## Singular-value decomposition of a tomosynthesis system |

Optics Express, Vol. 18, Issue 20, pp. 20699-20711 (2010)

http://dx.doi.org/10.1364/OE.18.020699

Acrobat PDF (1543 KB)

### Abstract

Tomosynthesis is an emerging technique with potential to replace mammography, since it gives 3D information at a relatively small increase in dose and cost. We present an analytical singular-value decomposition of a tomosynthesis system, which provides the measurement component of any given object. The method is demonstrated on an example object. The measurement component can be used as a reconstruction of the object, and can also be utilized in future observer studies of tomosynthesis image quality.

© 2010 Optical Society of America

## 1. Introduction

1. J. T. Dobbins III and D. J. Godfrey, “Digital x-ray tomosynthesis: current state of the art and clinical potential,” Phys. Med. Biol. **48**, R65–R106 (2003). [PubMed]

2. J. T. Dobbins III, “Tomosynthesis: at translational crossroads,” Med. Phys. **36**, 1956–1967 (2009). [PubMed]

4. G. Gennaro, A. Toledano, C. di Maggio, E. Baldan, E. Bezzon, M. La Grassa, L. Pescarini, I. Polico, A. Proietti, A. Toffoli, and P. C. Muzzio, “Digital breast tomosynthesis versus digital mammography: a clinical performance study,” Eur. Radiol. (2009). [PubMed]

8. A. S. Chawla, J. Y. Lo, J. A. Baker, and E. Samei, “Optimized image acquisition for breast tomosynthesis in projection and reconstruction space,” Med. Phys. **36**, 4859–4869 (2009). [PubMed]

9. T. Wu, R. H. Moore, E. A. Rafferty, and D. B. Kopans, “A comparison of reconstruction algorithms for breast tomosynthesis,” Med. Phys. **31**, 2636–2647 (2004). [PubMed]

10. Y. Zhang, H.-P. Chan, B. Sahiner, J. Wei, M. M. Goodsitt, L. M. Hadjiiiski, J. Ge, and C. Zhou, “A comparative study of limited-angle cone-beam reconstruction methods for breast tomosynthesis,” Med. Phys. **33**, 3781–3795 (2006). [PubMed]

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 **90**, 9758–9765 (1993). [PubMed]

2. J. T. Dobbins III, “Tomosynthesis: at translational crossroads,” Med. Phys. **36**, 1956–1967 (2009). [PubMed]

## 2. SVD analysis

*z, x*and

*y*direction. We present this case of limited object and image support as the most general, but will mainly consider the simplified case of infinite transverse extent. The support in the

*z*direction is per definition limited to at most the distance between the source and image planes. The variables used are

**r**= (

*x,y*) for transverse coordinates in object space and

*z*for longitudinal coordinate in object space. The image plane is placed at the origin (

*z*= 0), and has transverse coordinates

**r**

*= (*

_{d}*x*). The source plane is placed at

_{d},y_{d}*z*= −

*s*and position number

*m*given by

**r**

*= (*

_{m}*χ*). The source is assumed to be a point. The object

_{m}, ψ_{m}*f*(

**r**,

*z*) is identical to the absorption coefficient, the image is

*g*(

_{m}**r**

*) for the image taken using source position*

_{d}**r**

*, and*

_{m}**g**(

**r**

*) is the vector of all images.*

_{d}*I*(

_{m}**r**

*) is the irradiance,*

_{d}*A*is the strength of the point source in emitted flux per unit volume per unit solid angle,

**r**

_{1}= (

**r**

*, 0) = (*

_{d}*x*, 0) is the position of the image point in three dimensions and

_{d}, y_{d}**r**

_{0}= (

**r**

*, −*

_{m}*s*) = (

*χ*, −

_{m}, ψ_{m}*s*) the position of the source. The angle

*θ*is the angle of incidence of the ray from

**r**

*to*

_{m}**r**

*onto the detector plane. A change of variables to our chosen coordinate system, namely*

_{d}*z*= −

*l′s*/|

**r**

_{1}−

**r**

_{0}|, leads to

*θ*=

*s*/|

**r**

_{1}−

**r**

_{0}| and |

**r**

_{1}−

**r**

_{0}| = (|

**r**

*−*

_{d}**r**

*|*

_{m}^{2}+

*s*

^{2})

^{1/2}, and changing notation from the three-dimensional

**r**

_{1}and

**r**

_{0}to the two-dimensional

**r**

*and*

_{d}**r**

*, leads to*

_{m}*I*(

_{m}**r**

*) registered in the image plane and the image function*

_{d}*g*(

_{m}**r**

*):*

_{d}*g*(

_{m}**r**

*) is designed to yield a linear relation between object and image, and inserting Eq. (2) into Eq. (1) leads to the propagation integral from object to image*

_{d}*z*

_{1}and

*z*

_{2}limit the object extension in the

*z*direction, −

*s*<

*z*

_{1}<

*z*

_{2}< 0, M is the number of source positions,

*b*(

_{f}**r**,

*z*) and

*b*(

_{g}**r**

_{d}) define the object and image support respectively, the asterisk denotes complex conjugate, and the impulse-response function

*h*is the two-dimensional Dirac delta function

*b*(

_{f}**r**,

*z*) and

*b*(

_{g}**r**

*) can in general be adjusted according to specific situations, but the support in the axial direction can never extend beyond the source or image plane as indicated by the limits*

_{d}*z*

_{1}and

*z*

_{2}. Eqs. (3)–(5) give a complete description of the propagation, and consequently also of the reconstruction. Now the SVD method for multiple images, earlier presented for telecentric optical imaging systems [14], can be applied. However, this method requires shift invariance in the transverse direction, so we apply a change of variables [17]:

**r**

*of the source determines the inclination of the rays. The object is distorted by this transformation, but the image remains the same. In these coordinates, the impulse-response function becomes*

_{m}*h̃*(

_{m}**r̃**−

**r**

*) =*

_{d},z̃*h*(

_{m}**r**,

**r**

*). Since the Jacobian of the transformation is*

_{d},z*f̃*(

**r̃**,

*z̃*) =

*f*(

**r**,

*z*) is the distorted object function and

*b̃*(

_{f}**r̃**,

*z̃*) =

*b*(

_{f}**r**,

*z*) the distorted object-support function. The integration limits are

*z̃*

_{1}=

*sz*

_{1}/(

*s*+

*z*

_{1}) and

*z̃*

_{2}=

*sz*

_{2}/(

*s*+

*z*

_{2}) and fulfill −∞ <

*z̃*

_{1}<

*z̃*

_{2}< 0. We note that due to the change of variable, the distorted object- and image-space functions are defined on the Hilbert spaces with weighted inner products [18]

### 2.1. Infinite object and image

*z*

_{1}<

*z*<

*z*

_{2}, both support functions

*b*(

_{f}**r**,

*z*) and

*b*(

_{g}**r**

*) are identically equal to one. While this assumption is obviously not true, its effects on the resulting analysis are surprisingly small. Its main contribution is edge effects. Theoretically, the object and images are assumed to be infinite, but for the evaluation, finite object and images must be used. This leads to strange effects close to the edges, so the rim of the resulting images includes non-reliable results and should be discarded. The assumption simplifies the analysis considerably, since the system is now transversally shift-invariant and the kernels of the Hermitian operators may be written*

_{d}*k*

_{mm′}(

**r**

*−*

_{d}**r**

*′*) and

_{d}*p*(

**r̃**−

**r̃**′,

*z̃,z̃*′). Insertion of Eq. (8) into Eq. (13) yields, after performing the integral in

**r̃**,

*ρ*) is an

*M*×

*M*matrix with elements

*m*=

*m*′ or

*ρ*= (0,0), and

*m*≠

*m*′. Thus the elements of

**V**

*(*

_{j}*ρ*). The image-space singular functions have been found, in exactly the same manner as Burvall et al. [14]. The object-space eigenfunctions can then be found by projection of the image-space eigenfunctions using the adjoint operator in Eq. (10).

*Û*(

*ρ,z̃*)(

*s*−

*z̃*)

^{2}=

*U*(

*ρ,z̃*) we find the integral equation

**r̃**′,

*z̃*′) in the object will cause a distribution

*p*(

**r̃**−

**r̃**′,

*z̃,z̃*′) in the reconstruction. The impulse response function is a sum of

*M*lines whose intensity varies with

*z̃*, and that all meet at

**r̃**=

**r̃**′,

*z̃*=

*z̃*′.

## 3. Numerical results

### 3.1. Numerical SVD

*s*= 0.3 m,

*z*

_{1}= −0.1 m, and

*z*

_{2}= −0.01 m. Furthermore, it is assumed that all the sources are placed along a line or arc rather than in a 2D array. While it would be interesting to investigate other arrangements, this option simplifies the analysis significantly and is often employed in tomosynthesis. Assuming the source distribution is along the

*x*axis, we have

**r**

*= (ξ*

_{m}*, 0). The outermost source positions are at*

_{m}*χ*= ±0.1 m, and the source positions evenly distributed in between. The examples below are for 11 source positions, i.e.,

_{m}*M*= 11.

**r**

*= (*

_{m}*χ*, 0) is used to simplify the analysis by reducing large parts of it from 3D to 2D. A closer look at Eq. (18) reveals that it depends on

_{m}*ρ*only as

*ρ*· (

**r**

*−*

_{m}**r**

_{m′}) =

*ρ*(

_{x}*χ*−

_{m}*χ*

_{m′}), so the elements of

*ρ*and not on

_{x}*ρ*. Comparison to Eq. (16) shows that

_{y}**V**

*(*

_{j}*ρ*) =

**V**

*(*

_{j}*ρ*) or

_{x}*μ*=

_{ρ,j}*μ*will not depend on

_{ρx,j}*ρ*either.

_{y}*U*(

_{ρ,j}**r̃**,

*z̃*) can be found from the projection

*U*(

_{j}*ρ,z̃*) =

*U*(

_{j}*ρ*) only needs to be calculated for

_{x},z̃*ρ*= (

*ρ*, 0) and then the same 2D matrix can be used for all

_{x}*ρ*. Figure 3 shows the eigenvalues

_{y}*μ*while Fig. 4 contains the absolute values of some of the singular functions

_{ρx,j}*U*(

_{j}*ρ*). We note that the eigenvalues are approximately zero for low frequencies, and that this band of zero eigenvalues grows broader with increasing

_{x},z̃*j*. The eigenvalues are a combination of discrete and continuous spectra: discrete in

*j*which handles the axial dimension, but continuous in

*ρ*which handles the transverse dimensions.

_{x}*z̃*) and frequency (

*ρ*). It is easier to interpret the expansion of an example object into these eigenfunctions, i.e., its measurement component. This requires a relevant sample object.

### 3.2. Sample object

*A*is the amplitude of the nodule,

*R*its radius, and

*v*a real and non-negative shape factor that determines the sharpness of the edges. We have chosen

*R*= 3mm,

*A*= 0.03, and

*v*= 1.5. This value of

*v*is reported to give the best fit to average tumor profiles [21]. The 3D object that gives this projection is [22

22. I. Reiser and R. M. Nishikawa, “Task-based assessment of breast tomosynthesis: Effects of acquisition parameters and quantum noise,” Med. Phys. **37**, 1591–1600 (2010). [PubMed]

*K*is the numbers of clusters, each centered on

**r**

*, and*

_{k}*N*is the number of blobs or lumps in each cluster, their positions within the cluster given by

_{k}**r**

*. The function*

_{kn}*b*describes the shape of each blob,

*a*its strength, and

_{kn}**R**

_{θ}the rotation matrix of the random angle

*θ*. The theory of this 2D background is thoroughly described by Bochud et al. [23]. However, no proper 3D model exists and we need to model the 3D object rather than its projection. Since the intention in the current paper is only to provide an illustration, we have extended the model in the simplest possible way. The lumps are made three-dimensional, and placed in a volume by the 3D vectors

_{kn}**r**

*and*

_{k}**r**

*. They were also rotated in 3D rather than in 2D. Apart from that, we followed the classic CLB as introduced by Bochud [23]. The number*

_{kn}*K*was a Poisson random variable with expectation value 150 and

*N*a Poisson random variable with expectation value 20. The scale

_{k}*a*was set to 1 for all blobs, and the position of the cluster

_{kn}**r**

*a uniform random variable within the object area which is 40 × 40 × 90mm. The position within the cluster*

_{k}**r**

*is a Gaussian with standard deviation 3.6mm. The function*

_{kn}*b*is an asymmetrical exponential blob

*b*(

**r**,

**R**

*) = exp(−*

_{θkn}*α*|

**R**

_{θkn}

**r**|

^{β}/

*L*(

**R**

_{θkn}

**r**)) as given by Bochud [23], of length

*L*= 15mm and width

_{x}*L*=

_{y}*L*= 6mm. Parameters of the blob function are

_{z}*α*= 2.1 and

*β*= 0.5. The blob is rotated an angle

*θ*=

_{kn}*θ*(uniform random variable on [0,2

_{k}*π*]) in the xy-plane, and an angle

*ϕ*=

_{kn}*ϕ*(uniform random variable on [0,

_{k}*π*]) in the xz-plane.

*f ̃*(

**r̃**,

*z̃*) is shown in Fig. 5, in the distorted coordinates

**r̃**and

*z̃*. It is shown in two slices: one through the middle of the designer nodule in the

*x̃ỹ*plane, and one through the middle of the designer nodule in the

*x̃z̃*plane. The object is not a proper 3D breast model, since it is not connected enough for a breast CT slice [24

24. K. G. Metheany, C. K. Abbey, N. Packard, and J. M. Boone, “Characterizing anatomical variability in breast CT images,” Med. Phys. **35**, 4685–4694 (2008). [PubMed]

### 3.3. Numerical results

*A*(

_{j}*ρ*) are found as

*F̃*(

*ρ,z̃*) is the 2D Fourier transform of the object

*f̃*(

**r̃**,

*z̃*) with respect to

**r̃**. Two things should be noted, as they differ from the expression used in Ref. [14]. First, the complex conjugate in Eq. 23 was omitted in [14] since the singular functions were real, but must be included here. Second, the weighting function

*w*(

*x̃,z̃*) =

*s*

^{4}/(

*s*−

*z̃*)

^{4}follows from the earlier change of variables. Slices through the measurement component are shown in Fig. 6. The slices are taken at the same positions as those in Fig. 5. In part (b), the low axial resolution is showing up as elongation of both nodule and background.

## 4. Discussion and future work

27. S. Park, J. M. Witten, and K. J. Myers, “Singular vectors of a linear imaging system as efficient channels for the bayesian ideal observer,” IEEE Trans. Med. Imaging **28**, 657–668 (2009). [PubMed]

## 5. Conclusions

## Acknowledgments

## References and links

1. | J. T. Dobbins III and D. J. Godfrey, “Digital x-ray tomosynthesis: current state of the art and clinical potential,” Phys. Med. Biol. |

2. | J. T. Dobbins III, “Tomosynthesis: at translational crossroads,” Med. Phys. |

3. | L. T. Niklason, “Digital tomosynthesis in breast imaging,” Radiology |

4. | G. Gennaro, A. Toledano, C. di Maggio, E. Baldan, E. Bezzon, M. La Grassa, L. Pescarini, I. Polico, A. Proietti, A. Toffoli, and P. C. Muzzio, “Digital breast tomosynthesis versus digital mammography: a clinical performance study,” Eur. Radiol. (2009). [PubMed] |

5. | I. Andersson, D. M. Ikeda, S. Zackrisson, M. Ruschin, T. Svahn, P. Timberg, and A. Tingberg, “Breast tomosynthesis and digital mammography: a comparison of breast cancer visibility and BIRADS classification in a population of cancers with subtle mammographic findings,” Eur. Radiol. |

6. | W. F. Good, G. S. Abrams, V. J. Catullo, D. M. Chough, M. A. Ganott, C. M. Hakim, and D. Gur, “Digital breast tomosynthesis: a pilot observer study,” Am. J. Radiology |

7. | S. P. Poplack, T. D. Tosteson, C. A. Kogel, and H. M. Nagy, “Digital breast tomosyntheis: Initial experiance in 98 women with abnormal digital screening mammography,” Am. J. Radiology |

8. | A. S. Chawla, J. Y. Lo, J. A. Baker, and E. Samei, “Optimized image acquisition for breast tomosynthesis in projection and reconstruction space,” Med. Phys. |

9. | T. Wu, R. H. Moore, E. A. Rafferty, and D. B. Kopans, “A comparison of reconstruction algorithms for breast tomosynthesis,” Med. Phys. |

10. | Y. Zhang, H.-P. Chan, B. Sahiner, J. Wei, M. M. Goodsitt, L. M. Hadjiiiski, J. Ge, and C. Zhou, “A comparative study of limited-angle cone-beam reconstruction methods for breast tomosynthesis,” Med. Phys. |

11. | H. H. Barrett and K. J. Myers, |

12. | M. Bertero and P. Boccacci, |

13. | H. H. Barrett, J. N. Aarsvold, and T. J. Roney, “Null functions and eigenfunctions: tools for the analysis of imaging systems,” Lect. Notes. Comput. Sci. |

14. | A. Burvall, H. H. Barrett, C. Dainty, and K. J. Myers, “Singular-value decomposition for through-focus imaging systems,” J. Opt. Soc. Am. A |

15. | J. Yao and H. H. Barrett, “Predicting human performance by a channelized Hotelling observer,” Proc. SPIE |

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. | M. Y. Chiu, H. H. Barrett, R. G. Simpson, C. Chou, J. W. Arendt, and G. R. Gindi, “Three-dimensional radio-graphic imaging with a restricted view angle,” J. Opt. Soc. Am. |

18. | R. Pierri, A. Liseno, F. Soldovieri, and R. Solimene, “In-depth resolution for a strip source in the Fresnel zone,” J. Opt. Soc. Am. A |

19. | A. D. Polianin and A. V. Manzhirov, |

20. | C. Lanczos, |

21. | A. E. Burgess, “Visual signal detection with two-component noise: low-pass spectrum effects,” J. Opt. Soc. Am. A |

22. | I. Reiser and R. M. Nishikawa, “Task-based assessment of breast tomosynthesis: Effects of acquisition parameters and quantum noise,” Med. Phys. |

23. | F. O. Bochud, C. K. Abbey, and M. P. Eckstein, “Statistical texture synthesis of mammographic images with clustered lumpy backgrounds,” Opt. Express |

24. | K. G. Metheany, C. K. Abbey, N. Packard, and J. M. Boone, “Characterizing anatomical variability in breast CT images,” Med. Phys. |

25. | C. K. Abbey and J. M. Boone, “An ideal observer for a model of x-ray imaging in breast parenchymal tissue,” (E.A. Krupinski, Ed.): IWDM 2008, LNCS5116, 393–400 (2008). |

26. | C. Zhang, P. R. Bakic, and A. D. A. Maidment, “Development of an anthropomorphic breast software phantom based on region growing algorithm,” Proc. SPIE |

27. | S. Park, J. M. Witten, and K. J. Myers, “Singular vectors of a linear imaging system as efficient channels for the bayesian ideal observer,” IEEE Trans. Med. Imaging |

**OCIS Codes**

(000.1430) General : Biology and medicine

(100.3190) Image processing : Inverse problems

(340.7440) X-ray optics : X-ray imaging

**ToC Category:**

Image Processing

**History**

Original Manuscript: July 12, 2010

Revised Manuscript: September 8, 2010

Manuscript Accepted: September 10, 2010

Published: September 15, 2010

**Virtual Issues**

Vol. 5, Iss. 14 *Virtual Journal for Biomedical Optics*

**Citation**

Anna Burvall, Harrison H. Barrett, Kyle J. Myers, and Christopher Dainty, "Singular-value decomposition of a tomosynthesis system," Opt. Express **18**, 20699-20711 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-20-20699

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### References

- J. T. DobbinsIII, and D. J. Godfrey, "Digital x-ray tomosynthesis: current state of the art and clinical potential," Phys. Med. Biol. 48, R65-R106 (2003). [PubMed]
- J. T. DobbinsIII, "Tomosynthesis: at translational crossroads," Med. Phys. 36, 1956-1967 (2009). [PubMed]
- L. T. Niklason, "Digital tomosynthesis in breast imaging," Radiology 1997, 399-406 (1997).
- G. Gennaro, A. Toledano, C. di Maggio, E. Baldan, E. Bezzon, M. La Grassa, L. Pescarini, I. Polico, A. Proietti, A. Toffoli, and P. C. Muzzio, "Digital breast tomosynthesis versus digital mammography: a clinical performance study," Eur. Radiol. (2009), doi:10.1007/s00330-009-1699-5. [PubMed]
- I. Andersson, D. M. Ikeda, S. Zackrisson, M. Ruschin, T. Svahn, P. Timberg, and A. Tingberg, "Breast tomosynthesis and digital mammography: a comparison of breast cancer visibility and BIRADS classification in a population of cancers with subtle mammographic findings," Eur. Radiol. 18, 2817-2825 (2008). [PubMed]
- W. F. Good, G. S. Abrams, V. J. Catullo, D. M. Chough, M. A. Ganott, C. M. Hakim, and D. Gur, "Digital breast tomosynthesis: a pilot observer study," Am. J. Radiology 190, 865-869 (2008).
- S. P. Poplack, T. D. Tosteson, C. A. Kogel, and H. M. Nagy, "Digital breast tomosynthesis: Initial experience in 98 women with abnormal digital screening mammography," Am. J. Radiology 189, 616-623 (2007).
- A. S. Chawla, J. Y. Lo, J. A. Baker, and E. Samei, "Optimized image acquisition for breast tomosynthesis in projection and reconstruction space," Med. Phys. 36, 4859-4869 (2009). [PubMed]
- T. Wu, R. H. Moore, E. A. Rafferty, and D. B. Kopans, "A comparison of reconstruction algorithms for breast tomosynthesis," Med. Phys. 31, 2636-2647 (2004). [PubMed]
- Y. Zhang, H.-P. Chan, B. Sahiner, J. Wei, M. M. Goodsitt, L. M. Hadjiiiski, J. Ge, and C. Zhou, "A comparative study of limited-angle cone-beam reconstruction methods for breast tomosynthesis," Med. Phys. 33, 3781-3795 (2006). [PubMed]
- H. H. Barrett, and K. J. Myers, Foundations of Image Science (John Wiley, Hoboken, New Jersey, 2004).
- M. Bertero, and P. Boccacci, Inverse Problems in Imaging (Institute of Physics Publishing, Bristol, UK, 1998).
- H. H. Barrett, J. N. Aarsvold, and T. J. Roney, "Null functions and eigenfunctions: tools for the analysis of imaging systems," Lect. Notes Comput. Sci. 11, 211-226 (1991).
- A. Burvall, H. H. Barrett, C. Dainty, and K. J. Myers, "Singular-value decomposition for through-focus imaging systems," J. Opt. Soc. Am. A 23, 2440-2448 (2006).
- J. Yao, and H. H. Barrett, "Predicting human performance by a channelized Hotelling observer," Proc. SPIE 1768, 161-168 (1992).
- H. H. Barrett, J. Yao, J. P. Rolland, and K. J. Myers, "Model observers for assessment of image quality," Proc. Natl. Acad. Sci. U.S.A. 90, 9758-9765 (1993). [PubMed]
- M. Y. Chiu, H. H. Barrett, R. G. Simpson, C. Chou, J. W. Arendt, and G. R. Gindi, "Three-dimensional radiographic imaging with a restricted view angle," J. Opt. Soc. Am. 69, 1323-1333 (1979).
- R. Pierri, A. Liseno, F. Soldovieri, and R. Solimene, "In-depth resolution for a strip source in the Fresnel zone," J. Opt. Soc. Am. A 18, 352-359 (2001).
- A. D. Polianin, and A. V. Manzhirov, Handbook of integral equations (CRC Press, Florida, 1998) chapter 11.2.
- C. Lanczos, Linear differential operators (Van Nostrand, London, 1961).
- A. E. Burgess, "Visual signal detection with two-component noise: low-pass spectrum effects," J. Opt. Soc. Am. A 16, 694-704 (1999).
- I. Reiser, and R. M. Nishikawa, "Task-based assessment of breast tomosynthesis: Effects of acquisition parameters and quantum noise," Med. Phys. 37, 1591-1600 (2010). [PubMed]
- F. O. Bochud, C. K. Abbey, and M. P. Eckstein, "Statistical texture synthesis of mammographic images with clustered lumpy backgrounds," Opt. Express 4, 33-43 (1998).
- K. G. Metheany, C. K. Abbey, N. Packard, and J. M. Boone, "Characterizing anatomical variability in breast CT images," Med. Phys. 35, 4685-4694 (2008). [PubMed]
- C. K. Abbey, and J. M. Boone, "An ideal observer for a model of x-ray imaging in breast parenchymal tissue," (E.A. Krupinski, Ed.): IWDM 2008, LNCS 5116, 393-400 (2008).
- C. Zhang, P. R. Bakic, and A. D. A. Maidment, "Development of an anthropomorphic breast software phantom based on region growing algorithm," Proc. SPIE 6918, 69180V (2008).
- S. Park, J. M. Witten, and K. J. Myers, "Singular vectors of a linear imaging system as efficient channels for the Bayesian ideal observer," IEEE Trans. Med. Imaging 28, 657-668 (2009). [PubMed]

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