Procedia Engineering 38 ( 2012 ) 385 – 390
1877-7058 © 2012 Published by Elsevier Ltd. doi: 10.1016/j.proeng.2012.06.048
International Conference on Modeling Optimization and
Computing
Type-2 Fuzzy Thresholded Bandlet Transform for Image
Compression
R. Rajeswari∗
Department of Computer Applications, School of Computer Science and Engineering, Bharathiar University, Coimbatore-641046, Tamilnadu, India
R. Rajesh∗
Department of Computer Applications, School of Computer Science and Engineering, Bharathiar University, Coimbatore-641046, Tamilnadu, India
Abstract
Image compression techniques help in efficient storage and transmission of images. In this paper we propose a lossy compression technique for two dimensional images. The proposed technique uses bandlet transform to capture the anisotropic regularity of edge structures apart from capturing regularity information from smooth regions. Then type II fuzzy thresholding is performed to retain the important bandlet coefficients. The proposed method is applied to 2D images and the results are compared with bandlet based ordinary thresholding. Test results show that the proposed method gives better results in terms of PSNR and bits-per-pixel.
Keywords:
Bandlet Transform, Type II Fuzzy Thresholding 1. Introduction
Transform based image compression methods have been widely used in the state-of-art im-age compression, where a digital imim-age is represented by a set of coefficients, which are then coded. Transform methods like Discrete Cosine Transform (DCT) [1], Discrete Wavelet Trans-form (DWT) [2] can be used to represent the image as a set of coefficients. These transforms exploit correlation between pixels in the image data. They represent the image data as a set of
∗Corresponding Author:+91-9944126231, +91-9894850299
Email addresses: [email protected] (R. Rajeswari ), [email protected] (R. Rajesh)
less correlated coefficients and thus the coefficients are packed into a specific area of the trans-form domain. Image compression can be done by setting the coefficients below a given threshold to zero and then coding the coefficients. This property makes these transform techniques to be widely used in various image compression schemes [3, 4, 5, 6, 7]. The coefficients can be en-coded using Huffman coding [8] and arithmetic coding [9], which are popular variable length entropy coding techniques.
Standard Wavelet bases [2] are efficient to represent functions with pointwise singularities. But they do not capture the geometric regularity along the singularities of a surface, due to their isotropic support. To exploit anisotropic regularity of a surface along edges, the basis must in-clude elongated functions that are nearly parallel to the edges. There comes the need for bandlet transform. The bandlet decomposition [10, 11, 12, 13] is computed with a geometric orthogonal transform that is applied on orthogonal wavelet coefficients. Moreover, Type-2 fuzzy sets can handle uncertainty/ vagueness much better than Type-1 fuzzy sets. The threshold value in com-pression is used to distinguish significant coefficients (those which are retained) and insignificant coefficients (those which are discarded). In this paper we propose an approach for thresholding bandlet coefficients that are based on Type-2 fuzzy logic. The method uses the maximization of the measure of ultrafuzziness as proposed in [17]. The compression results of ordinary hard thresholding, type-1 thresholding and type-2 fuzzy thresholding show that type-2 thresholding gives much better Peak-Signal-to-Noise Ratio (PSNR) for given bits per pixel (bpp).
This paper is organized as follows. Section II gives an overview of bandelet transform. Sec-tion III provides a detailed descripSec-tion of the proposed type-2 fuzzy thresholding in the bandelet domain. Section IV provides the results of compressing various 2D images using the proposed thresholding technique. Section V discusses the conclusion and the scope for future work. 2. Review of Bandlet Transform
The bandlet decomposition [10, 11, 12, 13] is computed with a geometric orthogonal trans-form that is applied on orthogonal wavelet coefficients. Wavelet transform, when applied to an image of N pixels, computes the set of N dot products
< f, ψs jn> f or2−J≤ 2− j< √ Nand0≤ n1, n2< 2− j, (1) < f, φs Jn> f or0 ≤ n1, n2< 2−J, (2)
where the projection onφs
Jn functions produces a coarse approximation at scale 2J represents the level at which we stop the wavelet transform. Those values can be conveniently stored in an array of N pixels. A dyadic square is a square obtained by recursively splitting the original wavelet transformed image fs
j into four sub-squares of equal size. Let the width of the squares be L pixels with 4 ≤ L ≤ 2− j/2. For each dyadic square S at a given scale 2jand orientation s of the wavelet transform 1D reordering of the grid points is performed. The possible number of 1D reordering may be equal to the number of directions d joining pairs of points in square S of width L. 1D reordering is done by projecting the sampling location along d and sorting the resulting 1D points from left to right. To the resulting 1D discrete signal, fd, 1D wavelet discrete wavelet transform is performed. For a given threshold T , the direction d, which generated the less approximation error, is selected. Let bkdenote the coefficients of 1D wavelet transform of fd, and RBbe the number of bits needed to code the quantized coefficients QT(bk). To select the best geometry, the direction d that minimizes the Lagrangian
where fdRis the signal recovered from the quantized coefficients and RGis the number of bits needed to code the geometric parameter d with an entropy coder. λ is taken as 3/28 [14]. The bandlet transform can be applied to images in two ways. The first method is to apply the bandlet transform directly on an image and the second method is to apply the bandlet transform on each scale and orientation of the wavelet transform of an image. In this work we make use of the second method.
3. Proposed Type II Fuzzy Thresholding in Bandlet Domain
Fuzzy set theory and Fuzzy logic [15] offer us powerful tools to represent and process knowl-edge represented as fuzzy if-then rules. Type II fuzzy sets [16] overcome the disadvantages of type I fuzzy sets by efficiently representing uncertainties like the meaning of the words, the mea-surements which are noisy etc., which could not be efficiently represented by type I fuzzy sets. One of the simple ways of creating type II fuzzy sets are by using linguistic hedges like dilation and concentration.
Using fuzzy set theory, bandlet thresholding can be expressed as fuzzy bandlet thresholding. Let|bs,d(i, j)| be the absolute value of the bandlet coefficient, bs,d(i, j), at location (i, j) for the scale s and direction d and T be the threshold value. Any one of the membership functions μ is selected and α is initialized. The upper and lower membership degrees μU andμLof the membership function are given by
μU(x)= [μ(x)]0.5 (4)
μL(x)= [μ(x)]2 (5)
We use a triangular membership function for our experiments. A triangular membership function and its type II fuzzy set are shown in figure 1 whereα is 2. The frequency histogram for the bandelet coefficients, |bs,d(i, j)| in all scales and orientation except for the approximation scale,
is computed. The threshold is moved from 0 to max(|bs,d(i, j)|) and in each position the amount
of ultrafuzziness [17] is computed. The measure of ultrafuzziness is given by: γB= 1 MN max(|bs,d(i, j)|) g=0 h(g)min(μU(g), 1 − μL(g)) (6)
where M, N is the size of the 2D image, h(g) is the histogram value andμ(g) is the membership value of the bandlet coefficient g. The position where the amount of ultrafuzziness is maximum is used as the threshold. Let this threshold value be denoted by T . The bandlet coefficients whose absolute values are greater than T will remain as such and the remaining coefficients will be set to 0. The procedure mentioned above for type-2 thresholding is again repeated separately for all the bandlet coefficients in the approximation scale. Thus a threshold for bandlet coefficients in the approximation scale is found using the location where we have maximum measure of ultrafuzziness and thresholding is done based on this threshold. Then a threshold is found for the remaining coefficients again based on the maximization of the measure of ultrafuzziness and thresholding is performed.
4. Type II Fuzzy Thresholding on Images
The proposed type-2 thresholding can be performed on images in two steps. In the first step the 2D image Ix,yis forward bandlet transformed to obtain the bandlet coefficients. Let the
0 5 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y 0 5 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y
Figure 1: a) Triangular Membership Function (trimf in matlab) with P=[2 4 8] and b) its linguistic hedges with α = 2
(a) (b)
(c) (d)
(e)
Figure 2: Images used for Type-2 fuzzy thresholding a) lena b) barbara c) boat d) fingerprint e) brain
coefficients of 2D discrete bandlet transform of image Ix,y be represented as Fx,y. Orthogonal
bandlets uses an adaptive segmentation and a local geometric flow and is thus able to capture the anisotropic regularity of edge structures. The second step is to perform type II fuzzy bandlet thresholding as discussed in the previous section. The histogram of the bandlet coefficients in the approximation scale and a histogram of the bandlets in the remaining scales and orientations are computed. The threshold is moved and in each position the measure of ultrafuzziness is computed. The threshold T where the measure of ultrafuzziness is maximum is used as the threshold. The bandlet coefficients which are greater than T remain as such and the remaining coefficients are set as zero. Let FTx,ybe the fuzzy bandlet thresholded image. The fuzzy bandlet thresholded image can be encoded using any of the variable length coding techniques.
5. Experiments and Results
We have used four 2D images each of 512*512 dimension and one 2D image of 256*256 dimension. The images used in this experiment are shown in figure 2. The results of the com-pression of 2D images using the proposed Type-2 fuzzy thresholding are shown in table I. The table shows the Peak Signal to Noise Ratio (PSNR) values and the Bits-per-Pixel (BPP) using normal thresholding, type-1 fuzzy thresholding and type-2 fuzzy thresholding. The results for PSNR values versus Bits-per-Pixel are plotted in figures 3, 4, 5, 6 and 7 respectively for the 2D images.
It can be seen that the proposed scheme using bandlet and type II fuzzy thresholding tech-nique is better than the normal bandlet thresholding and bandlet type I fuzzy thresholding in terms of Peak Signal to Noise Ratio (PSNR) and Bits-per-Pixel.
1 1.5 2 2.5 3 3.5 4 4.5 5 34 35 36 37 38 39 40 41 42 43 Bits−Per−Pixel PSNR Bandlet+Thresholding Bandlet+Type−1 Thresholding Bandlet+Type−2 Thresholding
Figure 3: Bits-Per-Pixel and PNSR for lena
1.5 2 2.5 3 3.5 4 4.5 5 33 34 35 36 37 38 39 40 Bits−Per−Pixel PSNR Bandlet+Thresholding Bandlet+Type−1 Thresholding Bandlet+Type−2 Thresholding
Figure 4: Bits-Per-Pixel and PNSR for barbara
1.5 2 2.5 3 3.5 4 4.5 32 33 34 35 36 37 38 39 40 41 Bits−Per−Pixel PSNR Bandlet+Thresholding Bandlet+Type−1 Thresholding Bandlet+Type−2 Thresholding
Figure 5: Bits-Per-Pixel and PNSR for boat
1.5 2 2.5 3 3.5 4 4.5 28 29 30 31 32 33 34 Bits−Per−Pixel PSNR Bandlet+Thresholding Bandlet+Type−1 Thresholding Bandlet+Type−2 Thresholding
Figure 6: Bits-Per-Pixel and PNSR for fingerprint
1.5 2 2.5 3 3.5 4 4.5 31 32 33 34 35 36 37 38 39 40 41 Bits−Per−Pixel PSNR Bandlet+Thresholding Bandlet+Type−1 Thresholding Bandlet+Type−2 Thresholding
Figure 7: Bits-Per-Pixel and PNSR for brain
Table 1: PSNR and Bits-per-Pixel for 2D Images using Bandlet Transform and Type II Fuzzy Thresholding
Image Original Bandlet Transform+ Bandlet Transform+ Bandlet Transform+ Size Thresholding Fuzzy Thresholding Type II
Fuzzy Thresholding BPP PSNR BPP PSNR BPP PSNR lena 512*512 1.08 34.24 1.92 37.32 2.55 38.83 2.24 34.31 2.87 39.34 3.29 40.17 3.49 34.43 4.70 42.20 4.70 42.31 barbara 512*512 1.82 33.36 2.47 34.51 2.06 33.31 2.76 33.39 3.40 37.15 2.87 35.79 3.94 33.44 4.53 39.78 3.88 38.42 boat 512*512 1.68 32.71 2.82 37.19 2.46 36.34 2.99 32.73 3.61 39.07 3.61 39.18 3.61 32.79 4.39 40.71 4.39 40.81 fingerprint 512*512 1.95 28.04 2.82 30.38 2.72 30.26 2.96 28.11 3.64 32.05 3.83 32.15 3.56 29.97 4.28 33.22 4.29 33.33 brain 256*256 1.96 31.70 2.97 36.14 2.98 36.28 2.56 31.78 3.96 39.29 3.41 37.61 3.13 31.80 4.27 40.35 4.26 40.44
6. Conclusion and Future Work
This paper presents a novel type-2 fuzzy thresholded bandlet transform for image compres-sion. The power of bandlet transform to capture the geometric regularity along edges and the ability of type-2 fuzzy sets to handle uncertainty/ vagueness makes the compression better. The proposed type-2 fuzzy thresholding gives better results for 2D images in terms of PSNR and Bits-per-Pixel. The proposed Type-2 fuzzy thresholding can also be improved by making use of the local statistics of the wavelet coefficients in each subband.
Acknowledgement
This research is supported by University Grants Commission, India through Major Research Project Grant (UGC F.No. 33-67/2007(SR) dated 28 Feb 2008) and Minor Research Project Grant (UGC F.No. 33-429/2007(SR) dated 15 Apr 2008). The authors are thankful to Bharathiar University for valuable support.
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