Engineering, 2013, 5, 522-526 ttp://dx.doi.org/10.4236/eng.2013.510b107 Publised Online October 2013 (ttp://www.scirp.org/journal/eng) Two Modifications of Weigt Calculation of te Non-Local Means Denoising Metod Musab Elkeir Sali, Xuming Zang *, Mingyue Ding Scool of Life Science and Tecnology, Huazong University of Science and Tecnology, Wuan, Cina Email: * xmbosi.zang@gmail.com Received 2013 Abstract Te non-local means (NLM) denoising metod replaces eac pixel by te weigted average of pixels wit te surrounding neigboroods. In tis paper we employ a cosine weigting function instead of te original exponential function to improve te efficiency of te NLM denoising metod. Te cosine function outperforms in te ig level noise more tan low level noise. To increase te performance more in te low level noise we calculate te neigborood similarity weigts in a lower-dimensional subspace using singular value decomposition (SVD). Experimental comparisons between te proposed modifications against te original NLM algoritm demonstrate its superior denoising performance in terms of peak signal to noise ratio (PSNR) and istogram, using various test images corrupted by additive wite Gaussian noise (AWGN). Keywords: Non-Local Means; Singular Value Decomposition; Weigt Calculation 1. Introduction Image denoising is one of te most important tecniques and necessary preprocessing steps in image processing and computer vision. Denoising is to remove unwanted noise to enance and restore te original image. Many metods for image denoising ave been suggested, and a review of tem can be found in [1]. Among tem tat paper also proposes te NLM denoising metod, wic gives state-of-te-art results. Tis metod replaces eac pixel wit a weigted average of oter pixels wit similar neigboroods. Te basic idea is tat images contain repeated structures, and averaging tem will reduce te noise [2]. Tis new concept for image denoising is popular in oter image processing areas, suc as texture syntesis, were a new pixel is syntesized as te weigted average of known image pixels wit similar neigboroods [3-5]. Te issue of measuring similarity between pixels, or patces is a ot topic in many applications of image processing, macine learning, computer vision, information retrieval and document clustering. Document clustering is te measuring of similarity between documents. In tis te document is mapped into te vector space, and similarity is measured by te angles between te vectors using cosine function. Te smaller te angle between two documents, te larger te cosine of te angle is and te iger te similarity is; and vice versa [6]. So te cosine is a decaying function of te angle between two docu- * Corresponding autor. ments. Te original NLM filter uses te exponential decaying function as a weigting function for te similarity measurement and weigt calculation. In compare to cosine function te exponential function decays faster tan cosine function and tis will reduce te effect of averaging of te NLM filter. Because of tis, we proposed a NLM denoising metod using te cosine function. Te outperformance of cosine function mainly appears in ig level noise. To increase te performance more in te low level noise we combine te cosine weigting function and SVD. Te SVD belongs to a class of dimensionality reduction tecniques tat deal wit te uncovering of latent data structures and often associate noise to te least important components. Dimensionality reduction is a noise reduction process. Removing noise before calculate te similarities between patces around te pixels will enance te weigt calculation and increase te performance of NLM algoritm. Te rest of tis paper deals about te following. Section II introduces te classical NLM algoritm. In section III, te proposed metod followed to modify te classical NLM algoritm is introduced. In tis, we use a cosine weigting function for te weigt calculation, ten combine te cosine function wit SVD in te low level noise. Te results & discussions for four test images at various noise levels are sown in Section IV. Section V concludes te paper and proposed future work for more improvement to te NLM denoising metod.
M. E. SALIH ET AL. 523 2. Non-Local Means Filter Te NLM metod [1,7] estimates te intensity of eac pixel x in te noisy image u by a weigted average of all of te pixel intensities in te image (as a convention, we will refer to te pixel being denoised at any given time as te pixel of interest (POI), denoted x, and denote all oter pixels as y). Te weigts w(x, y) reflect te probability tat te POI (x) as te same intensity value as te pixel is being compared to (y). Tis probability is based on te similarity between te neigboroods around x and y. Te weigting function can be considered as a decreasing function depending on te similarity of te patces around te POI. If a particular local difference as a large magnitude ten te value of w(x, y) will be small and terefore tat measurement will ave little effect on te output image. A small neigborood, or patc, around eac pixel is used to compute te L2 norm. Te weigting factor w(x, y), is ten a normalized weigted Gaussian function of tis L2 norm. Consider a discrete noisy image u = f + n, in wic n is AWGN. Te NLM filter is written as f ( x,y) = w( x,y) u( y) y little bit dissimilar patces instead of zero or small values. Because of tis, our proposed metod was based on using te cosine function instead of te exponential one. For more investigation of different decay beaviors of te two decreasing functions, we calculated te weigt corresponding to te dissimilarity measurement of all 7 7 patces in 13 13 searc window around te pixel (20, 20) in te left top smoot sky background region of te ouse image corrupted by wite Gaussian noise, wit standard deviation (σ) = 40, sown in Figures 1(b). It appears tat cosine function as given iger suitable weigt values wile te exponential function as given small weigt values. Tis is te major advantage of using cosine function and results sown in Figures 1(c), (d) proved tis. Tis advantage as an important effect in te weigt calculation of te NLM denoising algoritm, so te cosine function was more robust weigting function, and we used it in NLM algoritm denoising for an efficient implementation of tis algoritm. 3.2. Weigt Calculation in te Reduced Dimensional Space Using Singular Value Decomposition In 1965 G. Golub and W. Kaan introduced te SVD as a were, decomposition tecnique for calculating te singular val- 1 1 ues, pseudo-inverse and rank of a matrix [9]. Te tec- ( ) ( ( )) ( ( )) 2 d d w x, y = exp u N x u N y 2 W ( x ) nique decomposes a matrix A into tree new matrices 2 T A = USV were N d (y) represents te square patc of size (2d + 1) were: (2d + 1) centered at x, and W is a normalizing term, W(x) = wxyu U is a matrix wose columns are te eigenvectors of (, ). Te parameter will be referred v te AA T matrix. Tese are termed te left eigenvectors. to as te filter parameter tat controls te decay of te S is a matrix wose diagonal elements are te singular exponential expression in te weigting sceme. Tis values of A. Tese are ordered in decreasing order along parameter is typically controlled manually in te algoritm. Coosing a very small leads to noisy results identical to te input, wile very large gives an overlysmooted image [8]. 3. Metodology 3.1. Weigt Calculation Using Cosine Function Te L2 norm of te difference between two pixels was used as input to te cosine function instead of te exponential function. Figure 1(a) demonstrates te beavior of te two functions to an arbitrary input. Te exponential function was decayed faster tan cosine function, wic lead to finising te weigt to zero value instantly for dissimilar patces, or to a very small weigt values even for te little bit dissimilar patces. Tis fast decay reduced te effect of averaging and smooting of te NLM filter. On te oter side, te cosine function s decay was slower tan exponential function, and so it gave a considerable weigt values for te dissimilar or te Figure 1. (a) Exponential and cosine weigting function; (b) Dissimilarity measurement of te pixel (20, 20) of ouse image; (c) and (d) Weigt calculation using exponential function and cosine function corresponding to similarity measurement respectively.
524 M. E. SALIH ET AL. te diagonal of S, i.e. s 1 > s 2 > s 3 > s n. V is a matrix wose columns are te eigenvectors of te A T A matrix. Tese are termed te rigt eigenvectors. Wen computing te SVD of a matrix is desirable to reduce its dimensions by keeping its first k singular values. T A = U SV K K K K Tis process is termed dimensionality reduction, and A K is referred to as te rank k Approximation of A, or te Reduced SVD of A. If we eliminate dimensions by keeping te tree largest singular values, tis is a rank 3 approximation [10]. Te top k singular values are selected as a mean for developing a latent semantics representation of A tat is now free from noisy dimensions. Tis latent semantics representation is a specific data structure in low-dimensional space in wic documents, terms and queries are embedded and compared. Tis idden or latent data structure is masked by noisy dimensions and becomes evident after te SVD. We replaced te distances by 1 d d u 2 ( N ( x) ) u N ( y) (3) ( ) 2 1 d d u 2 ( P ( x) ) u P ( y) (4) ( ) 2 were P d represents te projections of N d onto te lower-dimensional space determined by te SVD. If Nd is a particular row in A, ten te weigts for N d are just te corresponding row in U multiplied by diagonal elements of S. Te weigts for N d are referred to as projection of N d into te k-dimensional space. Better denoising is obtained wen similarity between pixels is computed using te dimension reduction introduced by te SVD. 4. Experiment Results and Discussions In te experiment, te size of patc, searc window, parameter values and rank k Approximation were selected corresponding to te best PSNR value. Our proposed NLM algoritm was applied on four test images (Lena, Cameraman, Pepper, and House); Using a 256 256 image size for all te images, and a 128 128 image size for Lena image. Te test images were corrupted by AWGN wit zero mean at σ = 10 (low noise level), and 40 (ig noise level). Te results were sown using PSNR in decibels (db) and istogram to demonstrate te superior performance of te proposed metod in noise reduction. We used te PSNR measurement defined as 2 PSNR 10log MAX = MSE were, MSE is Mean Square Error, and MAX is maxi- mum intensity value. Table 1 lists Te PSNR comparison results. Using cosine function produces better PSNR values wen compared to te original NLM algoritms for all te different noise levels. Te average performance is ig (0.7054 db) for te all images corrupted by a ig level noise, wile it is low (0.0440 db) for te all images corrupted by low level noise. Tat means te cosine function outperforms in te ig level noise and tis because of te dissimilarity between patces will increase more tan in te low level of noise te and so te exponential function will give zero or small weigt values wile cosine weigt function will give tem more and suitable weigt values. Tis is useful, and it enances te effect of averaging and smootening of te NLM filter. Te results sow tat te most cosine function superiority, were te performance is (1.12 db), mainly appears in te noisy (AWGN) ouse image, (σ = 40). To increase te average performance for te images corrupted by low level noise, wic is low as we mentioned previously, we combined te SVD wit cosine function to calculate te distances between te pixels in te low dimensional space rater tan te full space. Te results sowed an increased accuracy over using te full space. Te average performance increased form (0.0440 db to 0.2505 db). Tis is because te idden or latent data structure is masked by noisy dimensions and becomes evident after te SVD. And similarity computed in lower-dimensional space becomes more accurate because SVD remove noisy dimensions. Altoug we only applied te SVD on images corrupted by low level noise to increase te performance of te algoritm; but also it will work for images corrupted by moderate level noise. Because te noise will increase more tan in te low level of noise but SVD will remove it and te weigt calculation will be more efficiency. Our metod works well for all te images, but it works better for images (House and Pepper) wic ave a smoot region more tan images (Lena and Cameraman) wic ave more details. Table 1. Performance of te original nonlocal means denoising algoritm and te modified algoritm. PSNR Images 10 40 Exp Cos Cos + SVD Exp Cos Lena 32.0748 32.0174 32.1233 23.5971 24.1023 Cameraman 33.0014 33.0629 33.3857 25.3136 25.8697 Pepper 33.0034 34.0264 34.1926 25.7601 26.4025 House 35.1249 35.2667 35.4981 27.5251 28.6433 Average 33.5494 33.5934 33.7999 25.5490 26.2544 Performance 0.0440 0.2505 0.7054
M. E. SALIH ET AL. 525 (a) (b) (c) (d) (e) (f) (g) () Figure 2. Histograms of te: (a) Original image (b). Noisy image. (c) and (d) Denoised image by nonlocal means denoising algoritm using exponential and cosine function respectively. Figure 2 sows, tat using exponential or cosine functions wit te NLM filter retrieves a similar istogram and distinguises tree main peaks of te istogram of te original image. However, te cosine function increases te sarpness of te peaks and te contrast between tem, because tere is a small peak appears in te istogram of te filtered image using te cosine function and do not appear wen using exponential function. 5. Conclusion and Future Work Tis work sows tat using cosine weigting function and computing te similarity in lower-dimensional space increases te effect of averaging and denoising performance of te NLM denoising algoritm, in low and ig level noise especially for smoot images. Te next work is seeking for anoter and most appropriate weigting function tat will enance tis filter even for images containing more details. Furtermore, it will be very suitable if we use te weigting function locally instead of globally to suit te structure of te image. By selecting a function preserves te edges for image wit ig details, and for a smoot region coosing a function preserves te smootness. Also a future work is to find anoter robust dimensionality reduction metod to remove te noisy dimensions instead of SVD. Tis will be a good direction to improve te efficiency of NLM algoritm. References [1] A. Buades, B. Coll and J.-M. Morel, A Review of Image Denoising Algoritms, wit a New One, SIAM Journal on Multiscale Modeling and Simulation, Vol. 4, No. 2, 2005, pp. 490-530. ttp://dx.doi.org/10.1137/040616024 [2] W. Jin, et al., Fast Non-Local Algoritm for Image Denoising, 2006 IEEE International Conference on Image Processing, 2006, pp. 1429-1432. [3] A. A. Efros and T. K. Leung, Texture Syntesis by Nonparametric Sampling, Proceedings of te IEEE International Conference on Computer Vision, Corfu, Greece, September 1999, pp. 1033-1038. ttp://dx.doi.org/10.1109/iccv.1999.790383 [4] Y. Wexler, E. Sectman and M. Irani, Space-Time Video Completion, Proceedings of te IEEE International Conference on Computer Vision Pattern Recognition (CVPR), 2004. [5] L. Yatziv, G. Sapiro and M. Levoy, Ligt Field Completion, Proceedings of te IEEE International Conference on Image Processing, Singapore, 2004. [6] J. Zeng, et al., A Matcing Metod Based on SVD for Image Retrieval, 2009, pp. 396-398. [7] A. Buades, B. Coll and J.-M. Morel, Nonlocal Image
526 M. E. SALIH ET AL. and Movie Denoising, International Journal of Computer Vision, 2007, to Appear. [8] J. Orcard, et al., Efficient Nonlocal-Means Denoising Using te SVD, ICIP 2008, 15t IEEE International Conference on Image Processing, 2008, pp. 1732-1735. [9] G. Golub and W. Kaan, Calculating te Singular Val- ues and Pseudo-Inverse of a Matrix, SIAM Journal on Numerical Analysis, Vol. 2, No. 2, 1965. [10] E. Garcia, SVD and LSI Tutorial 3: Computing te Full SVD of a Matrix, 2006. ttp://www.miislita.com/information-retrieval-tutorial/sv d-lsi-tutorial-3-full-svd.tml