## An approach to the synthesis of biological tissue

Optics Express, Vol. 1, Issue 13, pp. 414-423 (1997)

http://dx.doi.org/10.1364/OE.1.000414

Acrobat PDF (310 KB)

### Abstract

Mathematical phantoms developed to synthesize realistic complex backgrounds such as those obtained when imaging biological tissue, play a key role in the quantitative assessment of image quality for medical and biomedical imaging. We present a modeling framework for the synthesis of realistic tissue samples. The technique is demonstrated using radiological breast tissue. The model employs a two-component image decomposition consisting of a slowly, spatially varying mean-background and a residual texture image. Each component is synthesized independently. The approach and results presented here constitute an important step towards developing methods for the quantitative assessment of image quality in medical and biomedical imaging, and more generally image science.

© Optical Society of America

## 1. Introduction

^{11. B.R. Hunt and T. M. Cannon, “Nonstationary assumptions for Gaussian models of images,” IEEE Trans. on Sys., Man, and Cybern. , 876–882 (1976).,22. R.N. Strickland and H.I. Hahn, “Wavelet transforms for detecting microcalcifications in mammograms,” IEEE Trans. on Med. Imaging 15, 218–229 (1996). [CrossRef] }We shall refer to the slowly, spatially varying mean-background as the mean background. We propose to model the mean background as a stochastic process known as the lumpy background.

^{3-53. K.J. Myers, J.P. Rolland, H.H. Barrett, and R.F. Wagner, “Aperture optimization for emission imaging: effect of a spatially varying background,” J. Opt. Soc. Am. A. 7, 1279–1293 (1990). [CrossRef] [PubMed] }The statistical properties of the lumpy background are well specified and will be summarized in section 3.

^{44. J.P. Rolland, “Factors influencing lesion detection in medical imaging,” Ph.D. Dissertation, University of Arizona, (1990).,55. J.P. Rolland and H.H. Barrett, “Effect of random background inhomogeneity on observer detection performance,” J. Opt. Soc. Am. A. 9, 649–658 (1992). [CrossRef] [PubMed] }Thus, we shall also compare the statistical properties of these two types of backgrounds. Some preliminary analysis of the first and second order statistical properties of these two types of backgrounds will be presented in section 6.

## 2. Synthesis of biological tissue via a two-component model

^{6–96. M.G.A. Thomson and D.H. Foster, “Role of second- and third-order statistics in the discriminability of natural images,” J. Opt. Soc. Am. A. 14(9), 2081–2090 (1997). [CrossRef] }It has been suggested that various classes of images, including mammograms, have power spectra of the form 1/f

^{α}. For mammograms, estimated values of α in the range of 1.5 to 2 have been reported.

^{8–98. F.O. Bochud, F. R. Verdun, C. Hessler, and J.F. Valley, “Detectability on radiological images: the influence of anatomical noise,” Proc. SPIE 2436, 156–165 (1995). [CrossRef] }A power-law spectrum exponent between 1.5 and 2 indicates that mammograms are not fractals. A two-dimensional fractal would yield an exponent greater than 2. This finding further suggests that such backgrounds cannot be synthesized using fractals.10 Therefore, while some investigations have demonstrated that different mammographic tissue types can be classified according to their estimated fractal dimension,7

7. C. Caldwell and M. Yaffe, “Fractal analysis of mammographic parenchemal pattern,” Phys. Med. Biol. **35**, 235–247 (1990). [CrossRef] [PubMed]

^{99. B. Zheng, Y.H. Chang, and D. Gur, “Adpative computer-aided diagnosis scheme of digitized mammograms,” Acad. Radiol. 3 (10), 806–814 (1996). [CrossRef] [PubMed] ,1111. J.W. Byng, MJ. Yaffe, G.A. Lockwood, L.E. Little, D.L. Tritchler, and N.F. Boyd, “Automated analysis of mammographic densities and breast carcinoma risk,” Cancer 80(1), 66–74 (1997). [CrossRef] [PubMed] }An additional complication with modeling biological tissue as a fractal is that it is difficult to accurately estimate a fractal dimension from digitized data.

^{12}

^{8–98. F.O. Bochud, F. R. Verdun, C. Hessler, and J.F. Valley, “Detectability on radiological images: the influence of anatomical noise,” Proc. SPIE 2436, 156–165 (1995). [CrossRef] }It has been demonstrated, however, that the power spectrum of a statistical complex background is not a complete descriptor of the required background statistics to predict human observer performance in various detection tasks: specifically, two studies demonstrate that two sets of images with equal power spectra, yet having Fourier spectra that differ in phase, yield different detectability performance and thus require different predictive mathematical models.

^{66. M.G.A. Thomson and D.H. Foster, “Role of second- and third-order statistics in the discriminability of natural images,” J. Opt. Soc. Am. A. 14(9), 2081–2090 (1997). [CrossRef] ,88. F.O. Bochud, F. R. Verdun, C. Hessler, and J.F. Valley, “Detectability on radiological images: the influence of anatomical noise,” Proc. SPIE 2436, 156–165 (1995). [CrossRef] }An ensemble of images with the same power spectrum as that of another ensemble of images, but with a Fourier spectrum of random phase, was obtained by filtering various realizations of white noise with the desired power spectrum.

^{1313. J.N. Wolfe, “Breast patterns as an index of risk for developing breast cancer,” Am. J.Roentgenol. 126, 1130–1139 (1976).}Radiographic contrast in mammography arises from differing attenuation between tissues that comprise the breast. The breast is made essentially of a mixture of fatty tissue, which appears dark on radiographs, connective and epithelial tissues which produce bright radiographic appearances also referred to as mammographic densities, and prominent ducts which yield cord-like structures or a beaded appearance.

^{1111. J.W. Byng, MJ. Yaffe, G.A. Lockwood, L.E. Little, D.L. Tritchler, and N.F. Boyd, “Automated analysis of mammographic densities and breast carcinoma risk,” Cancer 80(1), 66–74 (1997). [CrossRef] [PubMed] –1313. J.N. Wolfe, “Breast patterns as an index of risk for developing breast cancer,” Am. J.Roentgenol. 126, 1130–1139 (1976).}

^{11. B.R. Hunt and T. M. Cannon, “Nonstationary assumptions for Gaussian models of images,” IEEE Trans. on Sys., Man, and Cybern. , 876–882 (1976).,22. R.N. Strickland and H.I. Hahn, “Wavelet transforms for detecting microcalcifications in mammograms,” IEEE Trans. on Med. Imaging 15, 218–229 (1996). [CrossRef] }An example of the sample image and the resulting blurred image are shown in Figure 1a and 1b, respectively. The sample image is a 256 x 256 pixel section extracted from a mammogram from the database of N. Karssemeijer of University Hospital Nijmegen, The Netherlands.

^{1414. The Nijmegen database is available by anonymous FTP from ftp://figment.csee.usf.edu /pub/mammograms/nijmegen-images}We propose to model the mean background as a lumpy background that we know to be a wide-sense stationary stochastic process.

^{44. J.P. Rolland, “Factors influencing lesion detection in medical imaging,” Ph.D. Dissertation, University of Arizona, (1990).,1515. A. Papoulis. Probablity, Random Variables, and Stochastic Processes. (Mc Graw-Hill, NY, 1991).,1616. H.H. Barrett, J. Yao, J.P. Rolland, and K.J Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993). [CrossRef] [PubMed] }We thus assume that the stochastic process describing an ensemble of mean backgrounds extracted from several sub-images of a set of mammograms is wide-sense stationary as well. This assumption will be fully investigated in future work as it involves computation of an ensemble autocorrelation function over a large number of images and the careful study of its properties.

^{1515. A. Papoulis. Probablity, Random Variables, and Stochastic Processes. (Mc Graw-Hill, NY, 1991).}Such a validation is beyond the scope of this paper.

^{1}Our model differs from that of Hunt and Cannon as we hypothesize that the mean background arises from an underlying wide-sense stationary process. Moreover, as it relates specifically to mammography, several investigations by Byng and colleagues suggest that at least two parameters are required to characterize mammograms: one parameter to describe the distribution of breast tissue density as reflected by the brightness of the mammogram and another parameter to characterize the texture.

^{1111. J.W. Byng, MJ. Yaffe, G.A. Lockwood, L.E. Little, D.L. Tritchler, and N.F. Boyd, “Automated analysis of mammographic densities and breast carcinoma risk,” Cancer 80(1), 66–74 (1997). [CrossRef] [PubMed] }The mean background and texture components of the proposed decomposition are reminiscent of the first and second parameters in Byng’s model, respectively.

## 3. Synthesis of the slowly varying mean-background.

^{3–53. K.J. Myers, J.P. Rolland, H.H. Barrett, and R.F. Wagner, “Aperture optimization for emission imaging: effect of a spatially varying background,” J. Opt. Soc. Am. A. 7, 1279–1293 (1990). [CrossRef] [PubMed] }In the case of mammography, the mean background may account for the relative amount of fat and densities in the breast tissue.

^{33. K.J. Myers, J.P. Rolland, H.H. Barrett, and R.F. Wagner, “Aperture optimization for emission imaging: effect of a spatially varying background,” J. Opt. Soc. Am. A. 7, 1279–1293 (1990). [CrossRef] [PubMed] ,55. J.P. Rolland and H.H. Barrett, “Effect of random background inhomogeneity on observer detection performance,” J. Opt. Soc. Am. A. 9, 649–658 (1992). [CrossRef] [PubMed] ,1515. A. Papoulis. Probablity, Random Variables, and Stochastic Processes. (Mc Graw-Hill, NY, 1991).}The lumpy background assumes wide-sense stationarity, that is stationarity over the ensemble of images, where the autocorrelation function is only a function of the shift variable

**r**. The second assumption is that the background autocorrelation function is a Gaussian function. The power spectrum W(

**ρ**) is then defined as the Fourier transform of the autocorrelation function and is given by

**r**, r

_{b}is the correlation length of the autocorrelation function, and W(0) is the value of the power spectrum at zero frequency that we refer to as the lumpiness measure.

^{4}The first approach consisted of superimposing 2D Gaussian functions, referred to as Gaussian blobs, on a constant background of strength B

_{0}. To keep the mathematics simple, we assumed 2D Gaussian blobs of constant amplitude b

_{0}/π

_{b}. In this case, the lumpy background can be shown to be mathematically specified as

_{j}is a random variable uniformly distributed over the background area, and K is the number of Gaussian blobs in the background. Rolland further showed that to yield a Gaussian autocorrelation function, the number of Gaussian blobs K must also be a random variable with the mean of K equal to its variance. We thus chose K to be Poisson distributed for this condition to be satisfied.

^{4}A measure of lumpiness in the background is given by

_{d}is the mean number of Gaussian functions per mm

^{2}.

## 4. Synthesis of the residual texture image

^{1717. J.P. Rolland and L. Yu, “A four-layer pyramid framework for statistical texture synthesis,” (in preparation).}The synthesis of the texture of biological tissues using the proposed framework is presented here for the first time. We shall now describe each component of the framework.

*Pyramid transform*The proposed algorithm for the synthesis of the residual texture is based on a four-layer steerable pyramid transform. One layer of the pyramid is depicted in Figure 2. Layers are connected by a factor-of-two decimation of the image.

^{1818. P.P. Vaidyanathan. Multivariate systems and filter banks. (Prentice Hall, NJ, 1993).}Within each layer, the image is filtered by a set of bandpass filters and followed by a set of orientation filters. The algorithm adopts a 4 (scales) x 4 (orientations: 0 degree, 45 degree, 90 degree and 135 degree) steerable filter bank.

^{19–2219. D.J. Heeger and J.R. Bergen, “Pyramid-based texture analysis/synthesis,” Compt. Graph.. , 229–238 (1995).}Details of the filters employed for the synthesis will be detailed elsewhere.

^{1717. J.P. Rolland and L. Yu, “A four-layer pyramid framework for statistical texture synthesis,” (in preparation).}

*Image Decomposition*The texture image is processed through the left hand side of the pyramid transform shown in Figure 2. It is represented in Figure 2 as an input to the pyramid in the upper left corner. In parallel, a realization of uniformly distributed white noise, referred thereafter as white noise, is also processed by the same pyramid transform, that is, it is also fed independently to the pyramid transform in the upper left corner. The role of the white noise image is to provide a starting point for the synthesis.

*Histogram matching at multiple scales*After decomposition of a texture sample and a realization image of white noise, the histograms of the subband images (i.e. output images of the filters on the left hand side of the pyramid) of the texture image and of the noise image are matched.

^{2323. J. W. Woods. Subband Image Coding. (Kluwer Academic Publishers, MA,1991).}Histogram matching is an image processing technique, specifically a point operation, which modifies a candidate image so that its histogram matches that of a model image.

^{24–2524. W.K. Pratt, Digital Image Processing. (John Wiley & Sons, NY, 1991).}

*Image Synthesis*The histogram-matched noise subband-images obtained at multiple scales are then recombined according to the right hand side of the pyramid transform shown in Figure 2. This process yields a synthetic image such as that shown in Figure 2c. If another realization of white noise is processed instead, the synthesis yields another realization of the synthesized image. Such an example is shown in Figure 2d. In our implementation, the set of filters used in the decomposition and the reconstruction stages forms a quadrature-mirror filter bank.

^{1818. P.P. Vaidyanathan. Multivariate systems and filter banks. (Prentice Hall, NJ, 1993).}

^{1919. D.J. Heeger and J.R. Bergen, “Pyramid-based texture analysis/synthesis,” Compt. Graph.. , 229–238 (1995).}

## 5. A proposed mathematical phantom

_{i}(x,y) according to the described mathematical phantom can be established using an adaptive linear combination of realizations from the two model components: a realization of a lumpy background component denoted as L

_{i}(x,y) and a realization of the synthesized texture component denoted as T

_{i}(x,y). The resulting synthesized image will then be given by

^{1313. J.N. Wolfe, “Breast patterns as an index of risk for developing breast cancer,” Am. J.Roentgenol. 126, 1130–1139 (1976).}On a more theoretical basis, one can also study a wide range of combination of such backgrounds by varying β and the parameters associated with each component. Such a framework may naturally find application to other types of images beside mammographic tissue.

## 6. Preliminary investigation of the model’s first- and second-order statistics

^{1616. H.H. Barrett, J. Yao, J.P. Rolland, and K.J Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993). [CrossRef] [PubMed] }To address this issue, the ultimate test will be to conduct a set of psychophysical studies using real images and mathematically simulated images. While it has been shown that equalization of the first- and second-order statistics are insufficient to predict detectability in complex backgrounds, we propose to conduct a first investigation of such properties for the mammographic tissue, its proposed two components, and their syntheses, to draw parallels in the equivalence of first- and second-order statistics of the proposed synthesis model images compared to the original images.

^{1515. A. Papoulis. Probablity, Random Variables, and Stochastic Processes. (Mc Graw-Hill, NY, 1991).}The average histograms and power spectra for the five ensembles of images are shown in Figure 4 and Figure 5, respectively. The procedure used to estimate a power spectrum from an ensemble of images is the two dimensional extension of the method due to Welch that we adapted to two-dimensional structured backgrounds.

^{2626. P.D. Welch, “The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. 15, 70–73 (1967). [CrossRef] }An extension of Welch’s method to two dimensional noise images was detailed in Hanson.

^{2727. K.M. Hanson, “Detectability in computed tomographic images,” Med. Phys. 6(5), 441–451 (1979). [CrossRef] [PubMed] }The extension to structured backgrounds involves computation of an ensemble mean and subtraction of each image from the estimated ensemble-mean before computation of the power spectrum. Rolland and Barrett used a similar method for defining the power spectrum of the lumpy stochastic process, while the power spectrum of the lumpy background was analytically, rather than numerically, computed.

^{4–54. J.P. Rolland, “Factors influencing lesion detection in medical imaging,” Ph.D. Dissertation, University of Arizona, (1990).}The procedure we adopted can be summarized in six steps:

- Given an ensemble of images, compute the ensemble mean
- Subtract from each image of the ensemble the ensemble mean to form a new ensemble set
- 3. Take the FFT of each image i from the ensemble to yield X
_{i}(m,n) - 4. Compute the normalized periodogram as |X
_{i}(m,n)|^{2}/(NxN) - 5. Compute the average periodogram over the ensemble of images
- 6. Plot the Log
_{10}of the power spectrum along the x-dimension.

^{2626. P.D. Welch, “The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. 15, 70–73 (1967). [CrossRef] }

^{6}counts

^{2}/(sec

^{2}pixels) (i.e. mean number of blobs was 400, the strength of a blob b0 was 12800 counts/sec) and of correlation length rb equal to 15 pixels. For the synthesized texture images, eighteen syntheses were obtained using three of the residual texture images from the ensemble. The estimated power spectra of mammographic backgrounds, its two proposed components, and its two model components are shown in Figure 4.

## 7. Conclusion

## Acknowledgments

## References and links

1. | B.R. Hunt and T. M. Cannon, “Nonstationary assumptions for Gaussian models of images,” IEEE Trans. on Sys., Man, and Cybern. , 876–882 (1976). |

2. | R.N. Strickland and H.I. Hahn, “Wavelet transforms for detecting microcalcifications in mammograms,” IEEE Trans. on Med. Imaging |

3. | K.J. Myers, J.P. Rolland, H.H. Barrett, and R.F. Wagner, “Aperture optimization for emission imaging: effect of a spatially varying background,” J. Opt. Soc. Am. A. |

4. | J.P. Rolland, “Factors influencing lesion detection in medical imaging,” Ph.D. Dissertation, University of Arizona, (1990). |

5. | J.P. Rolland and H.H. Barrett, “Effect of random background inhomogeneity on observer detection performance,” J. Opt. Soc. Am. A. |

6. | M.G.A. Thomson and D.H. Foster, “Role of second- and third-order statistics in the discriminability of natural images,” J. Opt. Soc. Am. A. |

7. | C. Caldwell and M. Yaffe, “Fractal analysis of mammographic parenchemal pattern,” Phys. Med. Biol. |

8. | F.O. Bochud, F. R. Verdun, C. Hessler, and J.F. Valley, “Detectability on radiological images: the influence of anatomical noise,” Proc. SPIE |

9. | B. Zheng, Y.H. Chang, and D. Gur, “Adpative computer-aided diagnosis scheme of digitized mammograms,” Acad. Radiol. |

10. | M.F. Barnsley. |

11. | J.W. Byng, MJ. Yaffe, G.A. Lockwood, L.E. Little, D.L. Tritchler, and N.F. Boyd, “Automated analysis of mammographic densities and breast carcinoma risk,” Cancer |

12. | B. Dubuc, C.R. Carmes, C. Tricot, and S.W. Zucker, “The variation method: a technique to estimate the fractal dimension of surfaces,” Proc. SPIE |

13. | J.N. Wolfe, “Breast patterns as an index of risk for developing breast cancer,” Am. J.Roentgenol. |

14. | The Nijmegen database is available by anonymous FTP from ftp://figment.csee.usf.edu /pub/mammograms/nijmegen-images |

15. | A. Papoulis. |

16. | H.H. Barrett, J. Yao, J.P. Rolland, and K.J Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA |

17. | J.P. Rolland and L. Yu, “A four-layer pyramid framework for statistical texture synthesis,” (in preparation). |

18. | P.P. Vaidyanathan. |

19. | D.J. Heeger and J.R. Bergen, “Pyramid-based texture analysis/synthesis,” Compt. Graph.. , 229–238 (1995). |

20. | E.P. Simoncelli and E.H. Adelson, “Subband transforms”. |

21. | E.P. Simoncelli, W.T. Freeman, E.H. Adelson, and D.J. Heeger, “Shiftable multi-scale transforms,”. IEEE Trans. on Info. Theory , Special Issue on Wavelets |

22. | P. Perona, “Deformable kernels for early vision,” IEEE Trans. Pattern Analysis and Machine Intelligence , |

23. | J. W. Woods. |

24. | W.K. Pratt, |

25. | K.R. Castleman, |

26. | P.D. Welch, “The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. |

27. | K.M. Hanson, “Detectability in computed tomographic images,” Med. Phys. |

**OCIS Codes**

(100.0100) Image processing : Image processing

(170.3830) Medical optics and biotechnology : Mammography

**ToC Category:**

Focus Issue: Biomedical optics

**History**

Original Manuscript: October 20, 1997

Revised Manuscript: October 11, 1997

Published: December 22, 1997

**Citation**

Jannick Rolland and Robin Strickland, "An approach to the synthesis of biological tissue," Opt. Express **1**, 414-423 (1997)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-1-13-414

Sort: Journal | Reset

### References

- B.R. Hunt, and T. M. Cannon, "Nonstationary assumptions for Gaussian models of images," IEEE Trans. on Sys., Man, and Cybern., 876-882 (1976).
- R.N. Strickland, and H.I. Hahn, "Wavelet transforms for detecting microcalcifications in mammograms," IEEE Trans. on Med. Imaging 15, 218-229 (1996). [CrossRef]
- K.J. Myers, J.P. Rolland, H.H. Barrett, and R.F. Wagner, "Aperture optimization for emission imaging: effect of a spatially varying background," J. Opt. Soc. Am. A. 7, 1279-1293 (1990). [CrossRef] [PubMed]
- J.P. Rolland, "Factors influencing lesion detection in medical imaging," Ph.D. Dissertation, University of Arizona, (1990).
- J.P. Rolland, and H.H. Barrett, "Effect of random background inhomogeneity on observer detection performance," J. Opt. Soc. Am. A. 9, 649-658 (1992). [CrossRef] [PubMed]
- M.G.A. Thomson, and D.H. Foster, "Role of second- and third-order statistics in the discriminability of natural images," J. Opt. Soc. Am. A. 14(9), 2081-2090 (1997). [CrossRef]
- C. Caldwell, and M. Yaffe, "Fractal analysis of mammographic parenchemal pattern," Phys. Med. Biol. 35, 235-247 (1990). [CrossRef] [PubMed]
- F.O. Bochud, F. R. Verdun, C. Hessler, and J.F. Valley, "Detectability on radiological images: the influence of anatomical noise," Proc. SPIE 2436, 156-165 (1995). [CrossRef]
- B. Zheng, Y.H. Chang, and D. Gur, "Adpative computer-aided diagnosis scheme of digitized mammograms," Acad. Radiol. 3 (10), 806-814 (1996). [CrossRef] [PubMed]
- M.F. Barnsley, Fractals Everywhere. (Academic Press, San Diego, CA, 1988)
- J.W. Byng, M J. Yaffe, G.A. Lockwood, L.E. Little, D.L. Tritchler, and N.F. Boyd, "Automated analysis of mammographic densities and breast carcinoma risk," Cancer 80(1), 66-74 (1997). [CrossRef] [PubMed]
- B. Dubuc, C.R. Carmes, C. Tricot, and S.W. Zucker, "The variation method: a technique to estimate the fractal dimension of surfaces," Proc. SPIE 845, 241-248 (1987). [CrossRef]
- J.N. Wolfe, "Breast patterns as an index of risk for developing breast cancer," Am. J. Roentgenol. 126, 1130-1139 (1976).
- The Nijmegen database is available by anonymous FTP from <A HREF="ftp://figment.csee.usf.edu/pub/mammograms/nijmegen-images">ftp://figment.csee.usf.edu/pub/mammograms/nijmegen-images</A>
- A. Papoulis. Probablity, Random Variables, and Stochastic Processes. (Mc Graw-Hill, NY, 1991).
- H.H. Barrett, J. Yao, J.P. Rolland, and K.J Myers, "Model observers for assessment of image quality," Proc. Natl. Acad. Sci. USA 90, 9758-9765 (1993). [CrossRef] [PubMed]
- J.P. Rolland and L.Yu, "A four-layer pyramid framework for statistical texture synthesis," (in preparation).
- P.P. Vaidyanathan. Multivariate systems and filter banks. (Prentice Hall, NJ, 1993).
- D.J. Heeger, and J.R. Bergen, "Pyramid-based texture analysis/synthesis," Compt. Graph., 229-238 (1995).
- E.P. Simoncelli, and E.H. Adelson, "Subband transforms." In Subbands Image Coding, (Kluwer Academic Publishers, J.W. Woods, eds., MA 1991).
- E.P. Simoncelli, W.T. Freeman, E.H. Adelson, and D.J. Heeger, "Shiftable multi-scale transforms," IEEE Trans. on Info. Theory, Special Issue on Wavelets 38, 587607 (1992).
- P. Perona, "Deformable kernels for early vision," IEEE Trans. Pattern Analysis and Machine Intelligence, 17(5), 448-499 (1995). [CrossRef]
- J. W. Woods. Subband Image Coding. (Kluwer Academic Publishers, MA,1991).
- W.K. Pratt, Digital Image Processing. (John Wiley & Sons, NY, 1991).
- K.R. Castleman, Digital Image Processing. (Prentice Hall, NJ,1996).
- P.D. Welch, "The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms," IEEE Trans. Audio Electroacoust. 15, 70-73 (1967). [CrossRef]
- K.M. Hanson, "Detectability in computed tomographic images," Med. Phys. 6(5), 441-451 (1979). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.