality, but that the JND-index can be used as a surrogate for detection performance, i.e., for ROC studies. It has been recently shown that performance measured using the VDM and that measured using the ROC method correlated very well with each other [8], [9, 10]. If this claim were universally true a contradiction would result: on the one hand it is impossible to even define, let alone measure, detection performance by the ROC method on single images. On the other hand the VDM method yields a quantity that, while it is measured on single images, correlates very well with detection performance measured on ensembles of images using the ROC method. If the JND-index did correlate well with ROC detection performance, that would make the VDM method vastly preferable to performing time-consuming ROC experiments. Interest is rarely in measuring absolute detection performance, rather one is mostly interested in relative detection performance, and answering questions such as “is modality - A superior to modality - B for the detection of lung cancer”. Based on this premise, approximately 3-years ago the author got interested in applying VDM technology to medical imaging, and in collaboration with a team at Sarnoff, he was successful at getting funding from the National Institutes of Health (NIH) for a project titled “Perceptual Model for Workstation Display Optimization”. The rationale for the project was that if one could simulate lesions and backgrounds that were indistinguishable from real lesions and backgrounds, then one could exploit this ability to generate pairs of images, with and without a lesion, for analysis by the VDM software. The JND-index output of the VDM program could be used to optimize a workstation display for digital mammography. [At least one other project involving similar usage of the VDM was also funded by the NIH. Both projects had provisions for testing the assumption that the VDM measurement would correlate with the ROC measurements.] Initial work on our project focused on the simulation of mammographic abnormalities and testing that they were indistinguishable from real abnormalities, in which effort we were quite successful [11]. However, while performing initial tests with the VDM program using Gaussian signals superposed on mammographic image backgrounds, it became clear to us that the VDM was making an incorrect prediction regarding the relative visibility of small vs. large Gaussian shaped lesions [12], when both had the same peak-contrast. The VDM was predicting that the larger lesion should be more visible than the smaller one. However, it had recently been reported [13, 14] that for mammographic backgrounds, which have unusually strong noise power spectra components at low spatial frequencies, large lesions are actually less visible than small ones at the same peak contrast. The explanation of this initially non-intuitive finding was that the large lesion had more signal-power in the lower frequency bands, where it is more likely to be camouflaged by the low frequency noise (this finding was independently verified by the author [15]). The initial studies with Gaussian lesions showed that we had found at least one example where the VDM and ROC measurements did not agree, and the difference was so large, see Fig. 1, that some would question the need to perform an ROC study to confirm this finding. The image on the left in Fig. 1 is a normal mammogram region with a large superposed Gaussian lesion, and the one on the right is the same region with a small superposed Gaussian lesion with the same peak contrast. Examination of these images leaves little doubt that the small lesion has much higher visibility.

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A thought-experiment further illustrated the nature of the problem. Consider an ideal observer who has access to the test and reference image pixel values and subtracts the images. This would exactly cancel the deterministic background leaving behind the lesion superposed on the observer’s statistically random internal noise (which would not cancel out). In other words, the ideal observer could eliminate the statistically fixed mammographic background, and therefore eliminate the very reason for the unusual dependence of visibility on lesion size that is observed in mammographic images. This thought-experiment made it clear that the real problem was that the quasi-identical VDM presentation mode differed fundamentally from the clinical detection task. In the clinical task patient images are viewed one at a time, and abnormal cases must be distinguished from normal ones, so that two sets of images originating from different patients must be distinguished. While one can argue that real observers are not ideal, that real lesions are not Gaussianshaped and that for a given lesion shape and size the VDM might still predict the correct ordering of externally applied image processing, to the author this seemed like a significant hurdle for the current VDM usage paradigm. The author felt that the central assumption that one can predict ROC detection performance based on VDM measurements needed greater examination. This led to the present work. In the next section we describe the method we used to quantify the effects described above and to develop a new method of using the VDM to fix the lesion-size detectability anomaly noted above and resolve the clash with ROC methodology.

2. METHODS 2.1. Images We started with a set of 134 normal mammograms, which were digitized at 12-bits per pixel and 100-micron resolution using a Lumisys LS-100 digitizer. The digital images were displayed at full resolution on a 5 mega-pixel monitor (Siemens model SMM21200P). Using a commercially available photometer and commercially available software (DOME-TQA, Dome Imaging Systems) the 2048 x 2560 pixel (the pixel linear dimension was 14.3 microns) monitor was calibrated under low ambient lighting conditions according to the DICOM standard for a maximum luminance of 300 Cd/m2. This defined the transformation from display driving level (DDL) values (0 to 255) to CRT luminance (Cd/m2). Each displayed image was manually windowed by the author for optimal visibility of the breast. The windowlevel function was a standard linear look-up-table which converted all pixels below min to 0 DDL, and all values above max to 255 DDL (the display had an 8-bit DDL-range) and values in between were linearly interpolated. 2.2. Regions if interest (ROIs) Multiple non-overlapping regions of interest (ROIs) were extracted, around author-selected locations, from each digitized mammogram. The number of extracted regions per mammogram varied from 5 (for small breasts) to 50 (for large breasts). We restricted the regions so that they originated from parts of the breast with approximately constant thickness; specifically, the skin-line region was excluded. Each extracted ROI was a 256 x 256 matrix, and each was randomly, and with equal probability, assigned a truth status, T = 0 or T = 1. If T = 0 the extracted region was used unmodified to simulate a normal case. If T = 1 the original ROI was modified by adding one of four possible Gaussian shaped simulated lesions at the center of the ROI, thereby simulating an abnormal case. Symbolically, an abnormal region of interest (ROI) denoted by A (a 256 x 256 matrix) is given by

A= N

 −r  ) 1 + C exp( 2σ 2   2

Eqn. 1

Here r is the distance of the pixel from the ROI center and N is the original extracted ROI (also a 256 x 256 matrix). The parameter σ was set to 10 or 100 pixels and this represented the simulated lesion size. The contrast C of the lesioncenter was set to 0.1 or 0.2. In this manner we simulated abnormal regions containing large and small lesions at low and high contrasts, and an approximately equal number of normal regions. Table 1 tabulates the numbers of regions in the different categories.

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Table 1: Cross-tabulation of the numbers of ROIs in the different contrast (C = 0.1 and 0.2), size (σ = 10 and 100), normal (T = 0) and abnormal (T = 1) categories. All ROIs are from different cases or different regions. C = 0.2

C = 0.1

Status

Total σ = 100

σ = 10

σ = 100

σ = 10

T=0

1071

1091

1082

1055

4299

T=1

1076

1056

1065

1092

4289

Total

2147

2147

2147

2147

8588

All ROI pixel-values were transformed to DDL values using the min and max values determined for the individual images. Using the measured CRT luminance transfer characteristic, which was available from the monitor calibration software, the DDL values were transformed to luminance. All calculations from this point on used the luminance values of the pixels, not the actual pixel values. 2.3. Paired Analysis For each ROI in Table 1 we created the complementary case and region matched ROI: i.e., for each T= 0 region we created the matching T = 1 region by adding a lesion and for each T =1 region we created the matching T = 0 region by subtracting the lesion. Note that these images are identical except in the vicinity of the lesion: in fact at locations r >> σ, see Eqn. 1, the regions are precisely the same. These matching pairs of ROIs were analyzed by the VDM program (we use the program IDM, for Image Distortion Metric, version 9.10a (03/2)) which yielded the JND index. The individual channel outputs of the program were not used in this analysis. 2.4. Unpaired Analysis: For each normal or abnormal ROI represented in Table 1, we constructed a corresponding uniform 256 x 256 ROI, in which all pixels had the same value, equal to the mean pixel value of the corresponding normal or abnormal ROI. Each normal or abnormal region from Table 1 and the corresponding uniform region was regarded as a test-image - reference image pair, and submitted to the VDM software. We recorded the 26-channels of outputs of the VDM model provided by the program; the JND-index output was not used. The channel outputs are represented by the vector X, which is a 26element object, each component object of which is a mini-image, 16 x 16 pixels at 4 bytes per pixel (i.e., floating point values). Each mini-image was interpolated up to 256 x 256 pixels, and a normalized cross-correlation was calculated of this mini-image weighted by the known Gaussian lesion profile, as shown below.

Ωn =

∑ ∑

− rkl 2 ) 2σ 2 − rij 2 exp( 2 ) 2σ

Χ nkl exp(

kl

kl

Eqn. 2

The normalized cross-correlation was defined as the net output Ωn of the nth channel. Here n is the channel number (1, 2, …, 26), and k and l are integer indices (1, 2, …, 256) that index the pixels in the 256 x 256 mini-image and rkl is the distance between the pixel indexed by k and l and the center of the mini-image. This analysis was done using the appropriate value of σ for the region, as in Table 1. Specifically, note that σ depends on the size of the lesion. In this manner we characterized each region in Table 1 by a corresponding 26-element vector Ω.

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2.5.

Data Analysis 2.5.1.

Paired Analysis

A repeated measures ANOVA was used to assess the significance of the observed differences between the means of the JND-values corresponding to the 4 conditions (corresponding to C = 0.1 or 0.2 and σ = 10 or 100).

2.5.2. Unpaired Analysis We started with a logistic regression analysis of the data. Logistic regression allows one to find the optimal linear combination of the 26 output channels that best predicts the truth status of the region (T = 0 for normal and T = 1 for abnormal). A stepwise regression method implemented in SAS (SAS Institute Inc., Cary, NC, version 8.02) was employed that allowed us to determine which channels were most predictive in classifying the regions. The classification accuracy of the optimal logistic model was assessed by AUC, the trapezoidal area under the ROC curve, which describes the classification accuracy of the logistic regression model. This quantity is provided by the logistic regression SAS software. Logistic regression makes the assumption of independence between the 8588 different ROIs, but our data did not satisfy this condition, since some of the ROIs originated from the same mammogram. For this reason we did a second analysis using the generalized linear model (GLM) procedure available in SAS. In this analysis we included all predictor variables found to be significant in the above logistic regression analysis. We specified an exchangeable correlation structure between observations originating from the same case, and saved the predicted probabilities p for further analysis. The predicted probability p is a continuous variable in the range 0 to 1, which can be regarded as a rating. If one selects a threshold t, then all regions with p > t will be classified as abnormal regions by the generalized linear model, and the rest will be classified as normal regions. The absolute truth for these regions was known according to whether they originated from T = 0 or T = 1 regions. This allowed one to define ROC-like quantities, namely true positives and false positives and the ROC curve traced as the threshold is varied. The area under the ROC curve is a measure of the classification accuracy of the model. The predicted probability p was used to calculate the trapezoidal area under the ROC curve for each condition. [This was done in the standard manner by comparing the predictions for all possible pairs of regions. We incremented a variable by 1 if a normal rating was less than the abnormal rating and by 0.5 if they were equal. In this manner we calculated the probability that a normal region picked at random yielded a rating less than an abnormal region picked at random.] This was repeated for the 4 conditions (corresponding to C = 0.1 or 0.2 and σ = 10 or 100). To assess the significance of the differences between the observed AUC-values, we performed a jackknife analysis in which we removed each of the 134 cases, one-at-a-time, re-computing the AUC each time, and transformed this number to a pseudo-value. In this manner we reduced the data to 134-pseudovalues for each of the 4-conditions studied. A repeated measures ANOVA was used to assess the significance of the observed differences between the means of the pseudo-values corresponding to the different conditions.

3. RESULTS 3.1. Paired Analysis The JND-index outputs of the VDM-analysis were averaged over all images and regions, and standard deviations were calculated. Table 2 shows the mean JND index for the 4 conditions studied, and the corresponding standard deviations.

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Table 2: Paired analysis, JND and σ(JND) for the two contrasts and lesion size studied. Note that mean (JND) increases with lesion size, which is not the correct prediction, as evident from an examination of Fig. 1. Contrast

Size

mean(JND)

σ(JND)

0.1

10

6.45

.041

0.1

100

7.22

.041

0.2

10

8.66

.041

0.2

100

9.62

.041

As can be seen from this table, the mean JND-index increased with size and contrast. Using a mixed model ANOVA analysis both of these effects were determined to be highly significant (p < 0.001). The finding that the mean JND-index increased with size confirmed our initial findings, that the VDM predicts that lesion detectability increased with lesion size, which is the correct behavior for most tasks, except for mammographic backgrounds, where it is known that the opposite is true [13]. The contrast dependence of the VDM was as expected, namely detectability increases with contrast, which is not controversial.

3.2. Unpaired Analysis The result of this analysis is summarized in Table 3. In this table, instead of the AUC values we show the corresponding detectability, often referred to as d’ in the ROC literature. The transformation from AUC to d’ is well known [16]. This makes the comparison between the paired (Table 2) and unpaired (Table 3) analyses more transparent, since the JNDindex is like a detectability index for discrimination, and is unbounded, unlike the AUC which is less than 1. Table 3: Unpaired analysis, Logistic Regression Analysis, AUC for the two contrasts and lesion sizes studied. Contrast

Size

mass detectability d’

Channels

.1

10

1.83

26, 25, 24, 23, 16, 10, 8, 7, 2

.1

100

0.348

25, 8, 4, 2, 1

.2

10

3.25

26, 23, 20, 8, 7, 2, 1

.2

100

0.507

25, 18, 4, 2, 1

This table shows that AUC increases with contrast as expected. It also shows that AUC decreases with size, for both contrast values, which agrees with expectations for this type of background. This is the fundamental fix of the VDM model that we found. In the GEE model analysis we used the following predictor channels: 1, 2, 4, 7, 8, 10, 16, 18, 20, 23, 24, 25 and 26. The results are shown in Table 4.

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Table 4: Unpaired analysis, GEE Analysis, AUC for the two contrasts and lesion sizes studied. Contrast

Size

AUC

Channels

.1

10

1.83

26, 25, 24, 23, 16, 10, 8, 7, 2

.1

100

0.348

25, 8, 4, 2, 1

.2

10

3.25

26, 25, 24, 23, 16, 8, 7, 2, 1

.2

100

0.507

25, 4, 2, 1

Note that with a few exceptions the channels identified by the GEE procedure and those identified by the logistic regression procedure were identical. The same conclusions that were made about the dependence of the AUC on contrast and size apply with the GEE analysis. In particular, as lesion size increases, the ability of the linear classifier to discriminate between abnormal and normal regions decreases. This is consistent with the experimental observations (see Fig. 1). The results of the GLM analysis of the pseudo-values revealed that all differences in AUC observed in Table 2 were significant, except the difference between the two AUC values for the large lesions (0.5973 vs. 0.6400), corresponding to d’ values of 0.348 and 0.528, respectively.

4. DISCUSSION We have shown that the conventional way of using the VDM does not predict the correct behavior of Gaussian mass detectability when lesion size was varied. By using the single-ended analysis mode and using a linear combination of the VDM-channels to predict lesion presence, we found that we could predict the qualitative dependence of lesion detectability on lesion size. The general approach for using the VDM that is suggested by this work is as follows: Collect a set of normal images or regions. Superpose lesions simulating the clinical task of interest on about half the images, thereby simulating the abnormal set. Apply the VDM to all images in single-ended mode and calculate the channel output vector for each image. Determine the optimal channel combination that best predicts the truth status of the images. This completes the calibration of the VDM for the task of interest. This calibration may be used to predict the effect on ROC performance of different image processing algorithms, without having to do ROC experiments. One applies the algorithm of interest to all the images and recalculates the channel output vector for each image. Using the optimal linear combination determined above, one calculates the predicted truth status probability for each image. Finally, using these predicted probabilities as image ratings one measures ROC performance. Provided the calibration is invariant under the processing and the over-fitting has not occurred, the resulting VDM-predicted ROC performance should be correct – i.e., match that measured using an actual ROC experiment. Whether this method will always work is too early for us to say. Clearly more work is needed to probe this method further. However, the proposed approach does resolve the problem referred to in the Introduction – namely, it does not allow one to measure detection performance from a measurement on a single image.

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5. ACKNOWLEDGEMENTS This work was supported in part by a grant from the Department of Health and Human Services, National Institutes of Health, 8 RO1-EB002120. The author is grateful to Dr. Steve Hillis and Andrei Bandos for assistance with the statistical analysis, and to Dr. Jeffrey Lubin, Dr. Jeffrey Johnson and Dr. John Nafziger for assistance with the VDM analysis.

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Figure 1: The image on the left is a 256 x 256 region extracted from a normal mammogram with a superimposed Gaussian lesion with C = 0.8 and σ = 100. The one on the right is the same extracted region with a superimposed Gaussian lesion with C = 0.8 and σ = 10. Conventional VDM usage predicts that detectability of the lesion in the left image is greater than that on the right, see Table 2, by about 11%. The modified VDM usage predicts that the lesion on the right is about 6 x the detectability of the lesion on the left – see Tables 3 or 4. The lesion contrast used to generate these images is 4 x higher than the maximum value used in the study.

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