## Simulation of modulation transfer function using a rendering method |

Optics Express, Vol. 21, Issue 6, pp. 7373-7383 (2013)

http://dx.doi.org/10.1364/OE.21.007373

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### Abstract

We propose a novel simulation method based on rendering to evaluate the modulation transfer function (MTF) of optical imaging systems. The new simulation method corresponds to an experimental measurement of the MTF using imaging resolution test charts, and therefore allows an analysis of the resolving power of shift-variant optical systems that are difficult to evaluate with conventional methods based on the point spread function (PSF). Furthermore, the effects of stray light, such as from reflection or scattering, on the imaging performance can be analyzed. In contrast to methods based on illumination optics using Monte Carlo methods, the proposed method calculates the intensities on an image surface with rendering techniques used in three-dimensional computer graphics (3D CG), which results in calculations that are faster and have a higher precision. The proposed method is highly effective in analyzing the MTF of optical imaging systems through simulations.

© 2013 OSA

## 1. Introduction

## 2. Conventional methods

1. S. Yoshida, S. Horiuchi, Z. Ushiyama, and M. Yamamoto, “A numerical analysis method for evaluating rod lenses using the Monte Carlo method,” Opt. Express **18**, 27016–27027 (2010) [CrossRef] .

### 2.1. Obtaining the transfer function using the PSF

*I*is the intensity distribution in the

*u*–

*v*plane, and

*f*and

_{s}*f*are the spatial frequencies in the sagittal and tangential directions, respectively. The absolute value of the OTF is the MTF, which can be written in terms of the PSF as In simulations of geometrical optics, rays of light from a point source are traced to the image plane, and the spot diagram in the image plane is used to calculate the MTF.

_{t}### 2.2. Method based on illumination optics

## 3. New method using rendering

### 3.1. Formalism using the rendering equation

3. J. T. Kajiya, “The rendering equation,” SIGGRAPH 1986 , 143–150 (1986) [CrossRef] .

*L*,

_{o}*L*, and

_{e}*L*are the outgoing, emitted, and reflected radiances, respectively,

_{r}**p**is the spatial coordinate, and

*ω**is a vector in the outgoing direction. The reflected radiance can be described as the integral of the incident radiance on a hemisphere Ω at a point*

_{o}**p**: Here,

*L*is the incident radiance,

_{i}*f*is the bidirectional reflectance distribution function (BRDF),

_{r}

*ω**is a vector in the incident direction, and*

_{i}**n**is the normal vector. Many methods for solving the rendering equation have been proposed. Representative methods are the path tracing [3

3. J. T. Kajiya, “The rendering equation,” SIGGRAPH 1986 , 143–150 (1986) [CrossRef] .

4. T. Nishita, I. Okamura, and E. Nakamae, “Shading models for point and linear sources,” ACM Trans. Graph. **4**, 124–146 (1985) [CrossRef] .

6. T. Whitted, “An improved illumination model for shaded display,” Communications of the ACM **23**, 343–349 (1980) [CrossRef] .

*V*is the visibility function, which is 1 if the light source is visible from the observation point and 0 otherwise. Whitted’s recursive ray tracing method in Eq. (6) can be simplified further when applied to optical systems with test charts. If the test chart is uniformly irradiated, then the incident radiance

*L*is constant and there is no irregularity in the radiance. If the visibility function

_{i}*V*is defined as 1 on the test chart and 0 (vignetted) elsewhere, then

*L*is always constant, and therefore If the illumination from the light source is considered as Lambertian diffuse lighting model, the BRDF is 1 regardless of the direction vector, and hence Direct light from the light source does not exist in the optical system for test charts, and so the emitted radiance

_{i}*L*does not need to be considered. This reduces the rendering equation in optical systems with test charts to Eq. (8). The incident radiance

_{e}*L*can change with the radiant flux and distance to the test chart; however

_{i}*L*can be set to 1 when only the contrast ratio is relevant for obtaining the MTF. Furthermore, taking the direction normal to the test chart as the

_{i}*z*direction (

**n**= {0,0,1}) means that the necessary reflected radiance is determined only by the

*z*component of the incident light rays arriving at the test chart. In summary, the rendering equation becomes very simple in optical systems for test charts.

### 3.2. Test chart

7. J. W. Coltman, “The Specification of Imaging Properties by Response to a Sine Wave Input,” J. Opt. Soc. Am. **44**, 468–469 (1954) [CrossRef] .

*x*direction of a sine-wave chart is given as follows. Here,

*f*is the spatial frequency and

*ϕ*is the phase difference (brightness at each position). The square-wave chart can be denoted as follows in piecewise form. Here,

*T*is the period (

*T*= 1/

*f*), and

*a*is the position. The periodicity condition

*O*(

*x*+

*T*) =

*O*(

*x*) is satisfied.

### 3.3. Sampling and simulation procedure

*et al.*[11] covered 3D CG rendering through a camera lens, its description in a simplified model, and discussed sampling in detail. Appropriate filters must be used when considering the properties of the charge-coupled device (CCD) or film used in experimental optical systems. Various filters have been proposed to model the film properties [8, 12]; however, the effects of the filter are not thought to affect the results significantly in the test-chart optical systems in this paper.

## 4. Results

14. S. Horiuchi, S. Yoshida, Z. Ushiyama, and M. Yamamoto, “Performance evaluation of GRIN lenses by ray tracing method,” Opt. Quant. Electron. **42**, 81–88 (2010) [CrossRef] .

*y*axis is 0.6 mm.

## 5. Conclusion

## References and links

1. | S. Yoshida, S. Horiuchi, Z. Ushiyama, and M. Yamamoto, “A numerical analysis method for evaluating rod lenses using the Monte Carlo method,” Opt. Express |

2. | J. W. Goodman, |

3. | J. T. Kajiya, “The rendering equation,” SIGGRAPH 1986 , 143–150 (1986) [CrossRef] . |

4. | T. Nishita, I. Okamura, and E. Nakamae, “Shading models for point and linear sources,” ACM Trans. Graph. |

5. | H. W. Jensen, |

6. | T. Whitted, “An improved illumination model for shaded display,” Communications of the ACM |

7. | J. W. Coltman, “The Specification of Imaging Properties by Response to a Sine Wave Input,” J. Opt. Soc. Am. |

8. | M. Pharr and G. Humphreys, |

9. | C. Lemieux, |

10. | P. Shirley, “Discrepancy as a quality measure for sample distributions,” Proceedings of Eurographics |

11. | C. Kolb, D. Mitchell, and P. Hanrahan, “A realistic camera model for computer graphics,” |

12. | P. Shirley and R. K. Morley, |

13. | J. S. Warren, |

14. | S. Horiuchi, S. Yoshida, Z. Ushiyama, and M. Yamamoto, “Performance evaluation of GRIN lenses by ray tracing method,” Opt. Quant. Electron. |

**OCIS Codes**

(110.0110) Imaging systems : Imaging systems

(110.2760) Imaging systems : Gradient-index lenses

(110.4100) Imaging systems : Modulation transfer function

(330.1715) Vision, color, and visual optics : Color, rendering and metamerism

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: November 23, 2012

Revised Manuscript: March 5, 2013

Manuscript Accepted: March 7, 2013

Published: March 18, 2013

**Virtual Issues**

Vol. 8, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Shuma Horiuchi, Shuhei Yoshida, and Manabu Yamamoto, "Simulation of modulation transfer function using a rendering method," Opt. Express **21**, 7373-7383 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-6-7373

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### References

- S. Yoshida, S. Horiuchi, Z. Ushiyama, and M. Yamamoto, “A numerical analysis method for evaluating rod lenses using the Monte Carlo method,” Opt. Express18, 27016–27027 (2010). [CrossRef]
- J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company Publishers, 2005).
- J. T. Kajiya, “The rendering equation,” SIGGRAPH 1986, 143–150 (1986). [CrossRef]
- T. Nishita, I. Okamura, and E. Nakamae, “Shading models for point and linear sources,” ACM Trans. Graph.4, 124–146 (1985). [CrossRef]
- H. W. Jensen, Realistic image synthesis using photon mapping (AK Peters, Ltd., 2001).
- T. Whitted, “An improved illumination model for shaded display,” Communications of the ACM23, 343–349 (1980). [CrossRef]
- J. W. Coltman, “The Specification of Imaging Properties by Response to a Sine Wave Input,” J. Opt. Soc. Am.44, 468–469 (1954). [CrossRef]
- M. Pharr and G. Humphreys, Physically based rendering: From theory to implementation (Morgan Kaufmann, 2010).
- C. Lemieux, Monte Carlo and Quasi-Monte Carlo Sampling (Springer, 2009).
- P. Shirley, “Discrepancy as a quality measure for sample distributions,” Proceedings of Eurographics91, 183–193 (1991).
- C. Kolb, D. Mitchell, and P. Hanrahan, “A realistic camera model for computer graphics,” Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, 317–324 (1995).
- P. Shirley and R. K. Morley, Realistic ray tracing (AK Peters, Ltd., 2003).
- J. S. Warren, Modern Lens Design (Washington: SPIE Press, 2005).
- S. Horiuchi, S. Yoshida, Z. Ushiyama, and M. Yamamoto, “Performance evaluation of GRIN lenses by ray tracing method,” Opt. Quant. Electron.42, 81–88 (2010). [CrossRef]

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