Abstract

We introduce a near eye light field display proposal to reconstruct a light field in high synthesis speed by utilizing the multi-layer light field display technology and human visual features. The resolution distribution of the reconstructed light field is set to be identical to human visual acuity which decreases with the increasing visual eccentricity. We compress the light field information by using different sampling rates in different visual eccentricity area. A new optimization method for the compressed light field is proposed, which dramatically reduces the amount of calculation. The results demonstrate that the acceleration of the proposed scheme is obvious and escalates when the spatial resolution increases. The synthesis scheme is verified and its key aspects are analyzed by simulation and an experimental prototype.

© 2017 Optical Society of America

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References

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  1. E. Peli, “Visual and optometric issues with head-mounted displays,” in IS & T/OSA Optics & Imaging in the Information Age, (The Society for Imaging Science and Technology, 1996), pp. 364–369.
  2. D. Lanman and D. Luebke, “Near-eye light field displays,” ACM Trans. Graph. 32(6), 1–10 (2013).
    [Crossref]
  3. B. T. Schowengerdt, H. G. Hoffman, C. M. Lee, C. D. Melville, and E. J. Seibel, “57.1: Near‐to‐Eye Display using Scanning Fiber Display Engine,” SID Symposium Digest Tech. Papers 41(1),848–851 (2010).
    [Crossref]
  4. F. C. Huang, K. Chen, and G. Wetzstein, “The light field stereoscope: immersive computer graphics via factored near-eye light field displays with focus cues[J],” ACM Trans. Graph. 34(4), 60 (2010).
  5. R. B. Welch, T. T. Blackmon, A. Liu, B. A. Mellers, and L. W. Stark, “The effects of pictorial realism, delay of visual feedback, and observer interactivity on the subjective sense of presence,” Presence (Camb. Mass.) 5(3), 263–273 (1996).
    [Crossref]
  6. S. Uno and M. Slater, “The sensitivity of presence to collision response,” in Virtual Reality Annual International Symposium (IEEE, 1997), pp. 95–103.
    [Crossref]
  7. A. H. Chan and A. J. Courtney, “Foveal acuity, peripheral acuity and search performance: A review,” Int. J. Industrial Ergonomics 18(2), 113–119 (1996).
    [Crossref]
  8. Y. Takaki, “High-density directional display for generating natural three-dimensional images,” Proc. IEEE 94(3), 654–663 (2006).
    [Crossref]
  9. D. Lanman, M. Hirsch, Y. Kim, and R. Raskar, “Content-adaptive parallax barriers: optimizing dual-layer 3D displays using low-rank light field factorization,” ACM Trans. Graph. 29(6), 1–10 (2010).
    [Crossref]
  10. N. Ho, P. Van Dooren, and V. Blondel, “Weighted nonnegative matrix factorization and face feature extraction,” Image Vis. Comput. 2007, 1–17 (2007).
  11. R. Rosén, Peripheral Vision: Adaptive Optics and Psychophysics (KTH Royal Institute of Technology, 2013).
  12. S. T. Chung, J. S. Mansfield, and G. E. Legge, “Psychophysics of reading. XVIII. The effect of print size on reading speed in normal peripheral vision,” Vision Res. 38(19), 2949–2962 (1998).
    [Crossref] [PubMed]
  13. Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
    [Crossref]
  14. A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32(5), 153 (2013).
    [Crossref]
  15. A. Maimone and H. Fuchs, “Computational augmented reality eyeglasses,” in Mixed and Augmented Reality (ISMAR) (IEEE, 2013), pp. 29–38.
  16. S. Pannasch, J. R. Helmert, K. Roth, A. K. Herbold, and H. Walter, “Visual fixation durations and saccade amplitudes: Shifting relationship in a variety of conditions,” J. Eye Mov. Res. 2(2), 1–19 (2008).
  17. K. Rayner, “Eye movements in reading and information processing: 20 years of research,” Psychol. Bull. 124(3), 372–422 (1998).
    [Crossref] [PubMed]

2013 (2)

D. Lanman and D. Luebke, “Near-eye light field displays,” ACM Trans. Graph. 32(6), 1–10 (2013).
[Crossref]

A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32(5), 153 (2013).
[Crossref]

2010 (2)

F. C. Huang, K. Chen, and G. Wetzstein, “The light field stereoscope: immersive computer graphics via factored near-eye light field displays with focus cues[J],” ACM Trans. Graph. 34(4), 60 (2010).

D. Lanman, M. Hirsch, Y. Kim, and R. Raskar, “Content-adaptive parallax barriers: optimizing dual-layer 3D displays using low-rank light field factorization,” ACM Trans. Graph. 29(6), 1–10 (2010).
[Crossref]

2008 (1)

S. Pannasch, J. R. Helmert, K. Roth, A. K. Herbold, and H. Walter, “Visual fixation durations and saccade amplitudes: Shifting relationship in a variety of conditions,” J. Eye Mov. Res. 2(2), 1–19 (2008).

2007 (1)

N. Ho, P. Van Dooren, and V. Blondel, “Weighted nonnegative matrix factorization and face feature extraction,” Image Vis. Comput. 2007, 1–17 (2007).

2006 (1)

Y. Takaki, “High-density directional display for generating natural three-dimensional images,” Proc. IEEE 94(3), 654–663 (2006).
[Crossref]

2000 (1)

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

1998 (2)

S. T. Chung, J. S. Mansfield, and G. E. Legge, “Psychophysics of reading. XVIII. The effect of print size on reading speed in normal peripheral vision,” Vision Res. 38(19), 2949–2962 (1998).
[Crossref] [PubMed]

K. Rayner, “Eye movements in reading and information processing: 20 years of research,” Psychol. Bull. 124(3), 372–422 (1998).
[Crossref] [PubMed]

1996 (2)

R. B. Welch, T. T. Blackmon, A. Liu, B. A. Mellers, and L. W. Stark, “The effects of pictorial realism, delay of visual feedback, and observer interactivity on the subjective sense of presence,” Presence (Camb. Mass.) 5(3), 263–273 (1996).
[Crossref]

A. H. Chan and A. J. Courtney, “Foveal acuity, peripheral acuity and search performance: A review,” Int. J. Industrial Ergonomics 18(2), 113–119 (1996).
[Crossref]

Blackmon, T. T.

R. B. Welch, T. T. Blackmon, A. Liu, B. A. Mellers, and L. W. Stark, “The effects of pictorial realism, delay of visual feedback, and observer interactivity on the subjective sense of presence,” Presence (Camb. Mass.) 5(3), 263–273 (1996).
[Crossref]

Blondel, V.

N. Ho, P. Van Dooren, and V. Blondel, “Weighted nonnegative matrix factorization and face feature extraction,” Image Vis. Comput. 2007, 1–17 (2007).

Chan, A. H.

A. H. Chan and A. J. Courtney, “Foveal acuity, peripheral acuity and search performance: A review,” Int. J. Industrial Ergonomics 18(2), 113–119 (1996).
[Crossref]

Chen, K.

F. C. Huang, K. Chen, and G. Wetzstein, “The light field stereoscope: immersive computer graphics via factored near-eye light field displays with focus cues[J],” ACM Trans. Graph. 34(4), 60 (2010).

Chung, S. T.

S. T. Chung, J. S. Mansfield, and G. E. Legge, “Psychophysics of reading. XVIII. The effect of print size on reading speed in normal peripheral vision,” Vision Res. 38(19), 2949–2962 (1998).
[Crossref] [PubMed]

Courtney, A. J.

A. H. Chan and A. J. Courtney, “Foveal acuity, peripheral acuity and search performance: A review,” Int. J. Industrial Ergonomics 18(2), 113–119 (1996).
[Crossref]

Fuchs, H.

A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32(5), 153 (2013).
[Crossref]

Helmert, J. R.

S. Pannasch, J. R. Helmert, K. Roth, A. K. Herbold, and H. Walter, “Visual fixation durations and saccade amplitudes: Shifting relationship in a variety of conditions,” J. Eye Mov. Res. 2(2), 1–19 (2008).

Herbold, A. K.

S. Pannasch, J. R. Helmert, K. Roth, A. K. Herbold, and H. Walter, “Visual fixation durations and saccade amplitudes: Shifting relationship in a variety of conditions,” J. Eye Mov. Res. 2(2), 1–19 (2008).

Hirsch, M.

A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32(5), 153 (2013).
[Crossref]

D. Lanman, M. Hirsch, Y. Kim, and R. Raskar, “Content-adaptive parallax barriers: optimizing dual-layer 3D displays using low-rank light field factorization,” ACM Trans. Graph. 29(6), 1–10 (2010).
[Crossref]

Ho, N.

N. Ho, P. Van Dooren, and V. Blondel, “Weighted nonnegative matrix factorization and face feature extraction,” Image Vis. Comput. 2007, 1–17 (2007).

Huang, F. C.

F. C. Huang, K. Chen, and G. Wetzstein, “The light field stereoscope: immersive computer graphics via factored near-eye light field displays with focus cues[J],” ACM Trans. Graph. 34(4), 60 (2010).

Kim, Y.

D. Lanman, M. Hirsch, Y. Kim, and R. Raskar, “Content-adaptive parallax barriers: optimizing dual-layer 3D displays using low-rank light field factorization,” ACM Trans. Graph. 29(6), 1–10 (2010).
[Crossref]

Lanman, D.

D. Lanman and D. Luebke, “Near-eye light field displays,” ACM Trans. Graph. 32(6), 1–10 (2013).
[Crossref]

A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32(5), 153 (2013).
[Crossref]

D. Lanman, M. Hirsch, Y. Kim, and R. Raskar, “Content-adaptive parallax barriers: optimizing dual-layer 3D displays using low-rank light field factorization,” ACM Trans. Graph. 29(6), 1–10 (2010).
[Crossref]

Legge, G. E.

S. T. Chung, J. S. Mansfield, and G. E. Legge, “Psychophysics of reading. XVIII. The effect of print size on reading speed in normal peripheral vision,” Vision Res. 38(19), 2949–2962 (1998).
[Crossref] [PubMed]

Liu, A.

R. B. Welch, T. T. Blackmon, A. Liu, B. A. Mellers, and L. W. Stark, “The effects of pictorial realism, delay of visual feedback, and observer interactivity on the subjective sense of presence,” Presence (Camb. Mass.) 5(3), 263–273 (1996).
[Crossref]

Luebke, D.

D. Lanman and D. Luebke, “Near-eye light field displays,” ACM Trans. Graph. 32(6), 1–10 (2013).
[Crossref]

Maimone, A.

A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32(5), 153 (2013).
[Crossref]

Mansfield, J. S.

S. T. Chung, J. S. Mansfield, and G. E. Legge, “Psychophysics of reading. XVIII. The effect of print size on reading speed in normal peripheral vision,” Vision Res. 38(19), 2949–2962 (1998).
[Crossref] [PubMed]

Mellers, B. A.

R. B. Welch, T. T. Blackmon, A. Liu, B. A. Mellers, and L. W. Stark, “The effects of pictorial realism, delay of visual feedback, and observer interactivity on the subjective sense of presence,” Presence (Camb. Mass.) 5(3), 263–273 (1996).
[Crossref]

Pannasch, S.

S. Pannasch, J. R. Helmert, K. Roth, A. K. Herbold, and H. Walter, “Visual fixation durations and saccade amplitudes: Shifting relationship in a variety of conditions,” J. Eye Mov. Res. 2(2), 1–19 (2008).

Raskar, R.

A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32(5), 153 (2013).
[Crossref]

D. Lanman, M. Hirsch, Y. Kim, and R. Raskar, “Content-adaptive parallax barriers: optimizing dual-layer 3D displays using low-rank light field factorization,” ACM Trans. Graph. 29(6), 1–10 (2010).
[Crossref]

Rayner, K.

K. Rayner, “Eye movements in reading and information processing: 20 years of research,” Psychol. Bull. 124(3), 372–422 (1998).
[Crossref] [PubMed]

Roth, K.

S. Pannasch, J. R. Helmert, K. Roth, A. K. Herbold, and H. Walter, “Visual fixation durations and saccade amplitudes: Shifting relationship in a variety of conditions,” J. Eye Mov. Res. 2(2), 1–19 (2008).

Slater, M.

S. Uno and M. Slater, “The sensitivity of presence to collision response,” in Virtual Reality Annual International Symposium (IEEE, 1997), pp. 95–103.
[Crossref]

Stark, L. W.

R. B. Welch, T. T. Blackmon, A. Liu, B. A. Mellers, and L. W. Stark, “The effects of pictorial realism, delay of visual feedback, and observer interactivity on the subjective sense of presence,” Presence (Camb. Mass.) 5(3), 263–273 (1996).
[Crossref]

Takaki, Y.

Y. Takaki, “High-density directional display for generating natural three-dimensional images,” Proc. IEEE 94(3), 654–663 (2006).
[Crossref]

Uno, S.

S. Uno and M. Slater, “The sensitivity of presence to collision response,” in Virtual Reality Annual International Symposium (IEEE, 1997), pp. 95–103.
[Crossref]

Van Dooren, P.

N. Ho, P. Van Dooren, and V. Blondel, “Weighted nonnegative matrix factorization and face feature extraction,” Image Vis. Comput. 2007, 1–17 (2007).

Walter, H.

S. Pannasch, J. R. Helmert, K. Roth, A. K. Herbold, and H. Walter, “Visual fixation durations and saccade amplitudes: Shifting relationship in a variety of conditions,” J. Eye Mov. Res. 2(2), 1–19 (2008).

Welch, R. B.

R. B. Welch, T. T. Blackmon, A. Liu, B. A. Mellers, and L. W. Stark, “The effects of pictorial realism, delay of visual feedback, and observer interactivity on the subjective sense of presence,” Presence (Camb. Mass.) 5(3), 263–273 (1996).
[Crossref]

Wetzstein, G.

A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32(5), 153 (2013).
[Crossref]

F. C. Huang, K. Chen, and G. Wetzstein, “The light field stereoscope: immersive computer graphics via factored near-eye light field displays with focus cues[J],” ACM Trans. Graph. 34(4), 60 (2010).

Zhang, Z.

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

ACM Trans. Graph. (4)

D. Lanman and D. Luebke, “Near-eye light field displays,” ACM Trans. Graph. 32(6), 1–10 (2013).
[Crossref]

F. C. Huang, K. Chen, and G. Wetzstein, “The light field stereoscope: immersive computer graphics via factored near-eye light field displays with focus cues[J],” ACM Trans. Graph. 34(4), 60 (2010).

D. Lanman, M. Hirsch, Y. Kim, and R. Raskar, “Content-adaptive parallax barriers: optimizing dual-layer 3D displays using low-rank light field factorization,” ACM Trans. Graph. 29(6), 1–10 (2010).
[Crossref]

A. Maimone, G. Wetzstein, M. Hirsch, D. Lanman, R. Raskar, and H. Fuchs, “Focus 3D: Compressive accommodation display,” ACM Trans. Graph. 32(5), 153 (2013).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

Image Vis. Comput. (1)

N. Ho, P. Van Dooren, and V. Blondel, “Weighted nonnegative matrix factorization and face feature extraction,” Image Vis. Comput. 2007, 1–17 (2007).

Int. J. Industrial Ergonomics (1)

A. H. Chan and A. J. Courtney, “Foveal acuity, peripheral acuity and search performance: A review,” Int. J. Industrial Ergonomics 18(2), 113–119 (1996).
[Crossref]

J. Eye Mov. Res. (1)

S. Pannasch, J. R. Helmert, K. Roth, A. K. Herbold, and H. Walter, “Visual fixation durations and saccade amplitudes: Shifting relationship in a variety of conditions,” J. Eye Mov. Res. 2(2), 1–19 (2008).

Presence (Camb. Mass.) (1)

R. B. Welch, T. T. Blackmon, A. Liu, B. A. Mellers, and L. W. Stark, “The effects of pictorial realism, delay of visual feedback, and observer interactivity on the subjective sense of presence,” Presence (Camb. Mass.) 5(3), 263–273 (1996).
[Crossref]

Proc. IEEE (1)

Y. Takaki, “High-density directional display for generating natural three-dimensional images,” Proc. IEEE 94(3), 654–663 (2006).
[Crossref]

Psychol. Bull. (1)

K. Rayner, “Eye movements in reading and information processing: 20 years of research,” Psychol. Bull. 124(3), 372–422 (1998).
[Crossref] [PubMed]

Vision Res. (1)

S. T. Chung, J. S. Mansfield, and G. E. Legge, “Psychophysics of reading. XVIII. The effect of print size on reading speed in normal peripheral vision,” Vision Res. 38(19), 2949–2962 (1998).
[Crossref] [PubMed]

Other (5)

E. Peli, “Visual and optometric issues with head-mounted displays,” in IS & T/OSA Optics & Imaging in the Information Age, (The Society for Imaging Science and Technology, 1996), pp. 364–369.

A. Maimone and H. Fuchs, “Computational augmented reality eyeglasses,” in Mixed and Augmented Reality (ISMAR) (IEEE, 2013), pp. 29–38.

R. Rosén, Peripheral Vision: Adaptive Optics and Psychophysics (KTH Royal Institute of Technology, 2013).

S. Uno and M. Slater, “The sensitivity of presence to collision response,” in Virtual Reality Annual International Symposium (IEEE, 1997), pp. 95–103.
[Crossref]

B. T. Schowengerdt, H. G. Hoffman, C. M. Lee, C. D. Melville, and E. J. Seibel, “57.1: Near‐to‐Eye Display using Scanning Fiber Display Engine,” SID Symposium Digest Tech. Papers 41(1),848–851 (2010).
[Crossref]

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Figures (16)

Fig. 1
Fig. 1

The arrangement of viewpoints for original data rendering. (a) Illustration of the original data rendering in z-x plane. (b) Three-dimensional schematic of the arrangement of viewpoints, the eyeball is sketched as a sphere, and the schematic viewpoint located on the circular section of the eyeball.

Fig. 2
Fig. 2

Illustration of the proposed display system

Fig. 3
Fig. 3

Geometric relationship of our proposed system. (a)Illustration of the geometric relationship in z-x plane (b) Magnification of inset

Fig. 4
Fig. 4

Relationships between the critical point size and the eccentricity. The red line shows the linear relationship between critical print size (minute) corresponding to maximum reading speed and the eccentricity. The green polyline is the discrete function applied in this paper.

Fig. 5
Fig. 5

The compression process of L ̃ ( m r , m c , n x , n y ) . (a) The illustration of sampling perceived image of L ̃ ( m r , m c , n x , n y ) . (b) The illustration of S(p,q) .

Fig. 6
Fig. 6

The process of searching the corresponding pixels of L ' ̃ (p,q, n x , n y ) on P mf ' ( p f , q f ) and P mr ' ( p r , q r ) . (a) The original index ( m 0 , n 0 ) of sampling points B on reference plane is acquired by the S1(p,q) of reference plane, all of the original index of sampling points can be obtained in this way. (b) Since light ray is positioned by the index of sampling point B and the viewpoint, the location of intersections on multi-layer LCD is gained by geometric mapping. The intersection on front layer is illustrated as A( i 1 , j 1 ). (c) The index of A( i 2 , j 2 ) on P mf ' ( p f , q f ) is recorded by the pixel indexed as ( i 2 , j 2 ) on S2(k,l) . The index of A on P mf ' ( p f , q f ) is illustrated as ( m 2 , n 2 ).Index of C( i 2 , j 2 ) on P mf ' ( p f , q f ) can be obtained by the same way. (d) The corresponding pixel A on P mf ' ( p f , q f ) of light ray B is obtained by the index ( m 2 , n 2 ), the pixel C can be acquired by the same way.

Fig. 7
Fig. 7

The decompression process of the P mf ' ( p f , q f ) or P mr ' ( p r , q r ) . (a) The S2(k,l) of P mf (k,l) , or S2(i,j) of P mr (i,j) . (b) P mf ' ( p f , q f ) or P mr ' ( p r , q r ) . (c) P mf (k,l) or P mr (i,j)

Fig. 8
Fig. 8

The proposed reconstruction algorithm flow

Fig. 9
Fig. 9

Different perspective views of the experimental prototype.

Fig. 10
Fig. 10

The geometric positions of the model for display. (a) The illustration of geometric positions in X-Y plane. (b) The illustration of geometric positions in Z-X plane.

Fig. 11
Fig. 11

The sharpness of the perceived image of reconstructed light field varies with the focus depths and viewing directions. (a) The left column illustrates the discussed cubes. The perceived images in the right columns are the magnified insets in left boxes, which are taken when CCD focuses at different depths. (b) These experiment results are the magnified insets as illustrated in Fig. 11(a)’s left column. Perceived images of cube a and d recorded within depth of field but with different viewing directions. The images in the left and right columns are captured within and out of fovea, respectively. E.g. the top-right image was captured when CCD viewed toward cube c and kept the cube a within focus. So the cube is out of fovea and within focus.

Fig. 12
Fig. 12

Perceived images when corresponding cube within focus and within fovea.

Fig. 13
Fig. 13

Maps of each plane when gazing at cube a. (a) Maps of LCD1. (b) Maps of LCD2. (c) Maps of reference plane. The top and the bottom row correspond to S1 and S2, respectively.

Fig. 14
Fig. 14

Factorized patterns of multi-valued light field when the viewer concentrates at cube a. (a) The Factorized pattern for front LCD (b) The Factorized pattern for rear LCD

Fig. 15
Fig. 15

The time consumption of two algorithm with different spatial resolution and 2*2 viewpoints.

Fig. 16
Fig. 16

The time consumption ratio of conventional algorithm to the proposed algorithm with different spatial resolution and 2*2 viewpoints

Tables (1)

Tables Icon

Table 1 The runtime of two methods for four light fields.

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

x e =sin( n x θ x + Φ x )cos( n y θ y + Φ y )R
y e =sin( n y θ y + Φ y )R
z e =cos( n x θ x + Φ x )cos( n y θ y + Φ y )R
1 d ' - 1 d = 1 f
L[k,l,i,j]= P f [k,l] P r [i,j]
L m [k,l,i,j]= P mf [k,l] P mr [i,j]
argmin L mf, L mr 1 2 L ˜ L m 2 .
P mf P mf [(W L ˜ ) P mr T ] [(W( P mf P mr )) P mr T ]
P mr P mr [ P mf T (W L ˜ )] [ P mf T (W( P mf P mr ))]
x A = ( x B x e )( z A z e ) z B z e + x e
y A = ( y B y e )( z A z e ) z B z e + y e
R a = R ao ×(1+ E cc 1.39 )
R=col-255×floor(col÷255.00001)
G=row-255×floor(row÷255.00001)
B=floor(col÷255.00001)
A=255-floor(row÷255.00001)
x i = m,n=0 k xmn u i m v i n
y i = m,n=0 k ymn u i m v i n
x i ' = m,n=0 k xmn u i 'm v i 'n
y i ' = m,n=0 k ymn u i 'm v i 'n