Adaptive predictive multiplicative autoregressive model for medical image compression.

Abstract

In this paper, an adaptive predictive multiplicative autoregressive (APMAR) method is proposed for lossless medical image coding. The adaptive predictor is used for improving the prediction accuracy of encoded image blocks in our proposed method. Each block is first adaptively predicted by one of the seven predictors of the JPEG lossless mode and a local mean predictor. It is clear that the prediction accuracy of an adaptive predictor is better than that of a fixed predictor. Then the residual values are processed by the MAR model with Huffman coding. Comparisons with other methods [MAR, SMAR, adaptive JPEG (AJPEG)] on a series of test images show that our method is suitable for reversible medical image compression.

Full text

Page 1

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 18, NO. 2, FEBRUARY 1999181
Adaptive Predictive Multiplicative Autoregressive
Model for Medical Image Compression
Zuo-Dian Chen, Ruey-Feng Chang,* and Wen-Jia Kuo
Abstract—In this paper, an adaptive predictive multiplicative autore-
gressive (APMAR) method is proposed for lossless medical image coding.
The adaptive predictor is used for improving the prediction accuracy
of encoded image blocks in our proposed method. Each block is first
adaptively predicted by one of the seven predictors of the JPEG lossless
mode and a local mean predictor. It is clear that the prediction accuracy
of an adaptive predictor is better than that of a fixed predictor. Then
the residual values are processed by the MAR model with Huffman
coding. Comparisons with other methods [MAR, SMAR, adaptive JPEG
(AJPEG)] on a series of test images show that our method is suitable for
reversible medical image compression.
Index Terms—Image coding, lossless compression, multiplicative au-
toregressive model.
I. INTRODUCTION
There are many different kinds of medical digital images such
as magnetic resonance (MR), computerized tomography (CT), ul-
trasound (US), X-ray images, etc. Medical image compression can
be classified into two categories, lossless and lossy methods. For
example, differential pulse code modulation (DPCM) [1], hierarchical
interpolation (HINT) [2], and multiplicative autoregressive models
(MAR) [3]–[6] are lossless methods. The discrete-cosine transform
(DCT) [7] and vector quantization (VQ) [8] are lossy methods.
The major advantage of lossless methods is that the image can
be reconstructed exactly. This property is especially important for
medical images because of legal reasons.
In the MAR approach, pixel values that subtract the block mean
??from the original pixel values ???????? are predicted by the MAR
predictor. Subsequently, the residual values are encoded by entropy
coding. Although the correlation of the pixels can be reduced by
subtracting the block mean ?? before applying the MAR predictor,
the performance of other predictors to reduce the correlation of the
pixels is much better than that of using only the subtraction of block
mean ???
In this paper, a novel approach called adaptive predictive MAR
(APMAR) is introduced. Two predictive methods are used in the
decorrelation step of our method. First, adaptive prediction is used
to process the image blocks. One out of eight predictors, viz., the
seven predictors in the JPEG lossless standard [9] and the local
mean predictor in the space-varying MAR (SMAR) method [10],
is adaptively chosen as the predictor for the first decorrelation step.
Then we use the MAR predictor to further reduce the redundancy of
the residual image. Entropy coding is used in the final coding step
of APMAR.
The rest of this paper is organized as follows. In Section II, we
introduce the JPEG lossless mode and the MAR method. The new
compression method is described in Section III. In Section IV the test
Manuscript received June 18, 1996; revised December 31, 1998. This work
was supported by the National Science Council under Grant NSC85-2213-E-
194-016. The Associate Editor responsible for coordinating the review of this
paper and recommending its publication was M. Vannier. Asterisk indicates
corresponding author.
Z.-D. Chen, *R.-F. Chang, and W.-J. Kuo are with the Advanced System
Integration Laboratory, Department of Computer Science and Information En-
gineering, National Chung Cheng University, Chiayi, Taiwan 62107, R.O.C.
Publisher Item Identifier S 0278-0062(99)03153-5.
Fig. 1.
and ??and the current predicted sample ???
The relational position of the four predictive samples ?????????
TABLE I
PREDICTORS OF THE EIGHT-PREDICTOR SELECTION
images and the experimental results are presented and conclusions
are shown.
II. THE SEQUENTIAL LOSSLESS MODE
OF JPEG AND THE MAR METHOD
A. The Sequential Lossless Mode of JPEG
The JPEG lossless mode uses four neighboring pixels ?????????
and ?? to predict the value of the current pixel indicated by ???
Fig. 1 shows the relation of these pixels. The first seven predictors
in Table I are the JPEG predictors. In order to reduce the bit rate
and improve the image quality, we will select a suitable predictor for
each encoded block.
B. Lossless Image Compression Using the MAR Method
There are many different kinds of MAR models. The consideration
of choosing a MAR model mainly depends on the tradeoff among
predictor performance, the computation complexity of the predictor,
and the overhead of the parameters. We have chosen the 3 ? 3
nonsymmetric half-plane support (NSHP) model shown in Fig. 2. In
this model, the two-dimensional (2-D) polynomial operator ????????
is
???????? ? ?? ? ?????
???? ? ?????
? ??? ? ?????
? ???
where ??
and ? units along the horizontal direction and ???????? are constant
coefficients.
Among the many different algorithms for finding the parameters
of the MAR model, a simple, and thus attractive, estimation method
is recursive pseudo-linear regression (RPLR). The RPLR method
represents the pixel value ?????? by the form ?????? ? ? ? ???????? ? ? ?
??????, where ? ? ? is the unknown parameter matrix, ? ? ?????? is the
regression matrix, and ?????? represents zero-mean white noise. The
recursive least squares (RLS) algorithm [11] is used to iteratively
update the parameter matrix ? ? ? by the following steps:
???
?represents a shift of ? units along the vertical direction
0278–0062/99$10.00  1999 IEEE

Page 2

182IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 18, NO. 2, FEBRUARY 1999
Fig. 2.
predicted pixel.
The support region of the 3 ? 3 NSHP model, where ? is the current
1) ? ? ???? ? ? ? ??? ? ?? ? ? ? ??????????? ? ? ? ???????? ? ??? ? ????
2) ? ????? ? ? ? ??? ? ??? ? ????????? ? ? ???????? ? ??? ? ??? ? ?????????;
3) ? ? ????? ? ?? ? ? ? ? ?????? ? ????????? ? ??? ? ??
where ? ????? is a 3 ? 1 matrix, ? ? ???? is a 3 ? 3 matrix, ? ? ???? ?
???????????is the parameter matrix at iteration ?, and ?????? will
affect the parameters in the next iteration
?????? ?
?????? ? ??
???? ? ???? ? ????? ? ??? ? ??
???? ? ??? ? ?? ? ????? ? ????
?????? ? ??? ? ?? ? ??????? ? ????
?
The recursive algorithm requires the initial values of ? ? ???? and ? ? ????
with ? ? ???? invertible. A good choice for ? ? ???? is generally ? ? ???? ? ?? ? ?,
where ? is a small positive scalar. At each iteration, the parameters
?????????? must be checked to see whether they are in the stable
parameter range for the 3 ? 3 NSHP model which is given by
?? ? ???????????? ??????????????????
Hence, the residual value of ??????, which is obtained by subtract-
ing the block mean ?? from the original pixel value ??????, will be
predicted by the equation ? ?????? ? ?? ? ???????????????? For the
3 ? 3 NSHP MAR model
? ?????? ????????????
?
? ???????
? ??? ???????
? ?????
? ???
?
? ???????
?
? ?????
?? ??? ?????
????????
???????????? ???? ? ??????? ? ??? ? ??
? ??????? ? ??? ? ?? ? ??????? ? ????
? ????? ? ???? ? ????? ? ??? ? ??
? ??????? ? ???
III. APMAR METHOD
We propose a new compression method, APMAR, in this paper.
Both MAR and SMAR use a fixed prepredictor to reduce the
correlation of the pixels before the actual prediction of the MAR
model. In SMAR [10], the local mean ??? ? ??? ?? ? ????? is
first subtracted from the pixel value for each pixel. Then, the residual
values are encoded by the MAR method. It is not appropriate to use
a fixed predictor for different medical images because of the widely
varying image characteristics. Hence, we admit all seven predictors
of lossless JPEG to reduce the correlation of the pixels. The local
mean predictor ??????????????? is added as an option without
increasing the additional overhead because three bits are already
needed for the seven original JPEG predictors. We will refer to this
prepredictive process as eight-predictor selection.
The first step of APMAR is to split the image into blocks.
Table I shows the eight predictors of eight-predictor selection. The
one with the lowest total error between original pixel values and
predicted values in a block is adopted as the predictor of the
block. After choosing the predictor, MAR is used for predicting the
residual values. Finally, the index of the eight-predictor selection, the
parameters of the MAR model, and the final residual values for each
TABLE II
THE SUMMATION OF ENTROPY AND OVERHEAD FOR APMAR
AND OTHER LOSSLESS METHODS FOR THE TEST IMAGES
block are transmitted to the receiver. The encoder of our APMAR
method is shown in Fig. 3.
The steps of the APMAR compression method are as follows.
1) Step 1: Splitting the image into blocks.
2) Step 2: Using the eight-predictor selection to reduce the cor-
relation of pixels.
3) Step 3: Using MAR prediction to further reduce the correlation
of the residual values.
4) Step 4: Encoding the final residual values by entropy coding.
IV. EXPERIMENTS AND RESULTS
Experiments to evaluate the proposed method have been performed
on a number of 2-D images or slices of three-dimensional (3-D)
images (see Fig. 4 for an overview).
The experiments showed that the best performance was obtained
when the image was segmented into 64 ? 64 blocks for a 512 ?
512 image or 32 ? 32 blocks for a 256 ? 256 image in the 3
? 3 NSHP MAR model. Three bits are used for the eight-predictor
selection in APMAR. The overhead of the MAR prediction is twenty-
four bits for each block where each parameter of the MAR model
is quantized with eight bits. The block mean for the original MAR
method and the background pixel value ??? are all quantized with
eight bits.
The RLS algorithm requires initial values ? ? ???? and ? ? ????? The
initial value of matrix ? ? ? must be a small positive matrix. In our
experiments, the initial parameters ???????? of the 3 ? 3 NSHP
model for each block are set to be 0.01 and ? ? ???? is set to be ????? ? ?
where ? ? ? is the identity matrix.
A. Experimental Results of APMAR
The summation of entropy and overhead for APMAR is compared
with that of MAR, SMAR, and AJPEG. The entropy is defined as
? ? ?
?
???
??? ??????
where ?? is the probability of gray level ? in the image and ? is the
largest gray level.
The experimental results are listed in Table II. We find that
APMAR is more suitable for reversibly compressing different medical
images than the other lossless compression methods. Although in
individual cases MAR or AJPEG may accomplish a slightly lower en-
tropy than APMAR, the overall conclusion is that APMAR generally
outperforms the other reversible compression techniques significantly.
REFERENCES
[1] K. Chen and T. V. Ramabadran, “Near-lossless compression of medical
images through entropy-coded DPCM,” IEEE Trans. Med. Imag., vol.
13, pp. 538–548, Sept. 1994.

Page 3

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 18, NO. 2, FEBRUARY 1999 183
Fig. 3.Encoder of the APMAR.
(a)(b)(c)
(d)(e)
(f) (g)
(h)(i)
Fig. 4.
(e) Knee-B: idem. (f) Chest: transaxial slice (512 ? 512) CT image. (g) Heart: 512 ? 480 US image. (h) Angio: 512 ? 480 angiographic image. (i)
Chest: 512 ? 512 X-ray image. All images were quantized in 8 bits/pixel.
Overview of the test images. (a) Brain A: sagittal slice (256 ? 256) of an MR image. (b) Brain B: idem. (c) Brain C: idem. (d) Knee-A: idem.

Page 4

184IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 18, NO. 2, FEBRUARY 1999
[2] M. A. Viergever and P. Roos, “Hierarchical interpolation,” IEEE Eng.
Med. Biol. Mag., vol. 12, pp. 48–55, Mar. 1993.
[3] M. Das and N. K. Loh, “New studies on adaptive predictive coding of
images using multiplicative autoregressive models,” IEEE Trans. Image
Processing, vol. 1, pp. 106–111, Jan. 1992.
[4] S. Burgett and M. Das, “Predictive image coding using multiresolution
multiplicative autogressive models,” Proc. Int. Elect. Eng., vol. 140, no.
2, pp. 127–134, Apr. 1993.
[5] M. Das and S. Burgett, “Lossless compression of medical images using
two-dimensional multiplicative autoregressive models,” IEEE Trans.
Med. Imag., vol. 12, pp. 721–726, Dec. 1993.
[6] M. Das and C. L. Li, “Simple space-varying least squares model for
lossless medical image compression,” Electron. Lett., vol. 30, no. 11,
pp. 849–850, May 1994.
[7] M. S. Rebelo, S. S. Furuie, A. C. L. Munhoz, L. Moura, and C. P.
Melo, “Assessment of a lossy image compression technique applied to
cardiac images in nuclear medicine,” in Proc. Computers Cardiology,
Sept. 1993, pp. 827–830.
[8] P. C. Cosman, C. Tseng, R. M. Gray, R. A. Olshen, L. E. Moses,
H. C. Davidson, C. J. Bergin, and E. A. Riskin, “Tree-structured vector
quantization of CT chest scans: Image quality and diagnostic accuracy,”
IEEE Trans. Med. Imag., vol. 12, pp. 727–739, Dec. 1993.
[9] W. B. Pennebaker and J. L. Mitchell, JPEG Still Image Data Compres-
sion Standard. New York: Van Nostrand Reinhold, 1993.
[10] M. Das, D. L. Neuhoff, and C. L. Lin, “Near-lossless compression of
medical images,” in Proc. ICASSP, 1995, pp. 2347–2350.
[11] L. Ljung and T. Soderstrom, Theory and Practice of Recursive Identifi-
cation.Cambridge, MA: MIT, 1983.
Correction to “Performance Characteristics of a
Compact Position-Sensitive LSO Detector Module”
J. J. Vaquero, J. Seidel, S. Siegel, W. R. Gandler, and M. V. Green*
In Section III (pp. 971–972) of the above paper,1the three italicized
paragraphs were incorrectly moved to Section IV, p. 973. The
corrected text is reprinted in the following.
We apologize to the authors and readers for this error.
III. RESULTS
Single photon field flood images acquired at 30, 140, and 511 keV,
and count profiles along the central row of each image, are shown
in Fig. 3. These images were created with the centroid algorithm (1)
and are corrected for the LSO background. The least restrictive event
selection criteria were applied during the creation of these images.
A version of Fig. 3(c), enhanced to reveal faint structures, is shown
in Fig. 4(a). Each crystal in this image appears to be joined to its
eight neighbors by faint horizontal, vertical, and diagonal straight
lines in a “connect-the-dot” pattern. The details of this pattern change
depending on whether the LSO array is illuminated from the front
[Fig. 4(a)] or from the left side [Fig. 4(b)].
Manuscript received March 4, 1999. Asterisk indicates corresponding au-
thor.
J. J. Vaquero, J. Seidel, S. Siegel, and W. R. Gandler are with the Nuclear
Medicine Department, National Institutes of Health, Bethesda, MD 20892
USA.
*M. V. Green is with the Nuclear Medicine Department, National Institutes
of Health, Building 10, Room 1C401 MSC 1180, Bethesda, MD 20892 USA
(e-mail: mike@nmdhst.cc.nih.gov).
Publisher Item Identifier S 0278-0062(99)04761-8.
1J. J. Vaquero, J. Seidel, S. Siegel, W. R. Gandler, and M. V. Green, IEEE
Trans. Med. Imag., vol. 17, pp. 967–978, Dec. 1998.
An LSO background image acquired with no external sources
present, and a count profile along the central row, are shown in
Fig. 5. The least restrictive event selection criteria were applied
during creation of this image.
Average apparent crystal widths at the half maximum and tenth
maximum levels, expressed as a fraction of the actual crystal width
(2 mm), are listed in Table I for crystals in the central row of Fig. 3.
Average widths, and the largest and smallest widths in the row, are
included for images created from all data within the dynamic range of
the ADC’s. Average widths calculated from these same image data
when subjected to energy windowing (W) are also shown. At 140
keV the energy window was 140 keV ?20 keV, and at 511 keV
the window was 511 keV ? 70 keV. Energy windowing was not
applied to the I-125 field flood data since nearly all 30-keV photon
interactions in LSO are photoelectric and windowing is unnecessary.
The fraction of detected coincidence events that occur within each
of three adjacent crystals scanned by the collimated F-18 line source
is shown as a function of source position for different exclusion
conditions in Fig. 6(a). The dotted curve portrays the fraction of
events that occur in ROI’s the size of an LSO crystal (L) when no
energy exclusion condition is applied. The dashed curve portrays this
fraction for events within the L-ROI and in the 511-keV ? 70-keV
energy range. The continuous curve portrays the fraction of events
assigned to a crystal when events outside small (S) ROI’s drawn
around each apparent crystal are excluded, as well as events outside
the energy window.
Accuracy and sensitivity are portrayed in the bar graph shown in
Fig. 6(b) for different energy and spatial exclusion conditions when
the scanning source is positioned directly over the center of the middle
crystal in Fig. 6(a). For L-ROI’s, energy exclusion conditions alone
reduce the fraction of accepted events from 100% to 55%, while the
S spatial exclusion condition combined with the energy exclusion
condition reduces the fraction of accepted events further to 34%. The
accuracy of event positioning increases from 72% to 95% as these
increasingly selective criteria are imposed.
The apparent separation between peaks along the main diagonal
of Fig. 3(c) is plotted in Fig. 7(a) where the actual diagonal crystal
separation is 3.07 mm. The largest deviation of peak separation
from the mean peak separation was 0.82 mm, while the average
deviation from the mean peak separation was 0.6 mm. The sinusoidal-
like variation in spacing between peaks in Fig. 7(a) is detectable in
Figs. 3, 4, and 5(a).
Fig. 7(b) shows apparent module gain before correction as a func-
tion of crystal location. Gain is defined as the normalized channel
number in which the photopeak maximum occurs for each crystal.
Energy spectra for three different incident photon energies are
shown in Fig. 8 for the same central crystal. These spectra have
been corrected for the LSO background, whose spectrum is shown
in Fig. 8(c) The fraction of LSO background events occurring in the
energy range 140 ? 20 keV [Fig. 8(c)] was 1.2% of the total LSO
background rate or about 12 counts/s.
Energy resolution and its variation within the UFOV are listed in
Table II for several different incident photon energies. For illustrative
purposes, energy resolution at 511 keV in an identical detector module
composed of 2 mm?2 mm?10-mm-long BGO crystals rather than
LSO crystals is also included.
The TAC spectrum obtained with the detector module in time
coincidence with a second LSO detector is shown in Fig. 9. The
second timing peak was acquired with an interposed time delay of 4
ns. The average FWHM of these peaks was 1.2 ns.
0278–0062/99$10.00  1999 IEEE