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.

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  1. 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]
  2. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company Publishers, 2005).
  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]
  5. H. W. Jensen, Realistic image synthesis using photon mapping (AK Peters, Ltd., 2001).
  6. T. Whitted, “An improved illumination model for shaded display,” Communications of the ACM23, 343–349 (1980).
    [CrossRef]
  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]
  8. M. Pharr and G. Humphreys, Physically based rendering: From theory to implementation (Morgan Kaufmann, 2010).
  9. C. Lemieux, Monte Carlo and Quasi-Monte Carlo Sampling (Springer, 2009).
  10. P. Shirley, “Discrepancy as a quality measure for sample distributions,” Proceedings of Eurographics91, 183–193 (1991).
  11. 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).
  12. P. Shirley and R. K. Morley, Realistic ray tracing (AK Peters, Ltd., 2003).
  13. J. S. Warren, Modern Lens Design (Washington: SPIE Press, 2005).
  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]

2010 (2)

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]

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]

1991 (1)

P. Shirley, “Discrepancy as a quality measure for sample distributions,” Proceedings of Eurographics91, 183–193 (1991).

1986 (1)

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

1985 (1)

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

1980 (1)

T. Whitted, “An improved illumination model for shaded display,” Communications of the ACM23, 343–349 (1980).
[CrossRef]

1954 (1)

Coltman, J. W.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company Publishers, 2005).

Hanrahan, P.

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).

Horiuchi, S.

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]

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]

Humphreys, G.

M. Pharr and G. Humphreys, Physically based rendering: From theory to implementation (Morgan Kaufmann, 2010).

Jensen, H. W.

H. W. Jensen, Realistic image synthesis using photon mapping (AK Peters, Ltd., 2001).

Kajiya, J. T.

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

Kolb, C.

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).

Lemieux, C.

C. Lemieux, Monte Carlo and Quasi-Monte Carlo Sampling (Springer, 2009).

Mitchell, D.

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).

Morley, R. K.

P. Shirley and R. K. Morley, Realistic ray tracing (AK Peters, Ltd., 2003).

Nakamae, E.

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

Nishita, T.

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

Okamura, I.

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

Pharr, M.

M. Pharr and G. Humphreys, Physically based rendering: From theory to implementation (Morgan Kaufmann, 2010).

Shirley, P.

P. Shirley, “Discrepancy as a quality measure for sample distributions,” Proceedings of Eurographics91, 183–193 (1991).

P. Shirley and R. K. Morley, Realistic ray tracing (AK Peters, Ltd., 2003).

Ushiyama, Z.

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]

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]

Warren, J. S.

J. S. Warren, Modern Lens Design (Washington: SPIE Press, 2005).

Whitted, T.

T. Whitted, “An improved illumination model for shaded display,” Communications of the ACM23, 343–349 (1980).
[CrossRef]

Yamamoto, M.

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]

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]

Yoshida, S.

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]

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]

ACM Trans. Graph. (1)

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

Communications of the ACM (1)

T. Whitted, “An improved illumination model for shaded display,” Communications of the ACM23, 343–349 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Express (1)

Opt. Quant. Electron. (1)

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]

Proceedings of Eurographics (1)

P. Shirley, “Discrepancy as a quality measure for sample distributions,” Proceedings of Eurographics91, 183–193 (1991).

SIGGRAPH 1986 (1)

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

Other (7)

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).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company Publishers, 2005).

H. W. Jensen, Realistic image synthesis using photon mapping (AK Peters, Ltd., 2001).

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).

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Figures (10)

Fig. 1
Fig. 1

Simulation method based on illumination optics. The number of light rays emitted from a test chart illuminated by a light source is determined by Monte Carlo methods. The irradiance is obtained from the number of light rays that arrive at the image surface via the optical imaging system, and the MTF is derived from the irradiance distribution.

Fig. 2
Fig. 2

Simulation method using rendering. The radiance of pixels is obtained by ray tracing from the light receiving surface instead of the light source. One-dimensional ray tracing for each pixel on the light receiving surface is used to obtain the one-dimensional radiance distribution necessary to calculate the MTF.

Fig. 3
Fig. 3

Block diagram of the MTF analysis simulation using rendering. The test chart (position, spatial frequency, type) and film (size, position, number of pixels) are set and then the pixels and rear-most element of the optical imaging systems are sampled. The light rays generated by the sampling are traced, and the radiance distribution is obtained from the rays that arrive at the test chart.

Fig. 4
Fig. 4

Coordinate system for an optical system with a test chart.

Fig. 5
Fig. 5

Simulation algorithm.

Fig. 6
Fig. 6

Specifications of the Tessar lens. Quantities nd and νd are the refractive index and Abbe number at the d-line (587.6 nm), respectively, and ap is the aperture size. Units are in mm.

Fig. 7
Fig. 7

Simulated images and radiance distributions of square-wave and sine-wave charts at various spatial frequencies. The spatial frequencies are (a) 6 lp/mm, (b) 30 lp/mm, and (c) 50 lp/mm. For the radiance distribution, the solid and dashed lines are those of the square-wave and sine-wave charts, respectively.

Fig. 8
Fig. 8

Optical system with a rod lens and the refractive index distribution of the GRIN lens.

Fig. 9
Fig. 9

Simulated images and radiance distributions of the square-wave and sine-wave charts using a rod lens. The spatial frequency is 6 lp/mm. For the radiance distribution, the solid and dashed lines are those of the square-wave and sine-wave charts, respectively.

Fig. 10
Fig. 10

Radiance distributions of the square-wave charts through a rod lens. The spatial frequency is 6 lp/mm. Solid and dashed lines are results of the conventional method based on illumination optics and the proposed method using rendering, respectively.

Equations (10)

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OTF ( f s , f t ) = I ( u , v ) exp { i 2 π ( f s u + f t v ) } d u d v I ( u , v ) d u d v .
MTF ( f s , f t ) = | OTF ( f s , f t ) | = | PSF ( u , v ) exp { i 2 π ( f s u + f t v ) } d u d v | .
MTF = I max I min I max + I min .
L o ( p , ω o ) = L e ( p , ω o ) + L r ( p , ω o ) .
L r ( p , ω o ) = Ω f r ( p , ω o , ω i ) L i ( p , ω i ) ( ω i n ) d ω i .
L r ( p , ω o ) = f r ( p , ω o , ω i ) L i ( p , ω i ) ( ω i n ) V ( p , ω i ) .
L r ( p , ω o ) = f r ( p , ω o , ω i ) ( ω i n ) V ( p ) L i .
L r ( p , ω o ) = ( ω i n ) V ( p ) L i .
O ( x ) = 1 2 { cos ( 2 π f x ϕ ) + 1 } .
O ( x ) = { 0 ( T 2 + a < x a ) 1 ( a < x T 2 + a ) .

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