Abstract

The theoretical and experimental demonstration of a multispectral Sagnac interferometer (MSI) is presented. The MSI was created by including two multiple-order blazed diffraction gratings in both arms of a standard polarization Sagnac interferometer (PSI). By introducing these high-order diffractive structures, unique spectral passbands can be amplitude modulated onto coincident carrier frequencies. Extraction of the modulated multispectral images, corresponding to each passband, is accomplished within the Fourier domain. This yields a unique multispectral sensor capable of imaging all the passbands in a single snapshot. First, the theoretical operating principles of a PSI are discussed to provide a context for the MSI. This is followed by the theoretical and experimental development of the MSI, which is an extension of a dispersion-compensated PSI. Indoor and outdoor testing and validation of the MSI are performed by observing vegetation, demonstrating the ability of our experimental setup to detect four distinct spectral passbands.

© 2010 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2010

2009

M. W. Kudenov, M. E. L. Jungwirth, E. L. Dereniak, and G. R. Gerhart, “White light Sagnac interferometer for snapshot linear polarimetric imaging,” Opt. Express 17, 22520–22534(2009).
[CrossRef]

J. Craven, M. Kudenov, and E. Dereniak, “False signature reduction in infrared channeled spectropolarimetry,” Proc. SPIE 7419741909 (2009).
[CrossRef]

2008

2007

A. Vogel, V. Chernomordik, J. Riley, M. Hassan, F. Amyot, B. Dasgeb, S. G. Demos, R. Pursley, R. F. Little, R. Yarchoan, Y. Tao, and A. H. Gandjbakhche, “Using noninvasive multispectral imaging to quantitatively assess tissue vasculature,” J. Biomed. Opt. 12, 051604 (2007).
[CrossRef] [PubMed]

M. Kise, B. Park, K. C. Lawrence, and W. R. Windham, “Compact multi-spectral imaging system for contaminant detection on poultry carcass,” Proc. SPIE 6503, 650305 (2007).
[CrossRef]

2006

R. M. Levenson and J. R. Mansfield, “Multispectral imaging in biology and medicine: slices of life,” Cytometry Part A 69748–758 (2006).
[CrossRef]

2004

2003

A. Gitelson, Y. Gritz, and M. Merzlyak, “Relationships between leaf chlorophyll content and spectral reflectance and algorithms for non-destructive chlorophyll assessment in higher plant leaves,” J. Plant Physiol. 160, 271–282 (2003).
[CrossRef] [PubMed]

K. Oka and T. Kaneko, “Compact complete imaging polarimeter using birefringent wedge prisms,” Opt. Express 11, 1510–1519 (2003).
[CrossRef] [PubMed]

1990

J. P. Curran, J. L. Dungan, and H. L. Gholz, “Exploring the relationship between reflectance red edge and chlorophyll content in slash pine,” Tree Physiol. 7, 33–48 (1990).
[PubMed]

1975

Amyot, F.

A. Vogel, V. Chernomordik, J. Riley, M. Hassan, F. Amyot, B. Dasgeb, S. G. Demos, R. Pursley, R. F. Little, R. Yarchoan, Y. Tao, and A. H. Gandjbakhche, “Using noninvasive multispectral imaging to quantitatively assess tissue vasculature,” J. Biomed. Opt. 12, 051604 (2007).
[CrossRef] [PubMed]

Aso, T.

Baleine, E.

Biradar, C. M.

C. M. Biradar, P. Thenkabail, A. Platonov, X. Xiao, R. Geerken, P. Noojipady, H. Turral, and J. Vithanage, “Water productivity mapping methods using remote sensing,” J. Appl. Remote Sens. 2, 023544 (2008).
[CrossRef]

Chernomordik, V.

A. Vogel, V. Chernomordik, J. Riley, M. Hassan, F. Amyot, B. Dasgeb, S. G. Demos, R. Pursley, R. F. Little, R. Yarchoan, Y. Tao, and A. H. Gandjbakhche, “Using noninvasive multispectral imaging to quantitatively assess tissue vasculature,” J. Biomed. Opt. 12, 051604 (2007).
[CrossRef] [PubMed]

Chernouss, S.

Craven, J.

J. Craven, M. Kudenov, and E. Dereniak, “False signature reduction in infrared channeled spectropolarimetry,” Proc. SPIE 7419741909 (2009).
[CrossRef]

Curran, J. P.

J. P. Curran, J. L. Dungan, and H. L. Gholz, “Exploring the relationship between reflectance red edge and chlorophyll content in slash pine,” Tree Physiol. 7, 33–48 (1990).
[PubMed]

Dasgeb, B.

A. Vogel, V. Chernomordik, J. Riley, M. Hassan, F. Amyot, B. Dasgeb, S. G. Demos, R. Pursley, R. F. Little, R. Yarchoan, Y. Tao, and A. H. Gandjbakhche, “Using noninvasive multispectral imaging to quantitatively assess tissue vasculature,” J. Biomed. Opt. 12, 051604 (2007).
[CrossRef] [PubMed]

Deehr, C. S.

Demos, S. G.

A. Vogel, V. Chernomordik, J. Riley, M. Hassan, F. Amyot, B. Dasgeb, S. G. Demos, R. Pursley, R. F. Little, R. Yarchoan, Y. Tao, and A. H. Gandjbakhche, “Using noninvasive multispectral imaging to quantitatively assess tissue vasculature,” J. Biomed. Opt. 12, 051604 (2007).
[CrossRef] [PubMed]

Dereniak, E.

J. Craven, M. Kudenov, and E. Dereniak, “False signature reduction in infrared channeled spectropolarimetry,” Proc. SPIE 7419741909 (2009).
[CrossRef]

M. Kudenov, L. Pezzaniti, E. Dereniak, and G. Gerhart, “Prismatic imaging polarimeter calibration for the infrared spectral region,” Opt. Express 16, 13720–13737 (2008).
[CrossRef] [PubMed]

Dereniak, E. L.

Dogariu, A.

Dungan, J. L.

J. P. Curran, J. L. Dungan, and H. L. Gholz, “Exploring the relationship between reflectance red edge and chlorophyll content in slash pine,” Tree Physiol. 7, 33–48 (1990).
[PubMed]

Dyrland, M.

Fletcher-Holmes, D. W.

Gandjbakhche, A. H.

A. Vogel, V. Chernomordik, J. Riley, M. Hassan, F. Amyot, B. Dasgeb, S. G. Demos, R. Pursley, R. F. Little, R. Yarchoan, Y. Tao, and A. H. Gandjbakhche, “Using noninvasive multispectral imaging to quantitatively assess tissue vasculature,” J. Biomed. Opt. 12, 051604 (2007).
[CrossRef] [PubMed]

Geerken, R.

C. M. Biradar, P. Thenkabail, A. Platonov, X. Xiao, R. Geerken, P. Noojipady, H. Turral, and J. Vithanage, “Water productivity mapping methods using remote sensing,” J. Appl. Remote Sens. 2, 023544 (2008).
[CrossRef]

Gerhart, G.

Gerhart, G. R.

Gholz, H. L.

J. P. Curran, J. L. Dungan, and H. L. Gholz, “Exploring the relationship between reflectance red edge and chlorophyll content in slash pine,” Tree Physiol. 7, 33–48 (1990).
[PubMed]

Gitelson, A.

A. Gitelson, Y. Gritz, and M. Merzlyak, “Relationships between leaf chlorophyll content and spectral reflectance and algorithms for non-destructive chlorophyll assessment in higher plant leaves,” J. Plant Physiol. 160, 271–282 (2003).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

Gorman, A.

Gritz, Y.

A. Gitelson, Y. Gritz, and M. Merzlyak, “Relationships between leaf chlorophyll content and spectral reflectance and algorithms for non-destructive chlorophyll assessment in higher plant leaves,” J. Plant Physiol. 160, 271–282 (2003).
[CrossRef] [PubMed]

Harvey, A. R.

Hassan, M.

A. Vogel, V. Chernomordik, J. Riley, M. Hassan, F. Amyot, B. Dasgeb, S. G. Demos, R. Pursley, R. F. Little, R. Yarchoan, Y. Tao, and A. H. Gandjbakhche, “Using noninvasive multispectral imaging to quantitatively assess tissue vasculature,” J. Biomed. Opt. 12, 051604 (2007).
[CrossRef] [PubMed]

Heia, K.

Holmes, J. M.

Jungwirth, M. E. L.

Kaneko, T.

Kise, M.

M. Kise, B. Park, K. C. Lawrence, and W. R. Windham, “Compact multi-spectral imaging system for contaminant detection on poultry carcass,” Proc. SPIE 6503, 650305 (2007).
[CrossRef]

Kudenov, M.

J. Craven, M. Kudenov, and E. Dereniak, “False signature reduction in infrared channeled spectropolarimetry,” Proc. SPIE 7419741909 (2009).
[CrossRef]

M. Kudenov, L. Pezzaniti, E. Dereniak, and G. Gerhart, “Prismatic imaging polarimeter calibration for the infrared spectral region,” Opt. Express 16, 13720–13737 (2008).
[CrossRef] [PubMed]

Kudenov, M. W.

Lawrence, K. C.

M. Kise, B. Park, K. C. Lawrence, and W. R. Windham, “Compact multi-spectral imaging system for contaminant detection on poultry carcass,” Proc. SPIE 6503, 650305 (2007).
[CrossRef]

Levenson, R. M.

R. M. Levenson and J. R. Mansfield, “Multispectral imaging in biology and medicine: slices of life,” Cytometry Part A 69748–758 (2006).
[CrossRef]

Little, R. F.

A. Vogel, V. Chernomordik, J. Riley, M. Hassan, F. Amyot, B. Dasgeb, S. G. Demos, R. Pursley, R. F. Little, R. Yarchoan, Y. Tao, and A. H. Gandjbakhche, “Using noninvasive multispectral imaging to quantitatively assess tissue vasculature,” J. Biomed. Opt. 12, 051604 (2007).
[CrossRef] [PubMed]

Lorentzen, D. A.

Mansfield, J. R.

R. M. Levenson and J. R. Mansfield, “Multispectral imaging in biology and medicine: slices of life,” Cytometry Part A 69748–758 (2006).
[CrossRef]

Merzlyak, M.

A. Gitelson, Y. Gritz, and M. Merzlyak, “Relationships between leaf chlorophyll content and spectral reflectance and algorithms for non-destructive chlorophyll assessment in higher plant leaves,” J. Plant Physiol. 160, 271–282 (2003).
[CrossRef] [PubMed]

Mujat, M.

Noojipady, P.

C. M. Biradar, P. Thenkabail, A. Platonov, X. Xiao, R. Geerken, P. Noojipady, H. Turral, and J. Vithanage, “Water productivity mapping methods using remote sensing,” J. Appl. Remote Sens. 2, 023544 (2008).
[CrossRef]

Ntziachristos, V.

Oka, K.

K. Oka and T. Kaneko, “Compact complete imaging polarimeter using birefringent wedge prisms,” Opt. Express 11, 1510–1519 (2003).
[CrossRef] [PubMed]

R. Suda, N. Saito, and K. Oka, “Imaging polarimetry by use of double Sagnac interferometers,” in Extended Abstracts of the 69th Autumn Meeting of the Japan Society of Applied Physics (Japan Society of Applied Physics, 2008), p. 877 (in Japanese).

Park, B.

M. Kise, B. Park, K. C. Lawrence, and W. R. Windham, “Compact multi-spectral imaging system for contaminant detection on poultry carcass,” Proc. SPIE 6503, 650305 (2007).
[CrossRef]

Pezzaniti, L.

Platonov, A.

C. M. Biradar, P. Thenkabail, A. Platonov, X. Xiao, R. Geerken, P. Noojipady, H. Turral, and J. Vithanage, “Water productivity mapping methods using remote sensing,” J. Appl. Remote Sens. 2, 023544 (2008).
[CrossRef]

Pursley, R.

A. Vogel, V. Chernomordik, J. Riley, M. Hassan, F. Amyot, B. Dasgeb, S. G. Demos, R. Pursley, R. F. Little, R. Yarchoan, Y. Tao, and A. H. Gandjbakhche, “Using noninvasive multispectral imaging to quantitatively assess tissue vasculature,” J. Biomed. Opt. 12, 051604 (2007).
[CrossRef] [PubMed]

Riley, J.

A. Vogel, V. Chernomordik, J. Riley, M. Hassan, F. Amyot, B. Dasgeb, S. G. Demos, R. Pursley, R. F. Little, R. Yarchoan, Y. Tao, and A. H. Gandjbakhche, “Using noninvasive multispectral imaging to quantitatively assess tissue vasculature,” J. Biomed. Opt. 12, 051604 (2007).
[CrossRef] [PubMed]

Saito, N.

R. Suda, N. Saito, and K. Oka, “Imaging polarimetry by use of double Sagnac interferometers,” in Extended Abstracts of the 69th Autumn Meeting of the Japan Society of Applied Physics (Japan Society of Applied Physics, 2008), p. 877 (in Japanese).

Sigernes, F.

Suda, R.

R. Suda, N. Saito, and K. Oka, “Imaging polarimetry by use of double Sagnac interferometers,” in Extended Abstracts of the 69th Autumn Meeting of the Japan Society of Applied Physics (Japan Society of Applied Physics, 2008), p. 877 (in Japanese).

Svenoe, T.

Tao, Y.

A. Vogel, V. Chernomordik, J. Riley, M. Hassan, F. Amyot, B. Dasgeb, S. G. Demos, R. Pursley, R. F. Little, R. Yarchoan, Y. Tao, and A. H. Gandjbakhche, “Using noninvasive multispectral imaging to quantitatively assess tissue vasculature,” J. Biomed. Opt. 12, 051604 (2007).
[CrossRef] [PubMed]

Themelis, G.

Thenkabail, P.

C. M. Biradar, P. Thenkabail, A. Platonov, X. Xiao, R. Geerken, P. Noojipady, H. Turral, and J. Vithanage, “Water productivity mapping methods using remote sensing,” J. Appl. Remote Sens. 2, 023544 (2008).
[CrossRef]

Turral, H.

C. M. Biradar, P. Thenkabail, A. Platonov, X. Xiao, R. Geerken, P. Noojipady, H. Turral, and J. Vithanage, “Water productivity mapping methods using remote sensing,” J. Appl. Remote Sens. 2, 023544 (2008).
[CrossRef]

Vithanage, J.

C. M. Biradar, P. Thenkabail, A. Platonov, X. Xiao, R. Geerken, P. Noojipady, H. Turral, and J. Vithanage, “Water productivity mapping methods using remote sensing,” J. Appl. Remote Sens. 2, 023544 (2008).
[CrossRef]

Vogel, A.

A. Vogel, V. Chernomordik, J. Riley, M. Hassan, F. Amyot, B. Dasgeb, S. G. Demos, R. Pursley, R. F. Little, R. Yarchoan, Y. Tao, and A. H. Gandjbakhche, “Using noninvasive multispectral imaging to quantitatively assess tissue vasculature,” J. Biomed. Opt. 12, 051604 (2007).
[CrossRef] [PubMed]

Windham, W. R.

M. Kise, B. Park, K. C. Lawrence, and W. R. Windham, “Compact multi-spectral imaging system for contaminant detection on poultry carcass,” Proc. SPIE 6503, 650305 (2007).
[CrossRef]

Wyant, J. C.

Xiao, X.

C. M. Biradar, P. Thenkabail, A. Platonov, X. Xiao, R. Geerken, P. Noojipady, H. Turral, and J. Vithanage, “Water productivity mapping methods using remote sensing,” J. Appl. Remote Sens. 2, 023544 (2008).
[CrossRef]

Yarchoan, R.

A. Vogel, V. Chernomordik, J. Riley, M. Hassan, F. Amyot, B. Dasgeb, S. G. Demos, R. Pursley, R. F. Little, R. Yarchoan, Y. Tao, and A. H. Gandjbakhche, “Using noninvasive multispectral imaging to quantitatively assess tissue vasculature,” J. Biomed. Opt. 12, 051604 (2007).
[CrossRef] [PubMed]

Yoo, J.

Appl. Opt.

Cytometry Part A

R. M. Levenson and J. R. Mansfield, “Multispectral imaging in biology and medicine: slices of life,” Cytometry Part A 69748–758 (2006).
[CrossRef]

J. Appl. Remote Sens.

C. M. Biradar, P. Thenkabail, A. Platonov, X. Xiao, R. Geerken, P. Noojipady, H. Turral, and J. Vithanage, “Water productivity mapping methods using remote sensing,” J. Appl. Remote Sens. 2, 023544 (2008).
[CrossRef]

J. Biomed. Opt.

A. Vogel, V. Chernomordik, J. Riley, M. Hassan, F. Amyot, B. Dasgeb, S. G. Demos, R. Pursley, R. F. Little, R. Yarchoan, Y. Tao, and A. H. Gandjbakhche, “Using noninvasive multispectral imaging to quantitatively assess tissue vasculature,” J. Biomed. Opt. 12, 051604 (2007).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

J. Plant Physiol.

A. Gitelson, Y. Gritz, and M. Merzlyak, “Relationships between leaf chlorophyll content and spectral reflectance and algorithms for non-destructive chlorophyll assessment in higher plant leaves,” J. Plant Physiol. 160, 271–282 (2003).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Proc. SPIE

M. Kise, B. Park, K. C. Lawrence, and W. R. Windham, “Compact multi-spectral imaging system for contaminant detection on poultry carcass,” Proc. SPIE 6503, 650305 (2007).
[CrossRef]

J. Craven, M. Kudenov, and E. Dereniak, “False signature reduction in infrared channeled spectropolarimetry,” Proc. SPIE 7419741909 (2009).
[CrossRef]

Tree Physiol.

J. P. Curran, J. L. Dungan, and H. L. Gholz, “Exploring the relationship between reflectance red edge and chlorophyll content in slash pine,” Tree Physiol. 7, 33–48 (1990).
[PubMed]

Other

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

“Radiometric calibration,” in Landsat 7 Science Data Users Handbook (NASA, 2009), Chap. 8.1.2.

R. Suda, N. Saito, and K. Oka, “Imaging polarimetry by use of double Sagnac interferometers,” in Extended Abstracts of the 69th Autumn Meeting of the Japan Society of Applied Physics (Japan Society of Applied Physics, 2008), p. 877 (in Japanese).

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Figures (17)

Fig. 1
Fig. 1

The DCPSI contains two blazed diffraction gratings, G 1 and G 2 , at each output of a wire-grid beam splitter (WGBS). These gratings generate a shear ( S DCPSI ) that is linearly proportional to the wavelength. The rays from the object are only depicted on-axis for clarity.

Fig. 2
Fig. 2

(a) Single-order blazed grating, where d is the period, n 1 and n 2 are the indices of refraction for the incident and blaze medium, respectively, and h 1 is the depth ( OP D 1 wave). (b) Multiple-order blaze grating, where h 2 is typically 3–10 times larger than h 1 .

Fig. 3
Fig. 3

Diffraction efficiency (percent) for (a) a single-order blazed grating in air with h 1 = 1.28 μm and n = 1.5 and for (b) a multiple-order blazed grating in air with h 2 = 4.07 μm and n = 1.5 .

Fig. 4
Fig. 4

Experimental setup of the MSI with a, b, and c set to 13.3, 51.4 , and 19.1 mm , respectively. An IR blocking filter keeps wavelengths > 750 nm from entering the system.

Fig. 5
Fig. 5

(a) Raw image of a uniformly illuminated diffuser demonstrating the fringe pattern in the MSI. (b) Fourier transformation of the raw image data in (a). Four carrier frequencies are observed, corresponding to m = 2 , 3 , 4 , and 5.

Fig. 6
Fig. 6

Spatial resolution trade space in 1D. M is the total number of carrier frequencies in the system and the channels are assumed to be uniformly spaced from ξ = 0 to ξ = ξ n y .

Fig. 7
Fig. 7

Normalized cutoff frequency versus the number of passbands present in the system. The DoF approach maintains higher spatial resolution since it utilizes both spatial dimensions x and y.

Fig. 8
Fig. 8

monochromator configuration for sending light into the MSI for verification of the fringe visibility. The bandwidth of the light exiting the monochromator was approximately 10.5 nm using a 3 mm exit slit.

Fig. 9
Fig. 9

(a) Measured monochromator output in W / m 2 . (b) Measured relative spectral response of the different passbands within the MSI. Also included (solid light-gray line) is the relative response of the FPA multiplied by the measured transmission of the IR blocking filter.

Fig. 10
Fig. 10

(a) System setup for U2S reflectance measurements. The incidence angle of the source (tungsten–halogen lamp) was approximately 37 ° with respect to the surface normal of the sample. (b) System setup for MSI reflectance measurements. The sample is defocused to average the spatial details over the observed area.

Fig. 11
Fig. 11

Relative reflectance of the healthy and unhealthy leaves, normalized to the irradiance in order 2, as measured with the MSI and U2S. The healthy vegetation experiences more absorption in order 3 due to the presence of chlorophyll.

Fig. 12
Fig. 12

Relative reflectance images of a healthy and unhealthy leaf. (a) Band-integrated image ( m = 0 ). (b)–(e) Images from orders m = 2 through m = 5 .

Fig. 13
Fig. 13

NDVI image of the leaf, calculated using Eq. (26). A higher signal is associated with the presence of chlorophyll, as illustrated with the healthy leaf in the lower right (quadrant 4) of the image.

Fig. 14
Fig. 14

Relative reflectance of a healthy leaf from Acacia crassifolia measured with the U2S (solid black curve). The “red edge” begins at 700 nm and peaks at approximately 765 nm . The Landsat 7 (band 4) spans 775 900 nm (dark dashed curve) while the MSI spans 685 755 nm (gray dotted curve).

Fig. 15
Fig. 15

Photo taken with a standard color digital camera of an outdoor scene. Healthy vegetation is present, in addition to brick, concrete, and a relatively clear sky.

Fig. 16
Fig. 16

Relative reflectance images of an outdoor scene. (a) Band-integrated image ( m = 0 ). (b)–(e) Images from orders m = 2 through m = 5 .

Fig. 17
Fig. 17

NDVI image of the outdoor scene. Vegetation appears as a higher value than the other objects in the scene due to chlorophyll.

Equations (26)

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

S DCPSI = 2 m λ d ( a + b + c ) ,
I DCPSI ( x i , y i ) = 1 2 m = 0 d / λ min S 0 ( m ) + 1 2 m = 1 d / λ min [ S 2 ( m ) cos ( 2 π f o b j 2 m d ( a + b + c ) x i ) S 3 ( m ) sin ( 2 π f obj 2 m d ( a + b + c ) x i ) ] ,
S 0 ( m ) = λ min λ max DE 2 ( λ , m ) S 0 ( λ ) d λ ,
S 2 ( m ) = λ min λ max DE 2 ( λ , m ) S 2 ( λ ) d λ ,
S 3 ( m ) = λ min λ max DE 2 ( λ , m ) S 3 ( λ ) d λ ,
U DCPSI = 2 m d f obj ( a + b + c ) .
DE ( λ , m ) = sinc 2 ( m OPD λ ) ,
OPD = h ( n 1 n 2 ) ,
S WGBS = 1 2 [ 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 ] [ S 0 , inc S 1 , inc S 2 , inc S 3 , inc ] = [ S 0 , inc + S 2 , inc 0 S 0 , inc + S 2 , inc 0 ] ,
S 0 ( m ) = S 2 ( m ) = λ min λ max DE 2 ( λ , m ) [ S 0 , inc ( λ ) + S 2 , inc ( λ ) ] d λ ,
S 3 ( m ) = 0.
I MSI ( x i , y i ) = 1 2 m = 0 Ce [ λ 1 / λ min ] [ S 0 ( m ) ] + 1 2 m = 1 Ce [ λ 1 / λ min ] [ S 0 ( m ) cos ( 2 π f obj 2 m d ( a + b + c ) x i ) ] ,
S 0 ( m ) = λ min λ max DE 2 ( λ , m ) [ S 0 , inc ( λ ) + S 2 , inc ( λ ) ] d λ .
F [ I MSI ( x i , y i ) ] = 1 2 m = 0 Ce [ λ 1 / λ min ] [ S 0 ( m ) ] δ ( ξ , η ) + m = 1 Ce [ λ 1 / λ min ] [ S 0 ( m ) δ ( ξ U MSI ( m ) , η ) ] + m = 1 Ce [ λ 1 / λ min ] [ S 0 ( m ) δ ( ξ + U MSI ( m ) , η ) ] ,
U MSI ( m ) = 2 m d f obj ( a + b + c ) ,
F 1 [ C 0 ] = 1 2 m = 0 Ce [ λ 1 / λ min ] S 0 ( m ) ,
F 1 [ C 1 ( m ) ] = 1 4 S 0 ( m ) exp ( j 2 π U MSI ( m ) x i ) .
| F 1 [ C 1 ( m ) ] | = 1 4 S 0 ( m ) .
ξ c = ξ n y / ( 2 M + 1 ) .
ξ c , DoF = ξ n y / M .
ξ c ξ c , DoF = M ( 2 M + 1 ) .
R U 2 S ( m ) = 400 800 I leaf ( λ ) T MSI ( m , λ ) d λ 400 800 I diffuser ( λ ) T MSI ( m , λ ) d λ ,
R MSI avg ( m ) = l n I leaf ( l , n , m ) l n I diffuser ( l , n , m ) ,
ε = 1 4 m = 2 5 ( R U 2 S ( m ) R MSI ( m ) ) 2 .
R MSI ( l , n , m ) = I leaf ( l , n , m ) I diffuser ( l , n , m ) .
NDVI ( l , n ) = I ( l , n , 2 ) I ( l , n , 3 ) I ( l , n , 2 ) + I ( l , n , 3 ) .

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