Performance Analysis of Bi-orthogonal Wavelets for Fingerprint Image Compression

Performance Analysis of Bi-orthogonal Wavelets for Fingerprint Image Compression K.Selvakumarasamy 1, Radhikadevi 2, Ramya 3 & Sagunthala 4 Department of ECE, A.M.S College of Engineering E-mail : selva_ksamy@rediffmail.com, 2 ramya26.1.92@gmail.com Abstract Fingerprint is the unique way of identification, but still the application of fingerprint identification is overlooked because of its large storage capacity of the data. Due to which the technique of data compression has been introduced to reduce the storage of image. In this paper, bi-orthogonal wavelets are used for compressing the image and determine the most appropriate bi-orthogonal wavelet type for better compression. The Huffman coding are used because of its directness, high speed, and lack of encumbrance by patents. Appropriate bi-orthogonal type is analyzed by attaining minimum MSE (mean square error) and PSNR (peak signal to noise ratio). The transform and encoding scheme has been implemented using MATLAB. The gray scale digitized fingerprint image of size 300x300 is used as a test image. Keywords Haar, Db, Sym, Coif, Bior, CR, MSE, PSNR I. INTRODUCTION Fingerprint is the exceptional way of identifying a person. This technique is one of the most personal identification methods which can be used in ATMs, Aadhars, Voting, etc. It plays a vital role in forensic and civilian application. Between 1924 to 1998, the US Federal Buerea of Investigation (FBI) has a credential, mainly consisting of inked impressions on paper cards. The challenge here is the storage capacity of the data, which led to the idea of digital storage. The improvement in technology paved a way for electronic storage of the fingerprint. Still the replica of these stored images is poor in digitization quality and compatibility. On the outlook of storage capacity single fingerprint image insist on 0.6 Mbytes of storage [1]. On the latest, from July 28, 1999 the FBI has a credential of 70 million sets of fingerprint accounting 2 peta bytes of storage with a cost of 2 billion dollars [2]. This problem gave rise to the advancement in digitization standards and compression using wavelets. Compression can be achieved using wavelet transform such as haar, daubechies, coiflets, symlets, biorthogonal, etc. It is not much sure that the fingerprint minutiae i.e., ridges endings and bifurcations are preserved or not [1] after the compression process. For that verification, the best way is to re-construct the compressed fingerprint image and comparing it with the original image. The main advantage of using wavelet is that, it simultaneous offers the localization in both time and frequency domain, speed in computation and it can able to separate the fine details in an image [3]. Biorthogonal type wavelet transform has been chosen based on mean square error (MSE) and peak signal to noise ratio (PSNR). The different wavelet transforms are discussed in section II. Section III converse about the biorthogonal wavelet transform. Difference between orthogonal and bi-orthogonal wavelets is presented in section IV. Section V talk about the subband decomposition process. II. WAVELET TRANSFORM Wavelets are developed from the Fourier transform to overcome the drawback of overall domain analysis for which wavelet uses a localized time and frequency analysis. Wavelet plays a vital role in image compression in the part of improving the signal strength. Hence wavelets are widely used in the field where the degradation is not tolerated. Wavelets can also effectively remove the noise in an image [4]. Wavelet transform is defined as the infinite set of various transforms. Which uses the function that are localized in both the real and Fourier space, it is given as, F (a, b) = f x φ a,b x d(x) 64

Where, * is the complex conjugate symbol and φ is some function that obeys certain laws. compressing the image [16]. Figure 2 present the wavelet functions. A. Haar Wavelet Haar is discontinuous, and resembles the step function [5]. It performs its basic operation of averaging and difference on a pair of value and then the algorithm shifts by two value and again calculate the average and difference for next two pairs. It results in obtaining set of coefficients. Haar is the first and simplest wavelet even though it has a limitation in removing the noise [15]. All the high frequency changes should be detected by the high frequency coefficient spectrum, but the size of haar window is 2. So, the higher frequency changes cannot be detected by haar. The Haar wavelet function can be described as Fig. 2 Wavelet function for daubechies C. Coiflets Wavelet Due to sampling approximation property, orthogonal Coiflet system is popular in numerical analysis and due to their associated near-linear phase filterbanks it is popular in digital signal processing also. An orthogonal wavelet system is a generalized orthogonal Coiflet (GOC) system of order L if it satisfies the generalized Coifman criterion given as [12], Figure 1 illustrate the wavelet function for haar wavelet. Coiflet is much more symmetrical with respect to support length [7]. Orthogonal Coiflet systems result in smaller phase distortion and can achieve a better sampling approximation property. It permits the design of nearly half-point symmetric filterbanks [12]. Figure 3 reveal the wavelet transform. Fig. 3 Wavelet function for coiflet Fig. 1 Wavelet function for haar B. Daubechies Wavelet Daubechies is similar to haar transform were it is also a simplest and imaginable wavelet [6]. The daubechies wavelet is similar to haar wavelet in computing the successive averages and differences by means of scalar products with scaling signals and wavelets. The difference between them lies in hoe the scaling signals and wavelets are defined [15]. This wavelet has an unbiased frequency response and non linear phase response. This wavelet also uses overlapping windows due to which the high frequency coefficient spectrum reflects all the high frequency changes. Thus the limitations in the haar wavelets are overcome in daubechies by removing the noise and D. Symlet Wavelet Symlet refers to the symmetric property [7]. It is evolved from daubechies for increase in symmetry and simplicity, which is obtained by reusing the function. Figure 4 display the wavelet transform. Fig. 4 Wavelet function for symlet III. BI-ORTHOGONAL WAVELETS A bi-orthogonal wavelet is one type of wavelet in which the associated transform is inversing but it is not necessary to be orthogonal. It gives freedom in designing bi-orthogonal wavelets than orthogonal 65

wavelets. Additional freedom is the option to create symmetric wavelet function [14]. It compactly supports symmetric analyzing and synthesis wavelets and scaling functions. There is quite a bit of freedom in designing the biorthogonal wavelets, as there are no set steps in the design process [13]. It has a property of linear phase which is needed for image reconstruction. The properties can be derived by using two wavelets Decomposition and Reconstruction instead of using a single wavelet. A. Algorithm The proposed algorithm for the analysis of biorthogonal wavelets is as follows Step 1: Read the original fingerprint image into MATLAB workspace. Step 2: Set the decomposition level and decomposition filter. Step 3: Obtain the decomposition components and decomposition vector. Step 4: Use decomposition vector and compress the image using encoding technique. Step 5: Reconstruct the image by decoding and applying inverse wavelet transform. Step 6: Calculate the Compression Ratio (CR) and Peak Signal to Noise Ratio (PSNR). Step 7: Analyze the quality of reconstructed image by comparing it with the original image. Step 8: Repeat step 2 to 6 for various decomposition levels and wavelets. IV. ORTHOGONAL VS BIORTHOGONAL WAVELET The two main categories in the wavelet families are orthogonal and Biorthogonal wavelets. The ideal functioning of the wavelets in compressing an image depends on Scaling function, Symmetry, Regularity, compact support, Wavelet balancing, vanishing moments and Degree of Smoothness [8]. A. Scaling Functions The scaling function filters the coarse level of the transform to a fine level. Considering a set of functions f k (t) spans a vector space F (i.e) Span k {f k } = F with all elements of the space is of here k The set f k (t) is called a basis set for a given space F for the set of { k} is unique for any. If the set < f k (t), f l (t)> = 0 for all k l, then it is called orthogonal basis/scaling functions (i.e) < f k (t), f l (t)> = δ(k l). Though orthogonal scaling functions are simple and easy it does not satisfy some desired properties (i.e) f 1 f 2 = 0, f 2 f 1 = 0 conditions has to be satisfied. Fig. 5 Biorthogonal Basis [ ] and [ ] ~ ~ Here the basis pairs [f 1, f 2 ] and f 1, f 2 form ~ ~ biothogonal basis. The vector f 1 and f 2 are obtained ~ ~ by rotating f 1 and f 2 to 90 [s].thus the f 1, f 2 are called as the dual basis of [f 1, f 2 ] [6]. Thus the expression of orthogonality can be made to suit ~ all the properties using a dual basis set fk ( t ) f k t, whose elements are not orthogonal to each other but to the corresponding element of the expansion set, <, > =. The above type requires two sets of vectors, the expansion set and the dual set which together constitute biorthogonal 66

V. SUBBAND DECOMPOSITION B. Symmetry Property Wavelets based on filter have been used in signal processing applications for noise removal and data compression, but these filters cannot be made both symmetric and orthogonal expect for Haar wavelet. So to design a symmetric wavelet the concept of orthogonality is dropped [9]. One of the desired properties in the Image Processing application is Conservation of Energy. This property is well satisfied by the orthogonal filters, but not by biorthogonal filters. To make the biorthogonal wavelet satisfy the above property we design symmetry biorthogonal wavelets with orthogonal behavior where low pass filters h o and g o obey the property. There are two approaches to the subband decomposition of two dimensional signals using two dimensional filters, or using separable transforms that can be implemented using one dimensional filters on the row first and then on the columns. Most approaches, use the second approach. Figure 6 shows how an image can be decomposed using subband decomposition. Of the four sub images, the one obtained by low pass filtering the rows and columns is referred to as the LL image; the one obtained by low pass filtering the rows and high pass filtering the columns is referred to as the LH image; the one obtained by high pass filtering the rows and low pass filtering the columns is called the HL image; and the subimage obtained by high pass filtering the rows and columns is referred to as the HH image [14]. Figure 6 demonstrate the subband decomposition of an NxM image. Figure 7 shows the first level decomposition structure. Figure 8 give you an idea about three popular subband structures. So by nearly symmetric we mean that a filter h 0 with a subset of its coefficients being exactly symmetric. C. Filter Length In orthogonal filters the scaling filters are of same length and should be even, where as biorthogonal requires that both filters are either even length or of odd length. Let us have N n and N g are filter supports which are functions of k and L. The biorthogonality conditions require (N n + N g )/4 and symmetry requires (N n + N g - 2k)/2 and we require L/2 equations to guarantee the zero derivative conditions in the filter functions. Fig. 6 Subband decomposition of an N M image = Degree of freedom Fig. 7 First level decomposition Thus if K = odd, l =odd; if K = even, l = even. Another desirable property in Image Processing is that the uncorrelated input x (n) (i.e.) In case of orthogonal filters the resulting output remains uncorrelated i.e. the white notice preserves its nature, where as in biorthogonal filters the noise becomes correlated. This is over come by approximating orthogonality of symmetric filters and thus the input x (t) is nearly uncorrelated [9][11]. Fig. 8 Three popular subband structures 67

VI. RESULTS The dramatization of image compression and quality of reconstructed image for various wavelets is analyzed using two factors which discusses how much data can be compressed in such a way it can be reconstructed without loss in information and efficiency of reconstructed image. 1) Compression Ratio 2) PSNR A. Compression Ratio It is the ratio of the size of compressed data set to the size of the original image. It measures the ability of data compression. CR= x 100 B. PSNR The qualitative performance is achieved using PSNR of a reconstructed image. PSNR= Where, 255 is the maximum possible value can be obtained by the image MSE is defined as Where, M*N is the image size. X (M, N) = original image pixel value. Y (M, N) = recovered image pixel value [8]. Figure 9 present the Original Image and Reconstructed Image for coiflet, daubechies, haar and symlet wavelets. Figure 10 displays the reconstructed image for biorthogonal wavelets bior1.1, bior1.3, bior1.5, bior2.2. TABLE I COMPRESSION RATIO AND PSNR FOR ALL THE WAVELETS Wavelet Type bior1.1 Level Compression MSE PSNR Ratio (%) 1 77.3600 0.1193 57.3632 2 71.3756 0.1125 57.6176 bior1.3 1 80.3400 0.0942 58.3890 bior1.5 bior2.2 bior2.4 bior2.6 bior2.8 bior3.1 bior3.3 bior3.5 bior3.7 bior3.9 bior4.4 bior5.5 bior6.8 haar db1 db4 sym4 coif1 2 75.6956 0.0951 58.3484 1 82.4089 0.0972 58.2553 2 79.2978 0.0985 58.1943 1 69.4978 0.0984 58.2022 2 64.2267 0.1024 58.0289 1 71.2911 0.0996 58.1481 2 67.2689 0.1120 57.6401 1 73.0667 0.0971 58.2582 2 70.1444 0.1008 58.0983 1 74.8133 0.1003 58.1159 2 73.0244 0.1105 57.6958 1 65.0000 0.1974 55.1769 2 61.4044 0.2094 54.9209 1 66.2511 0.1509 56.3432 2 63.6267 0.1561 56.1968 1 67.8111 0.1418 56.6136 2 66.2000 0.1464 56.4761 1 69.4378 0.1432 56.5717 2 69.0089 0.1577 56.1531 1 71.1178 0.1391 56.6974 2 71.8867 0.1426 56.5911 1 69.6622 0.0845 58.8637 2 64.1244 0.0877 58.7029 1 71.6000 0.0928 58.4570 2 65.5889 0.0991 58.1689 1 72.3867 0.0911 58.5350 2 68.7756 0.1167 57.4599 1 78.7422 0.1193 57.3632 2 72.6822 0.1125 57.6176 1 78.7422 0.1193 57.3632 2 72.6822 0.1125 57.6176 1 68.8467 0.0817 59.0090 2 64.3044 0.0820 58.9914 1 70.3444 0.0816 59.0121 2 65.3867 0.0819 58.9967 1 73.0733 0.0824 58.9698 2 67.9178 0.0850 58.8386 68

Depending upon the various performance factor, the efficieny of various wavelets is analyzied and it is instituted that BIOR4.4 has the greatest efficiency in compressing the fingerprint image. The results showed (fig 12 & table 1), BIOR perform better than daubechies, symlet and coiflet wavelets for fingerprint image but extracted image quality is poor in the comparison. Biorthogonal extend the wavelet family. It is well known in the subband filtering community that symmetry and exact reconstruction are incompatible (except for the Haar wavelet) if the same FIR filters are used for reconstruction and decomposition. Fig. 9 Original Image and Reconstructed Image for Various Wavelets Fig. 10 Reconstructed Image for Biorthogonal Wavelets Thus depending on these factors the efficiency and performance of the reconstructed finger print image for various wavelets is tabulated below, The below graph represents the Compression Ratio and PSNR value for the decomposition of Level 1 (fig. 11, fig. 12) and Level 2 (fig. 13, fig. 14) respectively for various biorthogonal wavelets. Fig. 12 PSNR for Level 1 Decomposition Fig. 13 Compression Ratio(%) for Level 2 Decomposition Fig. 11 Compression Ratio(%) for Level 1 Decomposition 69

Fig. 14 PSNR for Level 2 Decomposition VI. CONCLUSION AND FUTURE WORK This paper aimed at developing computationally efficient and effective algorithm for fingerprint image compression. The proposed algorithm is developed to compress the image better PSNR and compression ratio. The result is obtained by concerning the reconstructed image quality as well as with compression ratio. The result clearly shows that Biorthogonal 4.4 is well suited for the finger print image compression. This work, in future can be elaborated for faster recovery and high quality. So that the finger print can be used as a unique identification in many fields to avoid immoral activity. VII. REFERENCES [1] Md. Rafiqul Islam, Farhad Bulbul and Shewli Shamim Shanta, Performance analysis of Coiflettype wavelets for a fingerprint image compression by using wavelet and wavelet packet transform, International Journal of Computer Science & Engineering Survey (IJCSES) Vol.3, No.2, April 2012. [2] http://www.fbi.gov/ [3] M. Sifuzzaman, M.R. Islam and M.Z. Ali, Application of Wavelet Transform and its Advantages Compared to Fourier Transform, Journal of Physical Sciences, Vol. 13, 2009, 121-134 ISSN: 0972-8791. [4] Karen Lees, Image Compression Using Wavelets. May 2002. [5] Nidhi Sethi, Ram Krishna and Prof R.P. Arora, Image Compression Using Haar Wavelet Transform, Computer Engineering and Intelligent Systems ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online). [6] K.P.Soman, K.T.Ramachandran and N.G.Resmi, Insight into wavelets from theory to practice, [7] http://en.wikipedia.org/wiki/ [8] Sarita Kumari, Ritu Vijay, Analysis of Orthogonal and Biorthogonal Wavelet Filters for Image Compression, International Journal of Computer Applications (0975 8887) Volume 21 No.5, May 2011. [9] A.Farras Abdelnour and Ivan W. Selesnick, Symmetric nearly orthogonal and orthogonal nearly symmetric wavelets, The Arabian Journal for Science and Engineering, Volume 29, Number 2C, December 2004. [10] C.Sidney Burrus, Ramesh A. Gopinath, Haitao Guo, Introduction to Wavelets and wavelet Transforms: A Primer. [11] S.V.Narasimhan,Nandini Basumallick ans S.Veena, Introduction to Wavelet Transform A Signal Processing approach, Narosa Publishing House. [12] DongWei, Coiflet-type Wavelets:Theory, Design and Application, August 1998. [13] Rao & Bopardikar, Biorthogonal Wavelets, 2001. [14] Khalid Sayood, Introduction todata Compression, Third edition 2006. [15] James S. Walker, A Primer on Wavelets and Scientific Applications, 1999. [16] Ian Kaplan, Applying the Haar Wavelet Transform to Time Series Information, July 2001 [17] E. Yeung, Image compression using wavelets, Waterloo, Canada N2L3G1, IEEE, CCECE, 1997 70