Transcript THE DENOISING UTILITY OF WAVELET COMPRESSION …

THE DENOISING UTILITY OF
WAVELET COMPRESSION
ALGORITHMS IN MAGNETIC
RESONANCE IMAGING
G.E. Sarty1 and M.S. Atkins2
1Royal
University Hospital, Saskatoon, Saskatchewan, Canada
2School of Computing Science, Simon Fraser University,
Burnaby, British Columbia, Canada
Introduction
The wavelet transform has been found to be
useful in these areas of image processing:
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


Denoising
Data Compression
Texture Analysis
Feature and Pattern Recognition
In this work we investigate the relationship
between the denoising and data compression
applications.
Wavelet Denoising
Images can be denoised through the application of the
following steps:
•Transform the image into a wavelet coefficient domain. The
transformed coefficients will represent image information on
various scales.
•Apply soft or hard thresholding to the wavelet coefficients;
the threshold may or may not be scale dependent.
•Transform the wavelet coefficients back to the image
domain.
In this work, a discrete Fourier transform method of
Sarty (1) is used to compute the wavelet coefficients.
Wavelet Data Compression
Image data compression can be achieved through the
following steps:
•Transformation: Lossless (to machine precision) wavelet
transformation; decorrelates image information.
•Quantization: All information loss occurs here.
•Compression: Efficient lossless representation of quantized
data using zerotrees of wavelet coefficients to represent
significance maps.
For this work, Shapiro’s (2) embedded zerotrees of wavelet
coefficients (EZW) compression method is used. EZW encodes
the significance map in an embedded fashion so that the most
significant wavelet coefficients are coded first.
Similarities
•Compression, to a precise target, is achieved with EZW by simply
terminating the transmitted bit stream at the appropriate point. The
resulting image, when reconstructed, contains information only from
wavelet coefficients above some threshold.
•To denoise an image, the wavelet coefficients below a threshold are
discarded (hard thresholding) or the threshold is subtracted from all the
coefficients (soft thresholding). The threshold may be constant across
the wavelet scales or it may vary from scale to scale.
In both instances, information from the insignificant wavelet
coefficients is eliminated.
The appearance of wavelet
compressed images and wavelet denoised images should ,
therefore, be similar.
Comparison
A dataset of 30 clinically acquired magnetic resonance (MR) images of
ovarian masses were chosen for the purposes of comparing the wavelet
denoising algorithm with the EZW compression algorithm. The database
is illustrated in the next panel. Ovarian mass images were chosen
because they were obtained using a body radio frequency (RF) coil which
has a poor spatial filling factor relative to other, more specialized, RF
coils. The poor filling factor leads to relatively noisy MR images.
The efficacy of each algorithm in removing noise from the image was
quantified by computing the signal-to-noise ratios (SNR) of the original
and processed images. The SNR was computed by taking the signal as
the average of a block of pixels entirely contained within the abdomen
and taking the noise as the average of a block of pixels within the
background (black) part of the image. This SNR measure is necessarily a
relative one.
MR Image Dataset
1
2
9
16
3
10
17
24
4
11
18
25
5
12
19
26
6
13
20
27
7
14
21
28
8
15
22
29
23
30
Comparison Example - Image 5
Original MR image, SNR = 32.6
Denoised Image, m = 0.5, SNR = 33.5
Compressed Image, 20:1, SNR = 29.7
•Note how each method gives a different texture to the “snowy” background
noise.
•In the tests, denoised images with four shrinkage thresholds, m = 0.1, 0.5, 1.0 and
2.0 were computed. EZW compression to ratios of 10:1 and 20:1 were performed.
Comparison Example - Image 19
Original MR image, SNR = 12.8
Denoised Image, m = 1.0, SNR = 14.7
Compressed Image, 20:1, SNR = 12.1
With the denoising algorithm, the SNR was improved in almost all tested
cases; only one case showed a decrease in SNR. With the EZW
algorithm, the SNR was improved in slightly more that half the tested
cases.
Difference Images
Original Image 19
Original - Denoised, m = 1.0
Original - Compressed, 20:1
Note how the differences are concentrated around the edges in the image
processed with the denoising algorithm. The differences are less coherent in
the EZW processed image.
Results
Image#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Orig.
13.4
13.5
13.7
18.7
32.6
17.4
32.5
4.7
4.6
41.5
41.1
40.9
40.3
27.8
28.0
Denoising shrinkage
m=0.1 m=0.5 m=1.0 m=2.0
13.5
13.5
13.5
13.7
13.7
14.0
14.4
15.1
13.9
14.1
14.4
15.1
19.1
19.7
20.5
22.4
34.0
33.5
32.3
30.5
18.4
19.6
21.1
23.6
35.0
37.8
40.9
46.0
4.8
5.1
5.4
6.2
4.7
5.0
5.4
6.1
43.0
42.4
42.1
41.8
42.7
42.3
42.1
42.2
42.4
42.2
42.0
42.5
42.0
41.8
42.1
43.1
29.1
29.5
30.3
32.1
29.3
29.6
30.3
32.0
EZW
10:1
13.2
13.6
13.9
18.8
30.6
17.7
31.6
4.7
4.6
41.8
41.8
40.5
40.4
27.6
28.4
20:1
13.4
13.5
13.9
18.7
29.7
18.2
35.3
4.7
4.5
44.6
43.4
39.6
40.4
28.9
29.1
Results
Image#
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Orig.
24.8
22.2
21.9
12.8
12.3
8.7
8.9
9.2
8.3
19.5
18.7
8.0
16.7
12.5
12.5
Denoising shrinkage
m=0.1 m=0.5 m=1.0 m=2.0
25.9
26.3
27.0
28.8
23.2
23.6
24.2
25.9
22.9
23.2
22.9
25.2
13.4
13.9
14.7
16.6
12.8
13.3
13.9
15.4
8.8
8.9
9.0
9.2
8.9
9.0
9.1
9.3
9.4
9.7
10.2
11.3
8.4
8.8
9.2
10.3
20.3
21.6
23.8
27.5
19.5
21.5
23.8
29.2
8.1
8.2
8.3
8.6
17.4
18.0
18.7
20.3
13.3
13.8
14.7
17.1
13.3
14.2
15.4
18.5
EZW
10:1
25.2
22.1
22.0
12.6
12.3
8.7
8.9
9.2
8.3
19.5
18.8
8.0
17.0
12.7
12.6
20:1
25.6
22.4
22.3
12.4
12.1
8.8
8.8
9.2
8.3
19.6
18.8
8.0
17.2
11.8
12.7
Discussion
According to Donoho, et al. (3), near minimax removal of image noise is possible by
using soft thresholding on the type of wavelet coefficients used in the EZW
algorithm. Except for the issue of quantization, the image delivered by EZW
should be an image in which the wavelet coefficients had been subjected to hard
thresholding. From the abstract work of Donoho, et al. it is not clear how one
choice of wavelet over another can lead to better denoised images (i.e. to better
estimates of the underlying data). Intuition, and the work of Mallat and Hwang (4),
would suggest that edge-detecting wavelets, such as the Haar wavelet, would be
better suited for denoising applications. The intuition being that an edge detecting
wavelet would tend to gather the edge information together into the more
significant wavelet coefficients, leaving the noise to the least significant wavelet
coefficients. Upon thresholding, no edge information would be lost and a better
reconstructed image should result.
The wavelet used in EZW here was the “Coiflet” (5). Coiflets tend to produce small
value wavelet coefficients in areas where the original image is smooth. Coiflets also
seem to have good data compression properties, perhaps because they have a good
“decorrelating” property (see earlier panel on wavelet compression). Perhaps by
replacing the Coiflet with an edge-detecting wavelet, compression performance can
be traded off with denoising performance.
Discussion
Examination of the table of results shows that there is no optimal choice for
shrinkage parameter m (which is used in conjuction with an image noise level
estimate to determine the threshold used). In practice, then, the threshold must be
found by a trial and error process. The use of the embedded encoding of data such
as that used with the EZW wavelet transform compression algorithm could be a
very useful tool for selecting the appropriate threshold. By adjusting the
compression rate in real time, an operator could view a continuum of denoised
images. With the threshold being so easily adjustable, an optimal denoised image
could be produced in short order.
The denoising numerical wavelet transform used in this work is different from the
wavelet transform used for compression; it has been designed specifically for
denoising (1). In addition, the denoising algorithm uses a scale-dependent
threshold. It is conceivable that a compression significance map could also use a
scale-dependent threshold to improve its denoising characteristics. Achieving a soft
threshold within an embedded code is also conceivable; however, there is some
anecdotal evidence (from some of our unpublished results) that hard thresholding
leads to better images than soft thresholding.
References
(1) Sarty, G.E. and Kendall, E.J., Self-Diffusion Maps from Wavelet De-Noised
NMR Images, Journal of Magnetic Resonance, Series B, 111, 50-60 (1996).
(2) Shapiro, J.M., Embedded Image Coding Using Zerotrees of Wavelet
Coefficients, IEEE Transactions on Signal Processing, 41, 3445-3462 (1993).
(3) Donoho, D.L., Johnstone, I.M., Kerkyacharian, G. and Picard, D., Wavelet
Shrinkage: Asymptopia?, Journal of the Royal Statistical Society B, 57, 301-369
(1995).
(4) Mallat, S. and Hwang, W.L., Singularity Detection and Processing with
Wavelets, IEEE Transactions on Information Theory, 38, 617-643 (1992).
(5) Daubechies, I., Ten Lectures on Wavelets, SIAM, Philadelphia, Pennsylvania
(1992).