Smoothing Low SNR Molecular Images Via Anisotropic Median-Diffusion

Transcription

1 Smoothing Low SNR Moleclar Images Via Anisotropic Median-Diffsion Jian Ling 1, Member, IEEE, and Alan C. Bovik 2, Fellow, IEEE 1 Sothwest Research Institte Bioengineering Department 6220 Clebra Road San Antonio, TX , USA jling@swri.ed 2 Laboratory for Image and Video Engineering (LIVE) Department of Electrical and Compter Engineering The University of Texas at Astin, Astin, TX , USA bovik@ece.texas.ed

2 2 Abstract We propose a new anisotropic diffsion filter for denoising low signal-to-noise (SNR) moleclar images. This filter, which incorporates a median filter into the diffsion steps, is called an anisotropic median-diffsion filter. This hybrid filter achieved mch better noise sppression with minimm edge blrring compared to the original anisotropic diffsion filter when it was tested on an image created based on a moleclar image model. The niversal qality index, proposed in this paper to measre the effectiveness of denoising, sggests that the anisotropic median-diffsion filter can retain adherence to the original image intensities and contrasts better than other filters. In addition, the performance of the filter is less sensitive to the selection of the image gradient threshold dring diffsion, ths making atomatic image denoising easier than with the original anisotropic diffsion filter. The anisotropic median-diffsion filter also achieved good denoising reslts on a piecewise-smooth natral image and real Raman moleclar images. Keywords moleclar imaging, image enhancement, nonlinear filter, anisotropic diffsion I. INTRODUCTION Moleclar imaging techniqes sch as florescent imaging, isotope radiation imaging, Raman imaging, and positron emission imaging are becoming increasingly important for visalizing and analyzing biophenomena. Moleclar imaging makes it possible to visalize the distribtions of interesting molecles or chemicals, which are transparent sing other modalities. It also becomes possible to measre qantitative information, sch as the moleclar concentration. However, becase recorded moleclar images often sffer from a low SNR, image denoising is an important prelde to visal interpretation or atomated analysis. It is highly desirable that the filtering process sppresses the noise, while simltaneosly retaining adherence to the original image intensities and contrasts, and in particlar, preserving information-bearing strctres sch as edges. Therefore, qality control in the denoising process is necessary. The anisotropic diffsion filter, first proposed by Perona and Malik [1], is a nonlinear filter which prports to smooth a noisy image withot blrring the edges. The diffsion eqation for image is given by = div [ c( ) ], (1) t

3 3 where is a local image gradient and c( ) is the diffsion coefficient, which is a fnction of the local gradient. The anisotropic diffsion filter has been broadly sed for edge detection [2-4], image segmentation [5, 6], and noise smoothing [7-15], often yielding better reslts than traditional filters sch as the moving average or median filters or other edge-preserving filters [16]. The idea behind the anisotropic diffsion filter is to evolve from a noisy image g(x,y) a family of increasingly smooth images (x,y,t), indexed by diffsion parameter t, to estimate the original image. The diffsion coefficient c( ) in eqation (1) is sometimes called the edge-stopping fnction, which largely dictates the behavior of the filter. The diffsion coefficient is set sch that the filter in eqation (1) diffses the image more in smooth areas and less near the edges. The diffsion coefficient has an argment, the local image gradient, which measres the local edge strength. However, the response of the gradient to the noise element may compete with or exceed the edge response, in which case the diffsion fnction cannot distingish between image strctre and noise contribtion. Therefore, the basic anisotropic diffsion filter in eqation (1) sally fails to deliver adeqate reslts in low SNR images; the reslt either fails to eliminate noise, or leaves the edges significantly blrred [13, 17]. In this paper, we propose an anisotropic median-diffsion filter which is specifically intended for denoising low SNR images, and which is particlarly sitable for piecewise smooth images sch as those encontered in moleclar imaging. In Section II, a moleclar image model is described. Based on this model, a cell phantom is created as the test image for the modified anisotropic diffsion filter. Section III proposes the se of the niversal image qality index [18] to measre the effectiveness of image denoising. The anisotropic median-diffsion filter is presented in Section IV. Comparisons of this modified filter with the traditional anisotropic diffsion filter are given in Section V. Finally, the conclsion and remarks are inclded in Section VI. II. MOLECULAR IMAGE MODEL A moleclar image illstrates the distribtion of a certain molecle. Since the molecles are sally limited within certain areas, a moleclar image can often be divided into several regions. A region with a high moleclar concentration appears bright while regions with low moleclar concentration appear dark. The intensity within a region sally changes gradally while the average intensities between the different regions are qite dissimilar. For sch an image, we can se a piecewise-smooth model to describe it, which we will write as

4 4 n f ( x, y) = si ( x, y), i (2) where the s i (x,y) indicates a smooth region of f(x,y). We create a 128x128-pixel phantom according to the model. This phantom, illstrated in figre 1, represents a moleclar image of a cell. In the phantom, the image is divided into five regions: the backgrond, the cytoplasm, the ncles, the mitochondria, and the endoplasmic reticlm (ER). In the phantom we assme the interesting molecles are distribted in the cell ncles, mitochondria, and ER regions, bt not in the cytoplasm region. The concentration of the molecles in the ncles region is Gassian distribted. The concentration of the molecles in the ER region is linearly distribted, with the highest concentration at the right end. The concentration of the molecles in the mitochondria region is niform. The entire image has a Gassian distribted backgrond illmination. A (noiseless) piecewise-smooth image is sally insensitive to a median filter. This is becase a median filter eliminates primarily sdden, transient spikes, while leaving sdden, sstained edges ndistrbed. Moreover, the median filter does not significantly pertrb the intensities in smooth regions. Figre 2 shows the reslt of the piecewise image shown in figre 1 following filtering by a 3x3 median filter. Except at the corners of the ER and mitochondria regions, no featres in this piecewise-smooth image were corrpted by the median filter. The qality index Q (discssed in Section III) of the filtered image relative to the original image is (on a range from -1 to 1, where 1 is the best possible). Moreover, the filtered image does not degrade mch if the median filter is applied repetitively. III. INDEX FOR MEASURING THE QUALITY OF DENOISE FILTER A simple noise model is sed in this paper. A recorded moleclar image g(x,y) is assmed to be corrpted by a zero mean Gassian white noise n(x,y): ( x, y) f ( x, y) + n( x y) g =,. (3) The goal of a denoising filter is to estimate an image (x) from the y(x) sch that the (x) is as close to the original image f(x) as possible. Many indices have been proposed to measre the efficiency of a denoising filter. The most poplar are the mean sqare error (MSE), the mean absolte error (MAE), the signal-to-noise ratio (SNR), and the peak signal-to-noise ratio. Recently, Wang and Bovik [18] proposed a niversal image qality index (Q) that

5 5 demonstrates a mch better correlation with sbjective qality as measred on hman sbjects. The index Q between the estimated image and the original image f is qite simply defined as: Q = Q Q Q, and (4) σ f 2µ f µ 2σ f σ Q1 =, Q2 =, Q =, σ σ f ( µ ) + ( µ ) σ + σ f f (5) where µ f and µ are the local image means in f and and where and σ f 2, σ f 2, and σ f are local variances and covariances of f and, respectively. The first item Q 1 measres the degree of local linear correlation between f and ; the second item Q 2 measres the similarity of the mean lminance between f and ; and the third item Q 3 measres the resemblance of the local contrast between the two images. The dynamic range of Q is between [-1, 1]. The best vale of 1 is achieved only when = f locally. The three components of Q provide a complete profile of the local qality of a denoising filter. In practice, Qs are first estimated in local regions sing a sliding window and then combine together to create an overall global image index. Therefore, Q is very sensitive to local, transient image degradations, as well as to global degradations. IV. THE ANISOTROPIC MEDIAN-DIFFUSION FILTER A discrete form of the anisotropic diffsion filter described in eqation (1) was proposed by Perona and Malik [1] as follows: [ c ] n N N + cs S + ce E + cw W i j n 1 n i, j = i, j + 4, + λ, (6) where λ [0,1] controls the rate of diffsion. Usally a small vale λ is sed to avoid destabilizing the diffsion process. The letters N, S, E, W (north, soth, east and west) describe the direction of the local gradient. The local gradient is calclated sing nearest-neighbor differences: N S E W i, j i, j i, j i, j = = = = i 1, j i + 1, j i, j + 1 i, j 1 i, j i, j i, j i, j (7) Research in the anisotropic diffsion filter has been oriented toward nderstanding the characteristics of the diffsion coefficient c( ) and then improving the performance of the diffsion filter [19-22]. In this paper, we se

6 6 the Tkey biweight norm as the diffsion coefficient proposed by Black et al. [22]. The normalized (magnitde) Tkey biweight diffsion coefficient is defined as: c 25 ( ) 1 ( 5K), K = 16K K otherwise (8) where K is a constant that is tned for a particlar application. This constant is the threshold of the local gradients, and determines if a local edge is detected. The flx fnction φ( ) is defined as: (, K) = c(, K ) φ. (9) The diffsion coefficient and the flx fnction are plotted in figre 3. This figre together with eqations (6) and (7) sggests that when the local gradient between a crrent pixel and its neighborhood pixel is smaller than the threshold K, the neighborhood pixel is classified as belonging to the same smooth region as the crrent pixel. Therefore, the neighborhood pixel is actively involved in the smoothing of the crrent pixel. However, when the local gradient is greater than K bt less than 5 K, the neighborhood pixel is considered to be more distant from the pixel being analyzed in the crrent smooth region, therefore, sch a neighborhood pixel will have less contribtion in the smoothing of the crrent pixel. When the local gradient is greater than 5 K, the neighborhood pixel is determined to belong to another region, possibly on the other side of an edge. Ths the involvement of this neighborhood pixel in the smoothing is eliminated to avoid edge blrring. Determining a good vale of K for a noisy image is critical for the performance of these filters. Perona and Malik sggested sing Canny's "noise estimator" [23] to determine K. Torkamani-Azar and Tait [13] sed the mean of the absolte gradient as K. Black et al. [22] determined K from the median absolte deviation. All of these methods intend to separate the gradient generated by the noise from the gradient generated by the edges. However, in low SNR images, the average gradient generated by the noise is comparable to or even larger than the edge gradients. Under sch a condition, determining a proper K to smooth the noise while retaining the edges is qite difficlt. Taking K to be too small will reslt in a filter that fails to satisfactorily eliminate the overall noise element, especially large noise spikes (see figres 5(b) and 9(b)). Taking K to be large can lead to edge blrring while still failing to redce large noise spikes (see figres 5(c), 5(d), 9(c) and 10(c)). We propose to se a median filter in combination with a small K in the diffsion process. In other words, after diffsion via eqation (6), a median operation (defined as follows) is sed to remove large noise spikes:

7 7 n n+ i, j = Median( i, j, W ) (10) where W is the window for the median operator (sch as a 3x3 sqare). This modified anisotropic diffsion filter we call the anisotropic median-diffsion filter. With each diffsion step, areas having small gradient are smoothed, while areas with large gradient (from edges or noise spikes) relative to srronding areas are left nchanged. When the large gradients are generated by large noise spikes, these spikes will be removed effectively by the sbseqent median filter. However, if the large gradients are generated by edges, the median filter will not affect them. With each iteration step, low-level noise is smoothed by diffsion, while implsive noise is removed by the median filter. This process is demonstrated in figre 4. A constant image with nit variance Gassian noise was created to illstrate the change of the image histogram (or eqivalent to noise histogram in this case) with the diffsion steps. The noise histogram (dotted line) after an anisotropic diffsion step indicates that the low-level noise increases and the median-level noise decreases, while the high-level noise remains the same. While the diffsion process smoothes the relative low-level noise, at the same time it eqivalently makes the relative high-level noise frther stand ot from the srrondings and ths easily removed by the following median filter. The noise histogram (dashed line) after the sbseqent median filter shows that all the relative high-level noise (than srrondings) was greatly eliminated. In a word, the anisotropic diffsion and the median filter work in a complementary way to gradally eliminate the overall noise element withot blrring the edges (refer to figres 5(f), 9(d) and 10(d)). The open-close and close-open filters were sggested by Acton [17] to be sed in a way similar to the median filter proposed here. These two morphological filters have the similar fnction of removing noise spikes. However, they do not perform as well as the median filter in this application (refer to figre 5(e)). The threshold K in the anisotropic median-diffsion filter is determined with the histogram of the gradient of a noisy image as the reference. However, the anisotropic median-diffsion filter is not sensitive to the exact vale of K. As long as K is mch less than the standard deviation of the gradient (hopeflly less than the edge gradient), while not too small to stop the smoothing process, the denoising reslts will be similar. V. EXPERIMENTAL RESULTS AND COMPARISONS Experiments were carried ot on the phantom images with different SNRs. The intensity of the phantom image was normalized into [0,1] for easy comparison. The SNR formla sed is:

8 8 σ SNR = 20log 10 σ [ f ( x, y) ] [ n( x, y) ] (11) where σ[f(x,y)] and σ[f(x,y)] indicate the standard deviations of the original image f(x,y) and noise n(x,y), respectively. Figre 5 compares the denoising capability with different anisotropic diffsion filters. The degraded image is shown in figre 5(a) with the SNR of 5 db. Figre 6 is the histogram of the gradient throghot the image in 5(a). As expected, there is no way to separate the gradient of the edges from the gradient of the noise. The standard deviation of the gradient is Figres 5(b), 5(c), and 5(d) illstrate images filtered sing the traditional anisotropic diffsion filter with different K vales. In figre 5(b) K is set to 0.09, which is approximately half of the standard deviation of the gradient. This filtered image shows both implsive noise spikes and large noise spots, which are de to improper smoothing of implsive noise. In figre 5(c) K is set to 0.19, which is the standard deviation of the gradient. The filtered image removed most of the implsive noise spikes bt not the large noise spots. In addition, edge blrring becomes evident. When K = 0.38, which is twice the standard deviation of the gradient, the edge blrring becomes significant. The big noise spots still remain in the filtered reslt. Clearly, traditional anisotropic diffsion filtering does not deliver adeqate reslts in low-snr images sch as these. A pair of open-close and close-open filters (alternatively sed in the diffsion) was sed to smooth the image in figre 5(a) as proposed in [17]. The filtered image in Figre 5(e) shows an obvios illmination distortion relative to the original image. In addition, several noise spots still remain on the filtered image. The trend of qality indexes Q, Q 1, Q 2, and Q 3 verss iterations is illstrated in figre 7. The plot of Q 2 obviosly illstrates that the mean illmination became worse after sing the open-close/close-open diffsion filter. Figre 5(f) illstrates the reslts obtained sing the anisotropic median-diffsion filter, where K = 0.03, which is mch smaller than the standard deviation of the gradient. This image yields an excellent estimate of figre 1. The qality indexes achieved by the different filters are compared in Table 1. Figre 8 illstrates the trend of qality indexes Q, Q 1, Q 2, and Q 3 verss iteration nmber. In contrast to the qality indexes in the open-close/closeopen diffsion filter, figre 8 shows that all three qality measrements: correlation, mean lminance and contrast are gradally improved with the progress of iterations. The anisotropic median-diffsion filter was tested again on the phantom degraded to SNR = 0 db. Becase of the large noise variation in this image, the standard deviation of the gradient is 0.34, larger than that of the 5 db

9 9 SNR image. However, the same K vale (0.03) was sed to smooth the image when sing the anisotropic mediandiffsion filter. The reslts shown in figre 9 illstrate improved performance of the anisotropic median-diffsion filter relative to the traditional algorithm. This example also sggests that selection of the threshold K is fairly robst, with the filter performance not being very sensitive to changes in K. The robstness of the K selection will make it easy for atomatic image denoising. The anisotropic median-diffsion filter was also tested on a piecewise-smooth natral image as shown in figre 10(a). This egg image was degraded to the SNR of 0 db by adding zero mean Gassian noise as illstrated in figre 10(b). The reslt of applying a traditional anisotropic diffsion filter (K = 0.2) is shown in figre 10(c). It also displays noise spots as in the phantom case. The filtered image sing the new filter is mch better at noise sppression, as illstrated in figre 10(d). Table 1. Smmary of the Qality Index Q of the filtered image after sing different filters Noisy Images Filters K Iterations Noisy Image Q Index* Filtered Image Q Index* Traditional anisotropic diffsion Phantom Image SNR = 5 db Traditional anisotropic diffsion Traditional anisotropic diffsion Anisotropic open-close-diffsion Anisotropic median-diffsion Phantom Image SNR = 0 db Egg Image Traditional anisotropic diffsion Traditional anisotropic diffsion Anisotropic open-close-diffsion Anisotropic median-diffsion Traditional anisotropic diffsion SNR = 0 db Anisotropic median-diffsion * The index Q was calclated relative to the original images.

10 10 Finally, we applied the anisotropic median-diffsion filter on Raman images depicting the drg distribtion in single tmor cells [24-25]. Drg distribtion in a cell can help to characterize the mechanisms of a drg as well as to evalate the drg efficacy. Figre 11(a) shows a breast cancer cell that has been exposed to Taxol for one hor (Taxol is an anticancer drg that is often sed to treat breast cancer). No drg information is shown in this conventional microscopic image. A Raman image was taken to show the Taxol distribtion within the boxed area of the cell. Figre 11(b) illstrates the recorded Raman image, which is very noisy. The traditional anisotropic diffsion filter was first applied to smooth the image with K = 0.1. The reslt is shown in figre 11(c), which still contains implsive noise spikes. Moreover, the edges were blrred somewhat compared to the filtered image in figre 11(d). However, the smoothed image sing the proposed anisotropic median-diffsion filter (K = 0.02) in figre 11(d) shows better noise sppression and edge retention. Figre 12(a) shows another breast cancer cell that has been exposed to Taxol for 1.75 hors. In this experiment, the Raman instrmentation was improved so that the field of view of Raman imaging was increased to cover the whole cell. Figre 12(b) is the corresponding Raman image of the cell to show the Taxol distribtion. The traditional anisotropic diffsion filter was first applied to smooth (50 iterations) the image with K = the standard deviation of the image gradient. The reslt is shown in figre 12(c), which still contains implsive noise spikes. The traditional anisotropic diffsion filter was then applied to smooth (still 50 iterations) the image with K = three times of the standard deviation of the image gradient. The reslt is shown in figre 12(d), which does not contain implsive noise spikes anymore. However, large noise spots remains, which was also seen in the experiment of synthetic images when K was increased. The reslt of sing the proposed anisotropic median-diffsion filter is shown in Figre 12(e) with K = the standard deviation of the image gradient. By only sing only five iterations, the smoothing reslt appears (althogh no objective criteria is available) better than the reslts in Figre 12(c) and (d). The smoothed Raman image in Figre 12(e) was frther processed by correcting the non-niform illmination, sbtracting the backgrond, deconvolting the 3-D blrring, and eliminating the florescence signal gives the Raman image in Figre 12(f) [25], it shows the Taxol distribtion within the cell. This experiment sggests that anisotropic median-diffsion is more efficient than the traditional anisotropic diffsion filter. This advantage cold allow the application of diffsion filter extend to video processing.

11 11 VI. CONCLUSION In this paper we have proposed a new anisotropic diffsion filter for denoising low SNR images, which incorporates a median filter into the diffsion step. This hybrid filter apparently achieves mch better noise sppression with minimm edge blrring compared to the original anisotropic diffsion filter. A niversal qality index sggests that this filter retains adherence to the original image intensities and contrasts better than other filters. In addition, this filter is less sensitive to the selection of the image gradient threshold dring diffsion, ths making atomatic image denoising easier than with the original anisotropic diffsion filter. This filter is also more efficient than the original anisotropic diffsion filter, ths making the application of diffsion filter in video processing feasible. This new filter is particlarly sefl for low-snr moleclar images sch as florescence, Raman, isotope radiation and positron emission images. References [1] P. Perona and J. Malik, "Scale-space and edge detection sing anisotropic diffsion," IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 12, pp , [2] S. T. Acton, "Edge enhancement of infrared imagery by way of the anisotropic diffsion pyramid," Proceedings. International Conference on Image Processing (Cat. No.96CH35919). IEEE. Part vol.1, 1996, pp vol. [3] G. Gallo, A. Zingale, and R. Zingale, "Detection of MRI brain contor sing nonlinear anisotropic diffsion filter," Proceedings of the 18th Annal International Conference of the IEEE Engineering in Medicine and Biology Society. `Bridging Disciplines for Biomedicine' (Cat. No.96CH36036). IEEE. Part vol.3, 1997, pp vol. [4] M. Garcia-Silvente, J. A. Garcia, J. Fdez-Valdivia, and A. Garrido, "A new edge detector integrating scalespectrm information," Image and Vision Compting, vol. 15, pp , [5] S. T. Acton, A. C. Bovik, and M. M. Crawford, "Anisotropic diffsion pyramids for image segmentation," Proceedings ICIP-94 (Cat. No.94CH35708). IEEE Compt. Soc. Press. Part vol.3, 1994, pp vol. [6] J. Maeda, T. Iizawa, T. Ishizaka, C. Ishikawa, and Y. Szki, "Segmentation of natral images sing anisotropic diffsion and linking of bondary edges," Pattern Recognition, vol. 31, pp , 1998.

12 12 [7] Y. L. Yo and M. Kaveh, "Forth-order partial differential eqations for noise removal," IEEE Transactions of Image Processing, vol. 9, pp , [8] L. Zhochen and S. Qingyn, "An anisotropic diffsion PDE for noise redction and thin edge preservation," Proceedings 10th International Conference on Image Analysis and Processing. IEEE Compt. Soc. 1999, pp [9] I. Bajla and M. Sramek, "Improvement of 3D visalization of the brain sing anisotropic diffsion smoothing of MR data," International Jornal of Healthcare Technology & Management, vol. 1, pp , [10] J. Monteil and A. Beghdadi, "A new adaptive nonlinear anisotropic diffsion for noise smoothing," Proceedings 1998 International Conference on Image Processing. ICIP98 (Cat. No.98CB36269). IEEE Compt. Soc. Part vol.3, 1998, pp vol. [11] Y. M. Chen, B. C. Vemri, and L. Wang, "Image denoising and segmentation via nonlinear diffsion," Compters & Mathematics with Applications, vol. 39, pp , [12] S. H. Lee, M. G. Kang, and K. T. Park, "CCD noise filtering based on 3-dimensional nonlinear partial differential eqation," IEEE Transactions on Consmer Electronics, vol. 44, pp , [13] F. Torkamani-Azar and K. E. Tait, "Image recovery sing the anisotropic diffsion eqation," IEEE Transactions of Image Processing, vol. 5, pp , [14] G. Gerig, O. Kbler, R. Kikinis, and F. A. Jolesz, "Nonlinear Anisotropic Filtering of MRI Data," IEEE Transactions on Medical Imaging, vol. 11, pp , [15] P. J. Verveer, M. J. Gemkow, and T. M. Jovin, "A comparison of image restoration approaches applied to three- dimensional confocal and wide-field florescence microscopy," Jornal of Microscopy-Oxford, vol. 193, pp , [16] J. Ling and A. C. Bovik, "Modeling and Restoration of Raman Microscopic Images," presented at IEEE 2000 International Conference on Image Processing (ICIP), Vancover, Canada, [17] S. T. Acton, "Diffsion-Based Edge Detectors," in Handbook of Image & Video Processing, A. C. Bovik, Ed. San Diego: Academic Press, 2000, pp [18] Z. Wang and A. C. Bovik, "A Universal Image Qality Index," Sbmitted to IEEE Signal Processing Letters, 2001.

13 13 [19] G. Abert and L. Vese, "A variational method in image recovery," Siam Jornal on Nmerical Analysis, vol. 34, pp , [20] S. Kichenassamy, "The Perona-Malik paradox," Siam Jornal on Applied Mathematics, vol. 57, pp , [21] Y. L. Yo, W. Y. X, A. Tannenbam, and M. Kaveh, "Behavioral analysis of anisotropic diffsion in image processing," IEEE Transactions on Image Processing, vol. 5, pp , [22] M. J. Black, G. Sapiro, D. H. Marimont, and D. Heeger, "Robst anisotropic diffsion," IEEE Transactions on Image Processing, vol. 7, pp , [23] J. Canny, "A comptational approach to edge detection," IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. PAMI-8, pp , [24] J. Ling, R. V. Moore, M. Miller, A. C. Bovik, and S. D. Weitman, "Application of Raman Imaging Microscopy to Evalate Drg Distribtion within Cancer Cells," presented at 91st Annal Meeting of American Association for Cancer Research, San Francisco, California, [25] J. Ling, The Development of Raman Imaging Microscopy To Visalize Drg Actions in Living Cells, Ph.D. Dissertation in Department of Biomedical Engineering. 2001, The University of Texas at Astin: Astin.

14 Endoplasmic reticlm (ER) Ncles Backgrond Mitochondria Cytoplasm Figre 1. A phantom to represent a celllar moleclar image. Figre 2. The reslt of the phantom in figre 1 filtered by a 3x3 median filter.

15 (a) (b) Figre 3. The diffsion fnction (a, Eq. 8) and the normalized flx fnction (b, Eq. 9) with K = 0.1. K

16 Figre 4. Histogram change of a constant image pls Gassian noise (with nit variance) dring anisotropic median-diffsion. Solid line: the original histogram of the Gassian noise image. Dotted line: the histogram after the first anisotropic diffsion step. Dashed line: the histogram after the sbseqent median filter.

17 17 (a) (b) (c) (d) (e) (f) Figre 5. Low SNR image denoising sing different anisotropic diffsion filters. (a) phantom from figre 1 degraded SNR = 5 db. (b), (c), (d) filtered images sing traditional anisotropic diffsion filter with K = 0.09, 0.19, and 0.38, respectively. (e) filtered image sing anisotropic open-close-diffsion filter with a 3x3 window, where K = (f) filtered image sing anisotropic mediandiffsion filter with a 3x3 window, where K = All the filters rn 100 iterations in the diffsion.

18 18 3 x Figre 6. Histogram of the gradient for the noisy image in figre 5(a) Q Iterations (a) Q Iterations (b) Q Iterations (c) Q Iterations (d) Figre 7. Qality indexes (a) Q, (b) Q 1, (c) Q 2 and (d) Q 3 vs. iteration nmber dring smoothing of the image in figre 5(a) with the anisotropic open-closediffsion filter.

19 Q Iterations (a) Q Iterations (b) Q Iterations (c) Q Iterations (d) Figre 8. Qality indexes (a) Q, (b) Q 1, (c) Q 2 and (d) Q 3 vs. iteration nmber dring smoothing of the image in figre 5(a) with the anisotropic mediandiffsion filter.

20 20 (a) (b) (c) (d) Figre 9. (a) phantom shown in figre 1 degraded to SNR = 0 db. (b), (c) filtered images sing traditional anisotropic diffsion filter with K = 0.17, 0.34, respectively. (d) filtered image sing the anisotropic median-diffsion filter with a 3x3 window, where K = All the filters rn 200 iterations in the diffsion. (a) (b) (c) (d) Figre 10. (a) natral piecewise smooth image of eggs. (b) image in (a) degraded to 0 db SNR. (c) filtered image sing traditional anisotropic diffsion filter with K = 0.2. (d) filtered image sing anisotropic median-diffsion filter with a 3x3 window and K = The filters rn 100 iterations in the diffsion.

21 21 (a) (b) (c) (d) Figre 11. (a) Conventional microscopic image of a breast cell which has been exposed to Taxol for one hor. (b) The recorded Raman image corresponding to the boxed area of the cell. (c) filtered image sing traditional anisotropic diffsion filter with K = 0.1. This filter ran 20 iterations in the diffsion. (d) filtered image sing anisotropic mediandiffsion filter with a 3x3 window and K = This filter ran 10 iterations in the diffsion.

22 22 (a) (b) (c) (d) (e) (f) Figre 12. (a) Conventional microscopic image of a breast cell which has been exposed to Taxol for 1.75 hors. (b) The recorded Raman image corresponding to the cell. (c) filtered image sing traditional anisotropic diffsion filter with K = the standard deviation of the image gradient. This filter ran 50 iterations in the diffsion. (d) filtered image sing traditional anisotropic diffsion filter with K = three times the standard deviation of the image gradient. This filter ran 50 iterations in the diffsion (e) filtered image sing anisotropic median-diffsion filter with a 3x3 window and K = the standard deviation of the image gradient. This filter ran 5 iterations in the diffsion. (f) Raman image shows the Taxol distribtion within the cell after frther processing (correction of the non-niform illmination, sbtract backgrond, 3-D deconvoltion, and eliminate florescence signal) the Raman image in (e).