Wavelet analysis is currently being investigated as an image enhancement tool for use in mammography. Although this approach to image processing appears to have great promise, there remain major uncertainties regarding an optimal form of wavelet based algorithms. It is, therefore, desirable to have a quantitative method for evaluating a wavelet based image processing algorithm. Optimization of algorithms prior to evaluation using standard Receiver Operating Characteristic method is made possible. A mathematical method has been developed where the input signal is a gaussian with added random noise. An enhancement factor (EF) is obtained from input and output signal-to-noise ratios, SNRi and SNRo, (EF equals SNRo/SNRi). The development and testing of this method is described, and a practical application in given showing the major features of a wavelet based image processing algorithm based on the Frazier-Jawerth transform

Fan, Jian

Huda, Walter

Jing, Zhenxue

Laine, Andrew F.

Xing, Yunong

Zheng, Yisheng

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<title>Quantitative evaluation of wavelet-based image processing algorithms</title>

Columbia University Academic Commons

Quantitative evaluation ofwavelet based image processing algorithmsZhenxue Jing1'2Yisheng Zheng3Walter Huda'2Andrew Laine,Jian FanYunong Xing4University of Florida, Departments of Nuclear Engineering 2Radiology, 3Computer and Information Sciences,and 4Electrical Engineering, Gainesville, Florida 32610ABSTRACTWavelet analysis is currently being investigated as an image enhancement tool for use in mammography. Although thisapproach to image processing appears to have great promise, there remain major uncertainties regarding an optimal formof wavelet based algorithms. It is, therefore, desirable to have a quantitative method for evaluating a wavelet based imageprocessing algorithm. Optimizationofalgorithmspriorto evaluationusingstand&dReceiver Operating Characteristic (ROC)methods is made possible.A mathematical method has been developed where the input signal is a gaussian with added random noise. Anenhancement factor (EF) is obtained from input and output signal-to-noise ratios, SNR and NRQ, (EF = NRQISNR ). The development and testing of this method is described, and a practical application is given showing the majorfeatures of a wavelet based image processing algorithm based on the Frazier-Jawerth transform.Key words: Mammography, wavelets, image processing, image quality, signal-to-noise ratio, image enhancement1. INTRODUCTIONMammographyis the most reliable methodfor earlydetection ofnonpalpable breast cancer Objects of diagnostic interestinclude tumors which differ from normal tissue by very small amounts, or microcalcifications which are very small. Theresultant low visibility of these objects in the mammograms makes accurate cancer diagnosis problematicaL Digitalenhancement ofmammograms, together with the use ofworkstations using computer aided diagnosis tools, maypermit a moreconfident interpretation ofdifficult cases without resorting to follow-up patient examinations Furthermore, the large numberof negative biopsies encountered in current practice may also be reduced if an enhanced mammogram was able to providea more certain diagnosisConventional image processing techniques generally do not perform well on mammographic images Large variationsin feature size and shape limit the effectiveness of classical fixed neighborhood techniques such as unsharp masking'7 Theuse of processing algorithms based on the wavelet transform has recently been proposed for use in mammography.5'9'10' Thisapproach to image enhancement is promising because it uses methods similar to those used in the human visual system10'12In this paper, we develop a mathematical model to investigate the behavior of a wavelet algorithm based on the Frazier-Jawerth Transform (FJT)13'140819416274/94/$6.OO SPIE Vol. 2303 / 569Downloaded from SPIE Digital Library on 25 Aug 2010 to 128.59.161.30. Terms of Use:  http://spiedl.org/terms2. METhOD2.1. Mathematical modelTwo-dimensional mathematical phantom images were generated which contained a gaussian signal and random noise.Each phantom image consisted of a 5122 matrixwith each pixel coded using 1 Byte, permitting the display of 256 gray levels.A two-dimensional gaussian shape signal was generated with a maximum intensity of I and a full width tenth maximum(FWTM) of W. The location of the signal was at the center of the image.Random background noise, N, with a gaussian distribution (a )about the mean (iz ),was generated using the expressionN = (-2alnY)"2cos(O) + I. (1)where Y is a random variable between 0 and 1, and e is a random variable between 0 and 2ir16 Figure la shows a profilethrough the central axis of a phantom image with a FWTM equal to 10 pixels and intensity I equal to 70, and Figure lb showsa profile through random noise remote from this gaussian signal.The signal S is the mean value of the signal intensity above the mean background level, where the signal was averaged overthe FWTM width W. The noise is the standard deviation, a ,ofrandom fluctuations obtained from a profile taken at a locationremote from the signal region. To evaluate improvements from the image processing algorithms investigated, an EF wasdefined by the expressionSNREF = ° (2)SNRwhere SNR and NRQ are the input and output signal-to-noise ratios, respectively.2.2. Wavelet algorithmThe algorithm evaluated in this study is based on the FJT1315 The inner product of the signal (S) with a wavelet ()reflects the character of S within the time-frequency region where is localized. Provided is spatially localized, two-dimensional features such as shape and orientation are preserved in transform space and may thus be used to characterizefeatures through scale space.For an isotropic function, a multiresolution representation divides the frequency spectrum of an image x into a low-passsub-band image y Ld a set of band-pass sub-band images yL,L = 1 , M where M denotes the number of levels. Witha 512 matrix size, the number of levels is 8. If FL is the equivalent filter for the Lth level, W L[X]denotes the operation offiltering x, and the sub-band image of an L-level multiresolution decomposition is given byyL =WL[x] (3)The FJT results in a multiresolution decomposition of the input image at each level L, where maxima of the waveletcoefficients may be determined14 Maxima above some threshold value T are multiplied by a selected gain factor, G, followedby the inverse transform to generate the processed image.570 / SPIE Vol. 2303Downloaded from SPIE Digital Library on 25 Aug 2010 to 128.59.161.30. Terms of Use:  http://spiedl.org/termsThe FiT image processing algorithm has three parameters whose values are selected by the operator: (1) level(s) (L =1 through L = 8) at which the modifications to the wavelet coefficients are to be performed, with L = 1corresponding to thehighest spatial frequencies and L = 8 to the lowest (DC cap); (2) the threshold value T above which wavelet coefficients aremodified; and (3) the gain factor, G, bywhich selectedwavelet coefficients (i.e. those corresponding to local maxima and abovethe threshold value T) are to be multiplied. The significance ofeach ofthese three parameters was systematically investigatedusing the mathematical phantom model described above.3. RESULT AND DISCUSSION3.1. Optimization strategyThe wavelet based image processing algorithm has three parameters: Level(L), Threshold (T) and Gain (G). In addition,the image signal and noise parameters could also be varied. For the purposes of this study, the noise level was fixed with amean value of 10 and with a standard deviation of 4. The peak signal intensity was fixed at a value 70 above the mean noiselevel, but five values of the signal width (W pixels) were investigated (4, 6, 10, 20 and 40).The initial parameter investigated was L using constant values of T =5 and G = 10. For each signal width, there is anoptimal value ofLwhich yields the highest EF. The optimal L value for each width was employed in subsequent investigationsof the T/G parameters. The next parameter to be studied was T, and the value observed to yield the highest EF was usedwhen investigating the G parameter. Although this method does not guarantee that the fmal algorithm parameters yields thebest possible EF value for this image, the significance and relative importance of each parameter can be investigated.32. L parameterFigure 2 shows results obtained for the EF vs. L experiments as the width of the gaussian (W) was systematically varied.The optimal value of L increased with increases in the W value which is to be expected. As the level L increases, thiscorresponds to lower spatial frequencies, and thus the larger objects are located at the levels (frequency bands) correspondingto large L values. The optimal L was 3 for W =4, increasing to 6 for W = 40. The absolute value ofthe EF was about 4 forthe narrow gaussian signals (W = 4, 6 and 10), but fell to about 2 for the wider gaussian signals (W = 20 and 40).3.3. T parameterFigure 3 shows the corresponding results obtained of the EF vs. T experiments. As expected, the value of EF approaches1 when a high threshold is set, since this would exclude all the wavelet coefficients from being modified, and the processedimage would then be identical to the original image. For smaller phantoms, the setting of a threshold is clearly important andincreases the EF value. This behavior is expected since there is a significant amount of noise at the lower levels whichcorresponds to the higher frequencies. This also explains the absence ofany effect ofthe threshold for the wider signals wherethe enhancement of coefficients occurs at higher L values since noise is primarily a high frequency phenomenon.3.4. G parameterFigure 4 shows the EF variation with G for each width of the gaussian signal. In each case, images were processed usingthe values of L and T that yielded the highest EF values. The narrowest signal appears to reach a plateau value (-12), butall the other signals show a monotonic increase in the EF value as G increases. The maximum value of EF observed was inexcess of3O, suggesting that the output signal-to-noise ratio may be significantly increased by the use ofan optimized waveletbased image processing algorithm.Figure 5 shows profiles through the gaussian signal with a W value of 10 processed with increasing values of the Gparameter. As G increases, the noise is reduced throughout the image, and it is seen that there is an overshooting at the edgeof the signal. The FJT algorithm enhanced edges by subtracting from the edge its second derivative, thus producing theovershooting observed in Figure 5.SPIE Vol. 2303 / 571Downloaded from SPIE Digital Library on 25 Aug 2010 to 128.59.161.30. Terms of Use:  http://spiedl.org/terms4. CONCLUSIONS1. The enhancement factor, given by the ratio of the output to input signal-to-noise ratios, can measure the performance ofan image processing algorithm.2. Choices for values ofthe L, T, and G parameters ofthe FJT wavelet algorithm have a marked impact on image processingperformance.3. Optimal values of the three parameters defming the image processing algorithm are a function of the width of the inputgaussian signal.4. Optimized algorithms suppress random noise and markedly enhance the visibility of the signal.5. Optimized algorithms can achieve an improvement as high as 30 in the processed signal-to-noise ratio.5. ACKNOWLEDGEMENTSThe authors wish to thank Drs. Janice C. Honeyman, Edward V. Staab, and Barbara G. Steinbach for their assistance inthis research project and Mrs. LindaWaters-Funk forpreparation ofthemanuscript. Valuable discussionswithSergio Schulerare also acknowledged.This work was sponsored in part by the Whitaker Foundation and the U.S. Army Medical Research and DevelopmentCommand, Grant No. DAMD 17-93-J-3003.6. REFERENCES1. Feig SA and Hendrick ER. "Risk, benefit, and controversies in mammographic screening," TechnicalAspects of BreastImaging, Haus AG, Yaffe MJ (eds), Radiological Society of North America (RSNA), Oak Brook, pp. 119-135, 1993.2. Vyborny CJ, Schmidt RA. "Mammography as a radiographic examination: an overview," RadioGraphics VoL 9, pp.723-764, 1989.3. Giger ML, Doi K, MacMohan H, et al. "An 'intelligent' workstation for computer-aided diagnosis," RadioGraphicsVol. 13, pp. 647-656, 1993.4. Hendrick ER, Parker SH. "Stereotaxic imaging," Technical Aspects of Breast Imaging, Haus AG, Yaffe MJ (eds),Radiological Society of North America (RSNA), Oak Brook, pp. 257-269, 1993.5. MorrowWM, Paranjape RB, RangayyanRM, DesautelsJEL. "Region-based contrast enhancement of mammograms,"IEEE Transactions on Medical Imaging, VoL 11, pp. 392-406, 1992.6. Prokop M, Galanski M, Oestmann JW, von Falkenhausen U, Rosenthal H, Reimer P, Nischelsky J, Reichelt S."Storage phosphor versus screen-film radiography: effect of varying exposure parameters and unsharp mask filtering on thedetectability of cortical bone defects," Radiology Vol 177, pp. 109-113, 1990.7. Higashida Y, Moribe N, Morita K, Katsuda N, Hatemura M, Takada T, Takahashi M and Yamashita J. "Detectionof subtle microcalcifications: comparison of computed radiography and screen-film mammography," Radiology VoL 183, pp.483-486, 1992.8. Lame A, Song S. "Multiscale wavelet representations for mammographic feature analysis," Proceedings of SPffi:Conference on Mathematical Methods in Medical Imaging, San Diego, July 23-25, 1992.572 ISPIE Vol. 2303Downloaded from SPIE Digital Library on 25 Aug 2010 to 128.59.161.30. 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Quantitative evaluation of wavelet-based image processing algorithms

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Quantitative evaluation of wavelet-based image processing algorithms

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