WAVELET FOR ULTRASONIC FLAW ENHANCEMENT AND
IMAGE COMPRESSION
W. Cheng, K. Tsukada, L.Q. Li, K. Hanasaki
Department of Earth Resources Engineering, Graduate School of Engineering,
Kyoto University, Kyoto, Japan, 606-8501
ABSTRACT. Ultrasonic imaging has been widely used in Non-destructive Testing (NDT) and
medical application. However, the image is always degraded by blur and noise. Besides, the pressure
on both storage and transmission gives rise to the need of image compression. We apply 2-D Discrete
Wavelet Transform (DWT) to C-scan 2-D images to realize flaw enhancement and image compression,
taking advantage of DWT scale and orientation selectivity. The Wavelet coefficient thresholding and
scalar quantization are employed respectively. Furthermore, we realize the unification of flaw
enhancement and image compression in one process. The reconstructed image from the compressed
data gives a clearer interpretation of the flaws at a much smaller bit rate.
INTRODUCTION
Ultrasonic testing has been an important and one of the mostly used method in
Non-destructive Testing and Evaluation
(NDT&E). It utilizes some particular
characteristics of the propagating stress waves [1], which broaden its application to a much
larger scope, industry and medical field, etc. With its expanding usage, here rises the
problems of getting more accurate flaw indication of the C-scan image, and of storing and
transforming the image more efficiently as well.
In this paper, we explore a method based on Discrete Wavelet Transform (DWT), to
achieve flaw enhancement and image compression. Furthermore, we discuss the possibility
and feasibility of solving the two problems in one process.
The rest of this paper is outlined as follows. The main part is divided into section 2
and section 3, which discusses Wavelet in C-scan flaw enhancement and image
compression respectively. A view looking into the combination of the two problems will be
presented in section 4 before a conclusion is drawn in section 5.
WAVELET IN C-SCAN IMAGE ENHANCEMENT
Ultrasonic C-scan images are raster scanned displays which use gray-scale or
pseudo-color to represent the reflected ultrasonic amplitude at each pixel in the image. The
resolution of the ultrasonic C-scan image is often degraded by blur and noise. The
size/shape or acoustic reflectivity of the flaw of low amplitude can be masked by noise.
While blur does not greatly affect flaw detection, it does limit the ability to characterize
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flaw size/shape, number, and spacing [2,3].
The solutions for these two problems at present include filtering (Wiener filter, etc),
multi-sensor data fusion, and so on. Most of them depend on the setting of the experiment
to acquire the data. Here we want to propose a post process method that mostly relies on the
statistical properties of the data.
Wavelet Method
Multi-Resolution Analysis (MRA), as implied by its name, analyzes the signal at
different frequencies with different time resolutions. MRA decomposes a given function/
into a series of spaces FmCL2(R), m • Z. These spaces describe successive approximation
spaces of /with resolution 2~m [4]. A space of Wm orthogonal to Vm in space Vm-i is defined
as details.
, VmLWm
(1)
Then the original /can be represented by
SmiM}
(2)
In wavelet transform, a scaling function 0 and a wavelet function If, are introduced,
which
r^(x) = 2-m/2r(2-mx-n)9 (m,fl)EZ 2
(3)
the (^ w w ) w e z and (V m «)«ez constitute an orthonormal basis for Vm and Wm respectively.
They are functions of the translation (time) parameter n and scale (frequency) parameter m
[5]. The results by the multiplication of /with the two functions are defined as the
approximation coefficient Am and the detail coefficient Dm.
^ =</,*.,.)
(4)
Dm=(f,Ym,,,)
(5)
The coefficients are an alternative mathematical representation of the signal / by which
DWT easily restructures a signal statistically. The / can be losslessly represented by
Equation (6),
__
(6)
2-Dimensional DWT
2-dimensional DWT is based on one scaling function 0(t) and three wavelets W
(where /c represents three spatial orientations: horizontal, vertical and diagonal ),which is
the products of a one-dimensional 0ft) and Wft). From here we can see that the details
extracted by the wavelets pick up the properties of the image in the three orientations. DWT
decomposes an image into bands that vary in spatial frequency and orientation, which
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0(x) <P(y)3 <P(x) W(y)3
V(x) <P(y)3 V(x) V(y) 3
<P(x) W(y)2
Y(x) <P(y)2
V(x) V(y)2
<P(x) V(y)l
V(x) V(y)l
Y(x) <P(y)l
FIGURE 1. Two-dimensional DWT
facilitates treating the image data differently according to different frequency and
orientation. The 2-D DWT is shown in Fig. 1 [4].
C-scan Image Simulation Model
We construct the C-scan simulation image model as having both blur and noise:
g(x, y)=f(x, y)*h(x5 a, y, £)+n(x, y)
(7)
where g(x, y) is the received image function, f(x, y) is the original image. The h(x, a , y, j3 )
is the point spread function associated with the ultrasonic system, expressing how much
the input value at position (x,y) influences the output value at position( a , j3 ). The n(x, y) is
additive noise. The symbol * denotes linear convolution. The noise is assumed to be
stationary and uncorrelated with the signal [2,5].
Wavelet Implementation in Image Enhancement
Here 'enhancement' refers to both de-noising and de-blurring. The scheme of the
implementation is outlined in Fig. 2. Here we take advantage of the DWT properties,
namely, multi-resolution and orientation selectivity, to set soft thresholds A (the
coefficients larger than A in absolute value are decreased by A if positive and increased
by A if negative [6]) to the DWT coefficients. The criteria used to evaluate how the
method works on eliminating blur and noise are SNR (Signal Noise Ratio =
101ogio[Var(signal)]/[Var(noise)]) and MSB (Mean Square Error between the image and the
original image). The experiments show that the wavelet method can not only eliminate the
blur and noise respectively, but also work well in cases that have both of them. Fig. 3 gives
an example of the case.
FIGURE 2. Wavelet in C-scan image enhancement
603
(a)
(b)
FIGURE 3. Wavelet in C-scan flaw enhancement, (a) flaw image with noise and blur. SNR=11.2216dB,
MSE=0.6605, (b) the processed image of (a) by wavelet enhancement method, SNR=65.2826dB,
MSE=0.5456
WAVELET IN IMAGE COMPRESSION
Compression Process
There are requirements for image compression and reasons why image can be
compressed. First, there is redundancy in the digital images, which comes form statistical
dependencies between pixels. Second, certain image degradations are not perceivable to
human visual system [7]. In the specific case of ultrasonic C-scan image, the flaw edges are
the most prominent feature that should be kept while some others can be omitted.
Image data are transformed by wavelet to de-correlate the data as much as possible.
Such transform is exactly invertible, therefore, lossless [6]. The edges are large coefficients
of high frequency scales, which is only a small portion of the whole coefficient set. By
emphasizing those coefficients that have much information about the flaw and by
discarding those that don't have, compression is realized. The method used here is 'scalar
quantization'. The compression scheme is outlined in Fig. 4.
Selection of Wavelet Base
When selecting a wavelet base to do image compression, some basic requirements of
it should be considered. They include those mentioned in [4,8], such as avoiding distortion
and artifacts in the reconstructed image, rapid convolution in DWT, smoothness of the
reconstructed image, and so on.
Unfortunately, not all of the requirements can be satisfied simultaneously. The highly
important linear phase constraint corresponding to symmetrical wavelets can be maintained
by relaxing the orthonormality and using biorthogonal bases [4]. The biorthogonal wavelets
are more often used because of their relatively short compact support and best trade-off
between regularity and vanishing moments [4,9]. By far, choosing wavelets for image com-
Decompression
Compressed
data
-*
Decoder
FIGURE 4. Wavelet method of compression
604
Q(x)
xO xl
x2
*M <x<Xi
x3
y4 x4 x5 x6 x7
y3
(yi>
x<xn
yi
FIGURE 5. A Non-uniform spatial quantizer
pression is still a somewhat inexact science. Based on the overall properties and
experiments, Bior4.4 (symmetric bi-orthogonal wavelet with 4 vanishing moment, linear
phase) is selected.
DWT Coefficients Quantization
As shown in Fig. 4, the key part of compression is 'Quantization'. Quantization refers
to the process of approximating the continuous set of original values in the image data with
a finite (preferably small) discrete set of values. Besides, the precision of the floating point
DWT output is also quantized by this step. Obviously, this is an irreversible, lossy part of
the compression.
For the sake of computational simplicity, Scalar Quantization, in which each input
coefficient is treated separately in producing the output, is employed here. We carry it out
by two approaches: spatial and spectral.
Fig. 5 explains how a Non-Uniform spatial Scalar Quantizer with input range divided
un-equally works. For common images, there is a method called Max Quantizer [4,10] to
part the input quantization steps. It is on the basis of the probability distribution of the
variable to be quantized. However, considering the flaw edge is mostly concerned in NDE,
we base our method on the amplitude of the DWT coefficients. Large coefficients embody
the flaw edges. To get a more concise view of this questioned parts, the steps between
bigger coefficients are more intensive than those between smaller ones. Those whose values
are among the smallest and thus set to 0 by quantization, is the largest in number.
The spectral approach, what we call 'Weighting', puts weights to different frequency
bands, shown in Fig. 6. Coefficients of some frequencies are emphasized by multiply a
large 'weight'. They are mostly high frequency ones caused by flaw edges and those that
human visual system is mostly sensitive to. Some are overlooked, and some others are
given weight '0', which mostly are of low frequencies. The reconstructed image only uses
the weighted coefficients of a particular frequency or frequencies [11].
In both approaches, by discarding these '0' pixels, compression can be realized. The
Original
DWT
Coefficients
W^D,
W 2 *D 2
D2
Weighted
Coefficients
Weighting
wm
FIGURE 6. Weighting —-one Spectral Quantization method
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FIGURE 7. Reconstructed image after compression of Fig.3(a) (PSNR=18.593, NCR=8bpp). (a) by Uniform
Quantization, quantization step No.=40, PSNR=14.455,NCR=0.0453bpp. (b) by Weighting, PSNR=15.371,
NCR=0.0769bpp
setting of the quantization step and weight in this process is trial and error. A mathematical
model for the function of this setting with frequency, orientation, compression rate and
display resolution will be a challenging task.
The quality criteria are PeakSNR (taking the peak value of 255 of the original signal
instead of its variance as the numerator in classical SNR) and NCR (nominal compression
rate, the unit is 'bpp', bit per pixel)[4]. Some examples are displayed in Fig. 7.
DISCUSSION ON TRADE-OFF AND UNIFICATION
Thus far, we have explored the DWT ability to do ultrasonic C-scan flaw
enhancement and image compression respectively. The ability is derived from the fact that
the wavelet transform gives the image an alternative mathematical representation by DWT
coefficients. Because they carry the information about time, frequency and orientation,
these coefficients are easier to be sorted and handled than the original gray level value.
In both the cases of enhancement and compression, one thing shared in common is
that some coefficients are set to zero, though by different rules. However, it can be found
that there is overlapping. More explicitly, some coefficients are small both in amplitude and
frequency, which are set to zero in both cases. From this point, we try to explore how to
realize the two tasks in one process.
It is obvious that there is trade-off between image enhancement and compression.
The threshold should be enlarged to get a higher compression ratio but at the expense of
clarity. Compression will be sacrificed in increasing the quantization steps to get a clearer
reconstructed image. Still, for the sake of image enhancement, the weighting should keep
the low frequency though the compression ratio will be lowered.
Fig. 8 shows the result of this process on the image given in Fig. 3(a). The effect of
the enhancement is not as good as Fig. 3(b), and the compression is less optimal than Fig. 7.
FIGURE 8. Unification and trade-off of compression and enhancement result of Fig. 3(a). PSNR=54.8173,
NCR=0.1796bpp
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But it realizes both of them in one process within the tolerance of NDE purpose. The data
of the criteria is acceptable and satisfactory.
CONCLUSION
The experiments prove that by proper soft thresholding, DWT can enhance the
Ultrasonic C-scan flaw image. The algorithms of non-uniform spatial quantization and
weigh quantization perform well in the C-scan image compression. We realize the
unification of flaw enhancement and image compression in one process. The reconstructed
image from the compressed data gives a clearer interpretation of the flaws at a much
smaller bit rate.
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