Alan H. Baydush, Ph.D., Carey E. Floyd Jr, Ph.D*
Thoracic Imaging Research Division
Department of Radiology
Duke University Medical Center
*and
Department of Biomedical Engineering
Duke University
Supporting grants:
National Cancer Institute
RO1-CA60849
Previously, it has been shown that Bayesian image estimation (BIE) can reduce the effects of scattered radiation and improve contrast-to-noise ratios (CNR) in digital radiographs of anthropomorphic chest phantoms by improving contrast while constraining noise. Here, the use of BIE as a noise reduction technique is reported.
An anthropomorphic phantom was imaged with a previously calibrated photostimulable phosphor system using standard bedside chest radiography protocols. The Bayesian technique was then used to process this image. BIE incorporates a radial exponential convolution scatter model with two adjustable parameters. In previous reports, these parameters were optimized to reduce the residual fraction of scattered radiation in the processed image. Here, the parameters were adjusted to evaluate the potential of BIE to reduce image noise. While the full-width at half-maximum of the scatter model was held constant, the magnitude was varied.
Evaluation was based on residual scatter fractions and CNR.
The magnitude of the kernel in the scatter model was varied from 0.0 to 2.5 in steps of 0.5. Previously, it was found that an "ideal" scatter kernel magnitude of 2.33 provided a minimum residual scatter fraction. This magnitude corresponds to the average scatter-to-primary ratio in the chest radiograph. As the magnitude was increased, the residual scatter fraction decreased and the CNR increased in both the lungs and the mediastinum. However, as the magnitude was decreased, the percent noise also decreased; therefore, a lower magnitude kernel reduces noise. By varying the magnitude of the kernel used, differing amounts of noise reduction and contrast enhancement can be obtained. These results demonstrate that Bayesian image estimation can be used to both increase contrast and decrease noise in digital chest radiography.
The detection of scattered x-ray photons in chest radiography can cause serious degradation of image contrast. In some regions of the chest, this contrast loss may be as much as 90%[1]. Several different physical techniques have been used to reduce scatter before detection. Anti-scatter grids and air gaps are examples of such devices[2-7]. Scanning and aperture techniques have been shown to be effective although mechanically complex[8,9]. A second approach is to remove the effects due to scatter post detection using digital image processing techniques. Initially, only linear techniques, such as deconvolution and convolution subtraction, were used[1,10,11]. Recently, investigation has begun into the use of non-linear statistical estimation techniques for scatter compensation. Bayesian image processing is one such statistical estimation technique for scatter reduction which can also constrain noise in the chest radiograph. Here, this noise reduction capability of BIE is described.
Previously, it has been shown that Bayesian image estimation (BIE) can improve the contrast-to-noise ratio (CNR) by improving contrast, while constraining image noise in digital chest radiography[12]. The technique can be viewed as a two step procedure, where the first step is scatter compensation and the second is noise constraint. First, scatter is removed from the image, which results in improved contrast. Scatter is modeled as a convolution of the scatter reduced image with a kernel that has two free parameters: magnitude and full-width at half-maximum (FWHM). Next, image noise is constrained by applying statistical priors to the image. The overall result is improved CNR. Previous results have been shown for full or "ideal" scatter compensation. Contrast was enhanced 233%, while noise was only increased 130%. Therefore, the net increase in CNR was 80%. For comparison, a typical deconvolution technique will increase contrast and noise equally for no net increase in CNR[13].
Previously, BIE was optimized only for its scatter reduction potential. For optimal scatter compensation, the scatter-to-primary ratio (SPR) was chosen as the magnitude of the kernel. Since a shift-invariant model was assumed, the average SPR over a large region of the chest was used. As stated above, the results of this study showed a large increase in contrast and a constrained or smaller increase in noise. The net result was an increase in CNR.
The present investigation examines the effect of partial scatter compensation. For this investigation, the magnitude of the scatter kernel was varied from the value which resulted in full scatter compensation. Chest radiographs were then processed using BIE with this scatter kernel. Partial scatter compensation yielded images with less noise, less contrast enhancement, and less CNR improvement than in the full scatter compensated case. While maximum CNR is appropriate for detection of low contrast pulmonary nodules, image noise is visually distracting to radiologists and overall radiographic impression may be enhanced by some noise reduction even at the cost of some contrast and CNR improvement.
This study provides a flexible method by which the operator, by choosing a certain kernel magnitude, can adjust the amount of scatter compensation and, hence, the amount of contrast improvement and noise reduction. While an observer preference or performance study would more accurately test the usefulness of the BIE technique, the authors believe that a more in depth study of the affect of BIE on quantitative image parameters should be investigated first. Inter-relations of image parameters should be carefully examined given the wide range of images that can be obtained by using the BIE technique with different algorithm parameters. Knowing the effect of algorithm parameters on image quality will help reduce the complexity of observer performance studies planned for the future.
A. Background
Bayesian image estimation is a non-linear statistical estimation
technique. As developed for chest radiography, this technique is a Maximum
a posteriori or MAP estimator of the scatter-reduced x-ray image.
This iterative algorithm maximizes the probability of the scatter reduced image
given the measured image. BIE is based on Bayes theorem, where Y represents
the measured image and
represents the estimate of the scatter
reduced image.
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yields the general BIE
equation. Realizing that maximizing a monotonic function of f is synonymous
with maximizing the function itself and that ln f(Y) is a constant with respect
to yields eq. 3.
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B. Models
Implementation of the Bayesian image estimation technique incorporates two
models for the proper statistical estimation of the scatter reduced image. The
first model is of the scattering process, while the second model is of the
prior information assumed about the image noise structure.
1 Scatter Model
Scattered radiation was modeled as a convolution of the primary radiation with
a radial exponential scatter kernel[1,15].
The radial exponential has two free parameters, which are the full-width at
half-maximum and the magnitude. The FWHM of the scatter kernel controls the
range of pixels which are assumed to contribute to the scatter estimation at
any spatial location. The magnitude of the kernel controls how much scatter,
on average, is assumed to be in the chest radiograph, and hence, how much
scatter is removed.
The maximum likelihood estimate is calculated using the Maximum Likelihood
Expectation Maximization (MLEM) algorithm[16],
as developed for emission computed tomography (ECT) by Shepp and Vardi[17]
and as applied to chest radiography by Floyd et. al.[18]
The MLEM algorithm reduces to the following:
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2 Noise Model
For the noise model, the image prior probability distribution was chosen to be
a simple Gibbs' prior, which was defined as follows:
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is an adjustable parameter which will be discussed later and
U is the energy function. The Gibbs' prior was chosen following similar work
in ECT[14,19-21].
U was defined as:
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The Gibbs prior presented above functions to constrain pixel fluctuations in the estimated image. By using a potential function that depends only on pixel differences within the first order neighborhood, constraints are placed on how any pixel varies from its neighbors. Small fluctuations in pixel values within a neighborhood are penalized and smoothed, while larger variations are not. Under the assumption that large fluctuations are more likely to be caused by image structure than by noise, this approach allows for noise constraint. Implementation showed that this noise constraint can be achieved without significant loss of resolution[12].
Incorporating equations 5 and 6 into equation 3 yields:
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, as stated previously, is an adjustable parameter
that regulates the dependence of the Bayesian estimate on the prior and on the
maximum likelihood estimate. As is increased towards infinity,
the Bayesian estimate approaches the MLEM estimate. As is
decreased, the Bayesian estimate approaches the constraints placed upon the
image by the prior.
C. Procedure
Quantitative digital chest radiographs of an anthropomorphic phantom
[Humanoid Systems, Carson, Ca.] were acquired both with and without
low-contrast tissue equivalent (12 mm thick polystyrene) disks in several
regions of the chest. These images were then processed by the BIE algorithm,
which takes approximately 50 seconds per iteration. The magnitude of the
scatter kernel was varied from 0.0 to 2.5 by steps of 0.5. This resulted in
differing amounts of scatter compensation, with varying levels of CNR
improvement. For each kernel magnitude, 20 iterations of the BIE algorithm
were performed with output images examined every 2 iterations. Contrast,
noise, CNR, and residual scatter fraction (RSF) were measured for each output
image. Resolution was also examined for each kernel magnitude.
D. Image Acquisition
Digital images were acquired using a photostimulable phosphor x-ray imaging
system (PPS) [Philips Computed Radiography Model 901: Philips Medical Imaging
of North America, Shelton, CT.] which has previously been evaluated for exposure[22] and scatter fraction[23]
measurements. Exposures were made at 95 kV and 1.25 mAs on Type II imaging
plates. Imaging plates were 36 cm x 43 cm and were sampled at 1760 by 2140
square pixels (0.2 mm/pixel side). Two images were obtained with each
exposure, to obtain 224 equally spaced (25 mm) scatter measurements using the
posterior beam stop technique[23](PBS).
Both images from the PBS acquisition were processed in the PPS system using a
fixed speed and latitude to facilitate ease of conversion of the digital images
to calibrated exposure values. Both images were then transferred to an image
processing system. The 1760 by 2140 pixel images were truncated in one
dimension and padded in the other to create a 2048 by 2048 pixel image. These
images were then resampled to 1024 by 1024 (0.4 mm/pixel side) by averaging
adjacent pixels. The full 10-bit intensity resolution was maintained
throughout this procedure. Once in this format, each pixel value was converted
to a floating point value that corresponded to the exposure.
E. Measurements
Measurements of contrast, noise, and CNR were obtained from two exposures; one
with and one without low contrast tissue equivalent disks superimposed on the
anatomical phantom. RSF was computed from scatter measurements which were
acquired by using the PBS technique.
1 Contrast
To measure contrast, the images acquired with and without contrast disks were
first registered to the nearest whole pixel. Perfect registration was
difficult to obtain. To minimize this problem, several images were acquired
and the best pair of images was chosen for the study. Interpolation or partial
pixel registration was not used since this would corrupt noise measurements.
The images were then subtracted. Average pixel values were measured in a 30 by
30 pixel region (12 by 12 mm) of interest (ROI) located inside the region of
the contrast disk in the subtracted image. This value was defined as the
contrast of the image at that region. Since the pixel values were proportional
to the log of exposure, this value is equivalent to
(signal-background)/background if the pixel values were converted to units of
exposure.
2 Noise
Noise was defined as the standard deviation within the ROI in the subtracted
image, divided by the square root of 2 to correct for image subtraction. This
value is equivalent to percent noise if pixel values were proportional to
exposure rather than to the logarithm of exposure. While these noise
measurements might not be quantitatively accurate when compared to other noise
measurement techniques, they are internally consistent when compared to each
other.
3 Contrast-to-Noise Ratio
CNR was defined as the ratio of the contrast value calculated in E.1 to the
noise value calculated in E.2. CNR, as defined here, reduces to the classical
measure of the signal-to-noise ratio.
4 Residual Scatter Fraction
RSF was defined to measure the scatter that remains in a
scatter-compensated image. For this measurement, the scatter exposure
measurements acquired using the PBS technique were used to calculate the
primary exposure as follows:
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5 Resolution
Resolution measurements are inherently difficult to make on non-linear
techniques such as BIE. Simple linear systems measurements, such as the
modulation transfer function, are not only difficult to measure but also
non-informative, since the frequency response in the processed image depends on
the input image. To describe the effect of BIE on image resolution in the
chest radiograph, the contrast improvement factor (CIF) was measured for
several different spatial frequencies (Nyquist, 0.5*Nyquist, and 0.25*Nyquist)
of a square wave pattern which was embedded in the chest image. This
measurement is sometimes referred to as the square wave response function. For
each frequency, the CIF was also measured at several different initial
contrasts (.01, .02, and .03). Specifically, to measure the CIF, a square wave
of known frequency and initial contrast was imbedded into the lung and
mediastinum regions of the chest radiograph. The test region was kept small to
minimize the perturbations on the BIE processing. This image was then
processed using the Bayesian technique for 20 iterations for each kernel
magnitude tested. The contrast of the square wave in the resultant image was
then measured and compared to the initial contrast. Therefore, a CIF value of
1.0 corresponds to no resolution loss for square waves at that frequency and
initial contrast. A value greater than 1.0 corresponds to the BIE improving
the output contrast of the square wave. Any CIF values less than 1.0
correspond to contrast loss which may be interpreted as resolution loss. The
measurement of CIF, as presented above, is independent of image sampling and
synonymous with measuring the modulation of square waves at several initial
contrasts and frequencies. Since the test pattern is embedded into the sampled
image at frequencies which are dependent upon pixel size, the CIF measurements
only examine the effect of the BIE technique on image resolution. Therefore,
any sub-sampling that was done previously should not inherently effect the
resolution results obtained by this measuring technique. As stated previously,
the CIF was measured for several different frequencies and initial contrasts to
give insight into the effects of BIE on resolution.
While the following results in figures 1, 2, 3, and 4 are shown for the lung region only, similar trends exist in the mediastinum. Some actual values of data from the mediastinum will be presented in the discussion.
A. Contrast
Fig. 1 shows a plot of contrast as a function of iteration in the lung region
of the chest for several different kernel magnitudes. The initial measured
value of contrast was 0.147. A kernel magnitude of 0 yields no increase in
contrast. This was expected, since no scatter compensation was performed. A
kernel magnitude of 2.5 yields a contrast increase of 142% at 20 iterations.
The use of kernel magnitudes in between these two end values results in varying
contrast enhancement depending on the magnitude of the kernel. As kernel
magnitude increases, so does contrast enhancement. The rate of contrast
enhancement decreases as iterations progresses.
Fig. 1: Contrast versus iteration in the lung region for several different
kernel magnitudes.
B. Noise
Fig. 2 shows a plot of noise as a function of iteration in the lung region of
the chest for several different kernel magnitudes. Noise was found to increase
as kernel magnitude was increased. A kernel magnitude of 0 yields a decrease
in noise by 18% from the original measured value (0.0104) at 20 iterations. A
kernel magnitude of 2.5 yields an increase in noise of 30% over the original
measured value. In most cases, for a low number of iterations, the noise
increases; however, as iteration number increases, noise values decrease slowly
and then do not vary significantly.
Fig. 2: Noise versus iteration in the lung region for several different kernel
magnitudes.
C. CNR
Fig. 3 shows a plot of CNR as a function of iteration in the lung region of
the chest for several different kernel magnitudes. As the algorithm iterates,
CNR increases and then becomes asymptotic. In some cases, CNR even decreases
again. The iteration at which the CNR value becomes asymptotic increases with
magnitude. The crossing of the CNR curves for the different kernel magnitudes
demonstrates the difficulty of quantifying non-linear systems. The initial
value of CNR in the uncompensated image was 14.1. A kernel magnitude of 0
yields an increase in CNR of 23% at 20 iterations. For a kernel magnitude of
2.5, this value is 87%.
Fig. 3: CNR versus iteration in the lung region for several different kernel
magnitudes.
D. RSF
Fig. 4 shows a plot of RSF as a function of iteration in the lung
region of the chest for several different kernel magnitudes. A kernel
magnitude of 0.0 yields no scatter compensation; hence, the RSF is the
uncompensated image scatter fraction (0.60). As kernel magnitude is increased,
RSF decreases and either remains constant or becomes asymptotic as iteration
progresses. For large kernel magnitudes, RSF is negative, corresponding to
over-compensation in the lung region. RSF is near 0 for a magnitude of 2.0 and
iterations above 6. Over-compensation in the lung region after several
iterations begins when the magnitude of the kernel is higher than 2.0.
Fig. 4: Residual scatter fraction versus iteration in the lung region for
several different kernel magnitudes.
E. Resolution
Fig. 5(a) and fig. 5(b) show contour plots of CIF results in the lung
and mediastinum regions, respectively. The contours of constant CIF=1.0 are
plotted as a function of spatial frequency and kernel magnitude at 20
iterations. A CIF of 1.0 corresponds to no resolution loss or gain for square
waves at that given frequency. The solid, dashed, and dotted lines represent
initial contrasts of .01, .02, and .03, respectively. The abscissa axis is
frequency (in units of the Nyquist frequency) and the ordinate is kernel
magnitude. For any point above a contour, CIF is less than 1.0. For points
below the contour, CIF is increased above 1.0.
Fig. 5: Resolution for BIE taken at 20 iterations (a) in the lung region and
(b) in the mediastinum. Lines show where CIF is equal to 1.0. Above lines,
resolution is degraded. Solid line, line with circles, and line with squares
represent initial contrasts of .01, .02, and .03, respectively.
Fig. 5(a) shows that in the lung region, square waves are partially degraded for all kernel magnitudes examined; however, a kernel magnitude of 1.5 or greater yields degradation for only initial contrasts of .01 and high frequencies (greater than 0.6 times the Nyquist frequency). In the mediastinum, shown in fig. 5(b), square waves show partial degradation for all kernel magnitudes below 2.0. In this region, kernel magnitudes of 2.0 or greater show no values of CIF less than 1.0.
The objective of this study was to investigate the interdependence of several image parameters. This was achieved by varying the magnitude of the scatter kernel and examining the effect on contrast, noise, CNR, RSF, and resolution. Using scatter kernel magnitudes less than the average SPR yields partial scatter compensation, since the magnitude of the scatter kernel represents an assumed SPR. While partial scatter compensation yields an increase in contrast, noise is not as magnified. Thus, partial scatter compensation gives the ability to produce a chest radiograph with increased CNR, decreased RSF, and minimal loss of resolution. It is assumed here that minimal loss of contrast for square waves corresponds to minimal loss of resolution, in that if a square wave containing many different frequencies is not distorted, then the resolution of that square wave has not been effected.
The usefulness of partial scatter compensation is that if a certain RSF were required, a kernel magnitude could be selected that would give this desired level. CNR is always increased by BIE (fig. 3). A larger kernel magnitude results in a larger CNR for high iterations. The CIF results demonstrate that kernel magnitudes of 1.5 or larger can be used without significant loss of resolution, as measured. A kernel magnitude of 1.5 at 10 iterations yields a decrease in scatter fraction of 75% in the lung, while CNR is improved by 67%. For the same kernel, results in the mediastinum show that RSF is reduced 21%, while CNR is improved 110%. For almost no resolution loss, a kernel magnitude of 2.0 or greater can be used. The authors believe that the BIE images also demonstrate improved subjective quality over that of the unprocessed images.
The significance of this work is that it shows that a user of BIE can adjust the trade-off between contrast enhancement and noise reduction. These values directly relate to how much scatter should be removed, which is based solely on the scatter kernel magnitude for this implementation of BIE.
For instance, if noise reduction was the required goal, a kernel magnitude of 0.5 could be used. RSF is reduced from .60 to .45 and from .91 to .86 in the lung and mediastinum, respectively. With this reduction in RSF, contrast is increased 35%, noise is reduced 6%, and CNR is increased 43% in the lung region. No resolution loss exists for square waves with initial contrasts of .03 or higher; however, at .01 and .02 initial contrasts, frequencies above 0.4 and 0.6 in units of the Nyquist frequency, respectively, are degraded.
If noise were required to remain unchanged and CNR improvement was desired, a kernel magnitude of 1.0 would be chosen. RSF is reduced from .60 to .29 and from .91 to .79 in the lung and mediastinum, respectively. With this reduction in RSF, contrast is increased 65%, noise is approximately unchanged, and CNR is increased 57% in the lung region. No resolution loss exists for square waves with initial contrasts of .03 or higher; however, at .01 and .02 initial contrasts, frequencies above 0.5 and 0.85 in units of the Nyquist frequency, respectively, are degraded.
If maximal CNR improvement in the lung were desired, a kernel magnitude of 1.5 would be used in the BIE processing. RSF is reduced from .60 to .15 and from .91 to .73 in the lung and mediastinum, respectively. With this reduction in RSF, contrast is increased 96%, noise is increased 10%, and CNR is increased 77% in the lung region. No resolution loss exists for square waves with initial contrasts of .02 or higher; however, at .01 initial contrast, frequencies above 0.6 times the Nyquist frequency are degraded.
By choosing a certain kernel magnitude, the user can decide how much CNR improvement, RSF reduction, and resolution loss is acceptable. By changing the kernel magnitude, the user can produce several different images, for simultaneous viewing, of varying appearance and image quality. More importantly, by choosing a certain kernel magnitude and iteration, BIE can produce images that have similar CNR improvement and scatter properties to that of other more commonly used scatter rejection techniques without having to use any additional hardware.
This work was supported in part by grant number CA60849 from the National Cancer Institute. Its contents are solely the responsibility of the authors and do not necessarily represent the official views of the National Cancer Institute.
