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·Aeronautics and Aerospace
Lossless compression of radiographic non-destructive testing images by region-based and entropy-optimal predictive codingL.-K. Shark, X. Y. Lin, M.R. Varley, B.J. Matuszewski
Department of Engineering and Product Design
University of Central Lancashire
Preston, PR1 2HE, UK
J. P. Smith
Warton, Preston, PR4 1AX, UK
|Fig 1: X-ray image of an aircraft component|
Applying the LOCO-I (Low Complexity Lossless Compression for Images) method which is the kernel technology used in the JPEG lossless scheme  to the X-ray image shown in Figure 1, the storage required by the compressed image is 636 Kbytes which is still a considerable amount. To further reduce storage requirement and transmission time, a region-based predictive coding method was previously proposed by the authors , whereby different predictors are used in different regions of the radiographic NDT image with each region having the same material structure. The storage required by the region-based predictive coding method with Huffman coding is 452.68 Kbytes, giving a compression ratio of 2.36. Compared with the compression ratio of 1.68 achieved by the LOCO-I method, the overall improvement about 40%. This paper extends the previously proposed method to entropy-optimal predictive coding by optimising the parameters of the predictors according to the specific image features contained in each region to satisfy the minimum entropy criterion.
The main processing stages in the proposed method can be described as:
Using the X-ray image shown in Figure 1, these stages are presented in detail in the following sections.
One possible way to achieve region-based image decomposition is to align the radiographic image acquired from the component under inspection to its corresponding wire-frame CAD model. An automatic image registration method for aligning NDT images with the CAD model has been previously proposed  and consists of four main processing stages, namely:
With the radiographic NDT image correctly aligned to the corresponding CAD model, it is a relatively easy matter to utilise the structural and material information available in the CAD data to divide the radiographic image into regions of similar image properties. For the X-ray image shown in Figure 1, Figures 2 and 3 show the two regions obtained with the former corresponding to the thin carbon fibre region and the latter corresponding to the thick honeycomb region. From the viewpoint of image compression, an immediate advantage of image alignment is that the number of pixels to be coded is reduced by ignoring those image pixels belonging to the background.
|Fig 2: Thin carbon fibre region||Fig 3: Thick honeycomb region|
Using Equation (1), the entropy value of the original X-ray image shown in Figure 1 is found to be 7.5073 bits/pixel. After image decomposition with the original X-ray image divided into two regions of similar image properties, the entropy value decreases to 6.8879 bits/pixel for the thin carbon fibre region and 7.1040 bits/pixel for the thick honeycomb region.
While the image of the thin carbon fibre region shown in Figure 2 is smooth with no obvious regular patterns, the image of the thick honeycomb region shown in Figure 3 is dominated by regular hexagon patterns. The spatial regularity of each region can be determined by computing the correlation coefficients of the image along the horizontal and vertical directions using
where f x,y and fx+Dx,y+Dy are the grey-level values of the two pixels at co-ordinates (x, y) and (x+Dx, y+Dy), and and are the expectation values(mean-grey level values of and respectively. While the lack of significant local maxima in RDx,Dy indicates no spatial regularity for the thin carbon fibre region, the periodicity in the occurrence of maximum RDx,Dy for the thick honeycomb region indicates a correlation with similar grey-level values between two image pixels separated by 11 pixels along the horizontal direction and by 8 pixels along the vertical direction.
To reduce the spatial redundancy, the thick honeycomb region can be more efficiently represented by a residual image which is the weighted difference between the grey-level values of the image pixels separated by either every 11 pixels in the horizontal direction or every 8 pixels in the vertical direction. That is
where w x and w y are weights for the subtraction to be performed either in the horizontal direction or in the vertical direction. To search for the optimum weight value, a convenient criterion is minimum entropy and a convenient weight for starting the search is the correlation coefficient. Based on a linear refinement strategy, the search interval is initially centred around the correlation coefficient, the entropy values are computed by stepping through the search interval with a relatively large step size, and the weight corresponding to the minimum entropy value over the search interval becomes the centre of the next search interval with a reduced width and a reduced step size. The process is repeated until the required accuracy is achieved for the weight.
For the thick honeycomb region shown in Figure 3, using equal weighting with wy = 1 yields an entropy value of 6.2000 bits/pixel for the residual image, whereas using entropy weighting with w y = 0.8910 produced by the search routine yields a lower entropy value of 6.1618 bits/pixel for the residual image.
A coding issue arises for those pixels lying along the image boundary which cannot be applied to the entropy weighted subtraction. With the subtraction operation performed along the vertical direction, the first 8 rows of the image pixels in the thick honeycomb region need to be stored. To reduce storage, they are coded using the predictive coding to yield a storage requirement of 8.49 Kbytes instead of 12.00 Kbytes using 8 bits/pixel.
where a1, a2, and a3 correspond to the prediction coefficients in the vertical, horizontal, and diagonal directions, respectively. The optimal predictive coding can be based on different criteria. Minimum mean square error (MMSE) is one of them. Based on the Orthogonality Principle , the optimal values (in term of MMSE) of the prediction coefficients are obtained by solving the three simultaneous linear equations expressed in matrix form as
Applying Equation (4) with the two different sets of the prediction coefficients obtained from Equation (5) yields an entropy value of 3.4619 bits/pixel for the error image of the thin carbon fibre region, and 5.1595 bits/pixel for the error image of the entropy weighted residual image of the thick honeycomb region.
A better criterion for optimal predictive coding is minimum entropy, and a numerical optimisation method based on the Nelder-Mead Simplex algorithm [6,7] is used to search the optimum prediction coefficients with the objective of minimising the entropy value for each decomposed region with similar image features. The prediction coefficients produced by Equation (5) was used as the initial guess for the first iteration. Each time the algorithm produces a new set of prediction coefficients with a reduced entropy value for the error image of a decomposed region by moving from one vertex to another in a multidimensional space containing possible solutions.
Based on the two sets of optimum prediction coefficients produced by the Nelder-Mead Simplex algorithm to satisfy the minimum entropy criterion, the final entropy value is 3.4545 bits/pixel for the error image of the thin carbon fibre region, and 5.1275 bits/pixel for the error image of the entropy weighted residual image of the thick honeycomb region. Using Huffman coding, to code error images results in 145.08 Kbytes required for the thin carbon fibre region and 280.36 Kbytes required for the thick honeycomb region. Taking into consideration the initial pixels required for prediction, which are 6.25 Kbytes for the thin carbon fibre region and 11.16 Kbytes for the thick honeycomb region (including those required for generating the entropy weighted residual image), the final storage required by the compressed image is 435.47 Kbytes, giving a compression ratio of 2.46. Compared with the compression ratio of 1.68 achieved by applying the LOCO-I method to the X-ray image shown in Figure 1, the incorporation of the entropy-optimal constraint to the texture-based pre-processing and the design of predictors is seen to extend the improvement on the compression ratio from about 40% offered by the previously proposed region-based predictive coding method to about 46%.
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