## Transformation of characteristic functionals through imaging systems

Optics Express, Vol. 10, Issue 13, pp. 536-539 (2002)

http://dx.doi.org/10.1364/OE.10.000536

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### Abstract

We describe how to transfer the characteristic functional of an object model through a noisy, discrete imaging system to arrive at the characteristic function of the images. Our method can also incorporate linear post-processing of the images.

© 2002 Optical Society of America

## 1 Introduction

1. H. H. Barrett, “Objective assessment of image quality: Effects of quantum noise and object variability,” J. Opt. Soc. Am. A , **7**, 1266–1278 (1990). [CrossRef] [PubMed]

## 2 Method

*is a sample object from the ensemble of functions that are being imaged,*

**f***H*is the system operator which maps a continuous function to a discrete image,

*is the noise in the imaging system, the statistics of which may depend on*

**n***, and*

**f***is the image. We assume that the mean image*

**g***for a fixed object*

**ḡ***is given by*

**f***the noiseless image.*

**ḡ**### 2.1 Noiseless Imaging Systems

*represents the Fourier conjugate of the function*

**ξ***. For now, let us envision a noiseless imaging system. The characteristic function of*

**f***is*

**ḡ***represents the expectation with respect to the PDF of*

**ḡ***, and*

**ḡ***is the Fourier conjugate of*

**ρ***. By using Eq. (2), the properties of the adjoint, and standard rules for transforming expectations, we can rewrite Eq. (4) as*

**ḡ***is known, then we also know the CF of any linear mapping of*

**f***by simply using the adjoint of the linear operator.*

**f**### 2.2 Noisy Imaging Systems

*=*

**g***+*

**ḡ***accounting for both object variability and noise. Two common noise models that researchers employ are Gaussian noise and quantum or Poisson noise.*

**n***K*is known to be Gaussian shaped as well with the form

*is a convolution of the PDF of*

**g***and the PDF of*

**ḡ***, which, using the Fourier-convolution theorem, yields,*

**n***i.e*.,

*ḡ*denotes the

_{m}*m*th component of the

*M*-vector

*. The probability of*

**ḡ***can then be obtained by marginalizing over the mean image*

**g***,*

**ḡ***is,*

**g***indicates a sum over all components of*

**g***from 0 to ∞,*

**g***except that the term in the exponent is not the same. We can relate the above expression to the CF of*

**ḡ***by defining a nonlinear operator*

**ḡ***(·) which maps an*

**Γ***M*vector to another

*M*vector using the following equation for each component

*m*,

*H*

^{†}and a known nonlinear operator to determine the CF for our noisy image data.

### 2.3 Filter Outputs

*ν*=

*T*=

_{g}*T*(

*H*+

_{f}*), you need only the adjoint*

**n***T*

^{†}. That is,

*is the Fourier conjugate of the filter outputs*

**ω***.*

**ν***T*could be Laguerre-Gauss channels for signal-detection tasks [3

3. H. H. Barrett, C. Abbey, B. Gallas, and M. Eckstein, “Stabilized estimates of Hotelling-observer detection performance in patient-structured noise,” in *SPIE Medical Imaging: Image Perception*, ed. H. L. Kundel, Proc. SPIE **3340**, 27–43 (1998). [CrossRef]

*as the sinogram data from a tomographic imaging system, and the*

**g***T*as a linear reconstruction operator. With this latter viewpoint, we arrive at the CF for the reconstructed images.

## 3 Conclusions

## Acknowledgments

## References and links

1. | H. H. Barrett, “Objective assessment of image quality: Effects of quantum noise and object variability,” J. Opt. Soc. Am. A , |

2. | A. Papoulis, |

3. | H. H. Barrett, C. Abbey, B. Gallas, and M. Eckstein, “Stabilized estimates of Hotelling-observer detection performance in patient-structured noise,” in |

**OCIS Codes**

(030.4280) Coherence and statistical optics : Noise in imaging systems

(110.2960) Imaging systems : Image analysis

(110.3000) Imaging systems : Image quality assessment

**ToC Category:**

Research Papers

**History**

Original Manuscript: April 16, 2002

Revised Manuscript: June 20, 2002

Published: July 1, 2002

**Citation**

Eric Clarkson, M. Kupinski, and H. Barrett, "Transformation of characteristic functionals through imaging systems," Opt. Express **10**, 536-539 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-13-536

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### References

- H. H. Barrett, �??Objective assessment of image quality: E.ects of quantum noise and object variability,�?? J. Opt. Soc. Am. A 7, 1266�??1278 (1990). [CrossRef] [PubMed]
- A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991).
- H. H. Barrett, C. Abbey, B. Gallas, and M. Eckstein, �??Stabilized estimates of Hotelling-observer detection performance in patient-structured noise,�?? in SPIE Medical Imaging: Image Perception, ed. H. L. Kundel, Proc. SPIE 3340, 27�??43 (1998). [CrossRef]

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