ITB J. Sci. Vol. 40 A, No., 008, 151-165 151 Tight Wavelet Frame Decomposition an Its Application in Image Processing Mahmu Yunus 1, & Henra Gunawan 1 1 Analysis an Geometry Group, FMIPA ITB, Banung Department of Mathematics, FMIPA ITS, Surabaya Abstract. This paper is evote to the formulation of a ecomposition algorithm using tight wavelet frames, in a multivariate setting. We provie an alternative metho for ecomposing multivariate functions without accomplishing any tensor prouct. Furthermore, we give explicit examples of its application in image processing, particularly in ege etection an image enoising. Base on our numerical experiment, we show that the ege etection an the image enoising methos exploiting tight wavelet frame ecomposition give better results compare with the other methos provie by MATLAB Image Processing Toolbox an classical wavelet methos. Keywors: ecomposition; ege etection; frame; image enoising; multiresolution analysis; wavelet. 1 Introuction In this paper, we provie an alternative metho for ecomposing functions in a multivariate setting. Extension from univariate function space to the multivariate setting can be accomplishe trivially by consiering the space generate by tensor prouct of univariate basis functions. Here, we propose the metho by using tight wavelet frames, instea of accomplishing a tensor prouct. Before going further to the main iscussion, let us introuce some notations use in this paper. For Z be a -copy of the set of integers an a finite subset Ψ of L (R ), let { : Ψ, Z Z } Λ ( Ψ) : = ψ, ψ,, / where for all Z an Z we write ψ, : = ψ ( ). We say that Λ(Ψ) is a frame if there exist two positive numbers A an B such that A f g Λ ( Ψ) f, g B f Receive January 1 st, 008, Revise July 4 th, 008, Accepte for publication July 14 th, 008.
15 Mahmu Yunus & Henra Gunawan for all f L ( R ). If A = B, then Λ (Ψ) is a tight frame [1]. Such frames, which in Hilbert spaces are recognize as a generalization of orthonormal bases, play a funamental role in signal processing, image processing, ata compression an transmission, sampling theory an more, as well as being an interesting subect of research in moern an applie mathematics. Ron an Shen [] introuce the construction of tight wavelet frame from multi resolution analysis (MRA). We now that in applications one uses tight wavelet frames ue to their efficiencies. Tight wavelet frames are efficient tools for separating several high-pass frequency parts from low-pass frequency parts. Also, frame reunancy is nown to be useful for recovering information from the corrupte one. In this paper, we follow the notion of MRA for wavelet in [] which is slightly a moifie version of the classical efinition of MRA. Let φ L ( R ) be a compactly supporte function satisfying the following conitions: iξ (i) There exists a trigonometric polynomial H in e such that ) ) φ ( ξ ) = H ( ξ / ) φ ( ξ / ) (1) If we write H iξ ( ξ ) : = he for a sequence { } Z h in l ( R ), then φ = hφ( ). () Z ) (ii) limφ ( ξ ) = 1. 0 ξ Next, we enote for all V Z / { ( ) Z } : = span φ :. Then V satisfy all conitions in classical efinition of MRA [3] except that an its shifts constitute a Riesz (or orthonormal) basis of V 0. In this efinition, the function φ is familiarly nown as the scaling function for the MRA { V }. Now, let φ L ( R ) be a compactly supporte function which satisfies the refinement equation (1). We call H in (1) a filter function an { h } a filter (sequence) associate with φ. The well nown metho for constructing φ
Tight Wavelet Frame Decomposition an Its Application 153 wavelet frame is the following conition calle Unitary Extension Principle (UEP) evelope by Ron an Shen in []: H ( ξ ) H ( ξ + η) + G 1, ( ξ + η) = 0, η = 0 r ( ξ ) G = 1 if η 0 if { 0, π} \ { } (3) In this case, we have to fin trigonometric functions G generators ψ s efine by s an wavelet frame ) ψ ) ( ξ ) : = G ( ξ / ) φ ( ξ / ), = 1, K, r. (4) As shown in [4], there are sequences{ g } in l, = 1, K, r, an if we write G = ψ = g iξ e Z then we have g φ( ) We call G as the filter function associate with an corresponing filter (sequence). ψ { } (5) g as its Throughout this paper we wor with wavelet frames constructe by using UEP of Ron an Shen s as above. In this case, we implement a fast algorithm of ecomposition an reconstruction of function by using compactly supporte tight wavelet frame relate to an MRA. In particular, we apply the wavelet frames constructe by Lai an Stöcler [4] an Sopina [5] in image processing. The rest of this paper is organize as follows. In Section, we briefly introuce the wavelet frame ecomposition an reconstruction, as well as efine the algorithm. In Section 3, we implement the algorithm evelope in Section an aapt it to ege etection an image enoising processes. In aition, the numerical results are presente in Section 3 an the conclusions are given in Section 4. Wavelet Frame Decomposition an Reconstruction Suppose that φ is a scaling function for MRA { V } with filter sequence an φ has an associate wavelet frame generator = { ψ 1, K, ψ r } sequences { g }, for = 1,K, r. { } h Ψ with filter
154 Mahmu Yunus & Henra Gunawan For all Z an Z, we enote / f, = f ( ) (6) so that we can write, for each Z, V = span { φ, : Z }, (7) Now, by setting W it follows that { : = 1,, r Z } = span ψ, K,, (8) V = V + W, (9) +1 Z (see for example [1, 3, 6, 7] for the etail of this ecomposition). Note that ecomposition (9) is not a irect sum ecomposition since in general V W 0 {}. Recall that, in fact, any family{ f i } i I span f i : i I (see e.g. [7, Theorem 5.1.3]). We then use this fact together with (7), (8), an (9), for any f V to get +1 : is a frame for { } f = r f, φ φ ψ,, + f,, ψ, Z = 1 Z (10) Now, for any f L ( R ) consier : = φ an, : = f,, = 1,K, r. (11) c, f,, For all Z, m Z, an we have φ /, m = h mφ + 1, Z ψ, = 1, K, r, an by two-scale relation () an (5) an ψ /, m g mψ + 1, = (1) By taing the inner proucts with f on both sies of the two equations in (1), we have a tight wavelet frame ecomposition:
Tight Wavelet Frame Decomposition an Its Application 155 c = h /, m m + 1, Z c an = g /, m m + 1, Z c (13) Using the fact that φ V an two-scale relation () an (5), from (10) we also have, r / φ + 1, m= hm φ, + gm ψ, (14) Z = 1 By taing inner proucts with f on both sies of (14), we have a tight wavelet frame reconstruction: c r / + 1, m = hm c, + Z = 1 g m } { m } hm g, (15) Note that all sequences { an are finite for all m, because an ψ, are compactly supporte for all Z, Z. φ, We summarize an illustrate the tight wavelet frame ecomposition an tight wavelet frame reconstruction in the Figure 1. H f +1 H G 1 1 +1 f G G +1 + G 1 ~ f...... G r r + 1 G r ecomposition reconstruction Figure 1 Tight wavelet frame ecomposition an reconstruction. It can be seen from the iagram above that in practical use of ecomposition an reconstruction by using tight wavelet frame are ust similar to the one using
156 Mahmu Yunus & Henra Gunawan orthonormal wavelet basis. In this case, the computational complexity of tight wavelet frame ecomposition is same as the computational complexity of orthonormal wavelet ecomposition (see e.g. [3]), that is of orer O(N) for an input N = M 1 M matrix. 3 Numerical Experiment We may consier the function f (in a iscrete form) escribe in Section, as an image (in a matrix form). The wavelet frame ecomposition separates the low frequency (smooth) parts from the high frequency (etail) parts of images effectively. It is well nown that all the eges an noise are relate to the high frequency parts of images. In this numerical experiment, we implement the tight wavelet frame ecomposition (TWFD) an reconstruction algorithms (13) an (15) by maing use of wavelet frames base on box splines in [4]. This implementation is applie for ege etection an image enoising on six test images (Figure ). Ban F16 Hea MRI Lena Partition Figure Test Images. Saturn
Tight Wavelet Frame Decomposition an Its Application 157 We choose these six ifferent test images from MATLAB Image Processing Toolbox, by virtue of ifferent patterns, namely: linearly eges, common long istance outoor picture, negative film picture, common inoor (face) picture, patterne image, an curvy image. The test image with a gray level intensity can be consiere as a matrix with each component varies from 0 to 55. Here, we consier 0 as blac an 55 as white. Let X : = [ x ] be a sub image of size, where I 1, I / J / an J are usually some power of an Z. We consier X + 0 as the original image an X as the image at the th level ecomposition. We call each value x as a pixel value at each th level. Let H be a matrix whose components are 1, obtaine by the finite sequence { h } 1, satisfying two-scale relation of the compactly supporte scaling function φ ( x, y) : φ( x, y) = 1, h 1, φ(x,y 1 We call the matrix H the low-pass filter which extracts the smooth parts of the image. Meanwhile, we call G the high-pass filter for =1, K, N which contains the etail parts of the image. Each component of the matrix G is from the two-scale relation of the wavelet frame generators ψ such that ψ ( x, y) = 1, g 1, φ(x,y 1 ) ), =1, K, N Then the image ecomposition is the matrix convolution of each H an G with the matrix X accompanying by own sampling, which is a proceure eleting all the even number of rows an columns. Let X +1 be the own sample matrix after we tae convolution of H with X an Y +1 be the own sample matrix after the convolution of G with X. Then, the size of matrices 1 N X +1, Y +1,, Y + 1 is a half of the size of the matrix X for each Z. We call X +1 a low-pass sub-image an the Y +1 s high-pass sub-images. The lowpass sub-image has low-pass (smooth) parts of the image an the high-pass subimages have high frequency (etail) parts of the image.
158 Mahmu Yunus & Henra Gunawan The reconstruction process is the reverse process of ecomposition. We insert zero rows an columns to the even number of rows an columns of 1 N matrices,,, respectively. X +1 Y +1 Y + 1 3.1 Ege Detection Our wavelet frame metho for ege etection can be escribe as in the following consecutive steps. (i) Decomposing the original image f, by using (13), into one low-pass sub image c1 an several high-pass sub images 1, K, r. (ii) Reconstructing the image by using (15), but without the low-pass component: f = 1 L r. As we mentione at the beginning of this section, since the abrupt changes an etail parts of the images are locate in high-pass components, the reconstructe image without low-pass component shows eges with noise. This is the noise containe in the original image. (iii) Removing the noise after we reconstruct the image to have clear eges: f = D(f ), where D is a Donoho s enoising operator (see [8]). (iv) Normalizing the reconstructe image into the stanar grey level from 0 to 55 an use a threshol to ivie the pixel values into two maor groups. That is, if a pixel value is bigger than the threshol, it is set to be 55; otherwise, it is set to be zero. (v) Removing all isolate pixel values. We use the tight wavelet frames base on the box spline φ 11 in [4] for ege etection. The box spline φ 11 has one low-pass filter an 8 high-pass filters. To compare the visual effectiveness of ege etection, we also use wavelet metho ege etection by using tensor proucts of Daubechies wavelet D6 an three commercial ege etection methos from MATLAB Image Processing Toolbox, namely Sobel, Prewitt, an Zero-crossing methos. For box spline tight wavelet frame, we choose the best ege representation among three levels of ecompositions to present here. Tensor proucts of Daubechies wavelet have three high-pass filters each of which etect eges of horizontal, vertical, an iagonal irection respectively. Thus they have alreay ha a isavantage of etecting curvy eges. Among the images, tight wavelet frame etects ege more effectively in all images. Especially, it etects ege better than the
Tight Wavelet Frame Decomposition an Its Application 159 MATLAB commercial ege etection methos in many images. We present the results visually in Figure 3 to Figure 8. Original image Prewitt Sobel Zero Crossing Wavelet D6 TWFD Figure 3 Ege etection for Image Ban using 5 ifferent methos. Original image Prewitt Sobel Zero Crossing Wavelet D6 TWFD Figure 4 Ege etection for Image F16 using 5 ifferent methos.
160 Mahmu Yunus & Henra Gunawan Original image Prewitt Sobel Zero Crossing Wavelet D6 TWFD Figure 5 Ege etection for Image Hea MRI using 5 ifferent methos. Original image Prewitt Sobel Zero Crossing Wavelet D6 TWFD Figure 6 Ege etection for Image Lena using 5 ifferent methos.
Tight Wavelet Frame Decomposition an Its Application 161 Original image Prewitt Sobel Zero Crossing Wavelet D6 TWFD Figure 7 Ege etection for Image Partition using 5 ifferent methos. Original image Prewitt Sobel Zero Crossing Wavelet D6 TWFD Figure 8 Ege etection for Image Saturn using 5 ifferent methos.
16 Mahmu Yunus & Henra Gunawan Accoring to the results presente visually above, we note some superiority of the application of tight wavelet frame ecomposition (TWFD) in ege etection compare with the other four methos. In general, TWFD affors sharper an smoother eges for all cases of the test images. For an image with linearly parts (Figure 3), TWFD results sharp an etaile shape of ege more than the other four methos. Especially for images containing curvy parts, TWFD shows the most reaable shape of ege (Figure 4, 5, 6, an 8). In aition, TWFD etects eges of patterne image clearly (Figure 7) without generating a new pattern within the resulting image. 3. Image Denoising In our numerical experiment on image enoising, we mae a noisy image first by aing a white noise to the pixel values x i, of the image. The white noise has a Gaussian istribution δ i, with mean zero an variance σ. Then the pixel values of noisy image can be presente as f i, f i, = xi, + σδ i, with δ i, N(0,1) where σ is chosen from values 5, 10, 15 an 0. We then ecompose the image into a low-pass part an several high-pass parts of the image by using wavelets or tight wavelet frames. Next, we apply the soft-thresholing to each high-pass part of the image. Finally, we reconstruct the image after we use noise softening. In this step, we choose optimal threshol value by using the metho introuce by Donoho [8]. To measure the quality of the enoise image, we use the pea signal to noise ratio (PSNR) compute by the following formula. For a given image x i, of size N N pixels an a reconstructe image x, where 0 x 55, PSNR = 10log10 N i, = 1 55N ( x i x', i, ) ' i, i, For image enoising, we use tight wavelet frame base on the box spline φ1111 in [4]. The other box spline tight wavelet frames in [4] give similar results. Comparing wavelets for enoising, tight wavelet frames give better results in the set test images we teste.
Tight Wavelet Frame Decomposition an Its Application 163 Since the most of the noise appears in the high-pass parts of the first level of ecomposition, we consier only the first level of ecomposition. In this case, for each given image an fixe noise we are able to fin the optimal threshol values of soft-thresholing (see [8] for the etail) for the sub-images ecompose by Haar, Daubechies, or biorthogonal wavelets an wavelet frame. We present four tables of average value of PSNR from 100 experiments accoring to each σ value. Each entry in the tables contains PSNR values of noisy images an their enosie images reconstructe by tensor proucts of Haar, Daubechies, biorthogonal wavelets an tight wavelet frame. The largest PSNR values in Table 1 to Table 4 (type in shae cell) inicate the best enoise image. We see that wavelet frame has the most of largest PSNR values. Table 1 The PSNR comparison for σ = 5. Image Noisy Haar Daubechies Biorthogonal TWFD Ban 34.7 36.1 36.06 36.06 35.97 F-16 34.16 36.09 36.45 36.49 36.50 Hea MRI 35.9 37.89 38.47 38.78 38.45 Lena 34.16 36.06 36.55 36.63 36.67 Partition 36.11 35.01 37.04 37.77 37.98 Saturn 36.33 40.00 41.40 41.40 43.14 It can be seen from Table 1, that ecomposition by using Haar wavelet gives the best enoise image for image Ban with low noise level. In this case, the image Ban contains linearly parts, an Haar wavelet is nown to be very goo for hanling such in of images. The other methos rener a low-level noisy image an result an over-smooth enoise image; without the exception of TWFD. Table The PSNR comparison for σ = 10. Image Noisy Haar Daubechies Biorthogonal TWFD Ban 8.31 31.16 31.15 31.16 31.3 F-16 8.16 31.0 31.69 31.76 3.05 Hea MRI 9.31 3.91 33.56 33.84 34.17 Lena 8.15 31.43 3.07 3.18 3.78 Partition 30.4 31.90 3.78 3.74 33.04 Saturn 30.33 35.0 35.9 36.01 39.31
164 Mahmu Yunus & Henra Gunawan For higher noise level, TWFD inicates much better enoise image even for image Ban. For curvy image (Saturn), TWFD results a highly significant PSNR improvement. Table 3 The PSNR comparison for σ = 15. Image Noisy Haar Daubechies Biorthogonal TWFD Ban 4.64 8.6 8.3 8.33 8.80 F-16 4.66 8.35 8.89 8.96 9.74 Hea MRI 5.8 30.03 30.56 30.77 31.84 Lena 4.63 8.75 9.33 9.41 30.8 Partition 5.4 7.54 31.1 30.76 31.78 Saturn 6.8 3.03 3.6 3.66 37.16 Table 3 still shows the expecte results. TWFD elivers the best quality of enoise image among the other methos, especially for totally curvy image (Saturn). Table 4 The PSNR comparison for σ = 0. Image Noisy Haar Daubechies Biorthogonal TWFD Ban.38 6.0 6.31 6.31 7.18 F-16.4 6.3 6.87 6.90 8.3 Hea MRI 3.36 7.96 8.35 8.50 30.15 Lena.14 6.80 7.7 7.34 9.55 Partition 4.33 6.8 8.7 8.1 30.00 Saturn 4.35 9.8 30. 30. 35.54 Again from Table 4, we can see that TWFD yiels the best enoise image reconstructe from quite high-level noisy image. Also, the highest PSNR improvement is achieve for the curviest image. 4 Concluing Remar We now that ege etection is only a small part of an image processing, so that we consier only the visual result. We fin that the ege etection metho using wavelet frame ecomposition (an reconstruction) provies the best result. Here, we get the most clear curvy image through the wavelet frame ecomposition as this ecomposition is one irectly --- without oing any tensor prouct of one imensional process. It shoul be notice that Donoho in [8] prove theoretically the superiority of using wavelet ecomposition in image enoising. At that time, wavelet frames have not been well-evelope. However, accoring to the results presente in
Tight Wavelet Frame Decomposition an Its Application 165 Table 1 to Table 4, in general, the image enoising using wavelet frame ecomposition gives a better result compare to that of classical wavelets ecomposition. For further wor, we might apply our fast algorithm to the image processing by using the other wavelet frames. We might also tae into account the algorithm complexity, incluing the time complexity as well as the usage of computer memory. Acnowlegement The authors woul lie to than Dr. Armein Z. R. Langi for his valuable comments an constructive suggestions on the earlier version of this paper. References [1] Christensen, O., Recent Developments in Frame Theory, in Moern Mathematical Moels, Methos an Algorithms for Real Worl Systems, A.H. Siiqi, I.S. Duff an O. Christensen, es., Anamaya Publishers, New Delhi, 159-195, 006. [] Ron, A. & Shen, Z., Affine Systems in L ( R ) : Analysis of the Analysis Operator, J. Funct. Anal., 148, 408-447, 1997. [3] Daubechies, I., Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Appl. Math., SIAM, Philaelphia, 199. [4] Lai, M.J. & Stöcler, J., Construction of Multivariate Compactly Supporte Tight Wavelet Frames, J. Appl. Comput. Harmonic Anal., 1, 34-348, 006. [5] Sopina, M., Multivariate Wavelet Frame, Tech. Report, Dept. of App. Math. an Control Processes, Saint Petersburg State University, 007. [6] Chui, C. K. & He, W., Compactly Supporte Tight Frames Associate with Refineble Functions, J. Appl. Comput. Harmonic Anal., 8, 93-319, 000. [7] Christensen, O., An Introuction to Frames an Riesz Bases, Birhauser, Boston, 003. [8] Donoho, D.L., De-Noising by Soft-Thresholing, IEEE Trans. Info. Theory, 43, 933-936, 1993.