Abstract

Coregistration errors in multi- and hyperspectral imaging sensors arise when the spatial sensitivity pattern differs between bands or when the spectral response varies across the field of view, potentially leading to large errors in the recorded image data. In imaging spectrometers, spectral and spatial offset errors are customarily specified as “smile” and “keystone” distortions. However these characteristics do not account for errors resulting from variations in point spread function shape or spectral bandwidth. This paper proposes improved metrics for coregistration error both in the spatial and spectral dimensions. The metrics are essentially the integrated difference between point spread functions. It is shown that these metrics correspond to an upper bound on the error in image data. The metrics enable estimation of actual data errors for a given image, and can be used as part of the merit function in optical design optimization, as well as for benchmarking of spectral image sensors.

© 2012 OSA

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References

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  1. P. Mouroulis, D. A. Thomas, T. G. Chrien, V. Duval, R. O. Green, J. J. Simmonds, and A. H. Vaughan, Trade Studies in Multi/hyperspectral Imaging Systems—Final Report (NASA Jet Propulsion Laboratory, 1998).
  2. P. Mouroulis, “Spectral and spatial uniformity in pushbroom imaging spectrometers,” Proc. SPIE 3753, 133–141 (1999).
    [CrossRef]
  3. P. Mouroulis, R. O. Green, and T. G. Chrien, “Design of pushbroom imaging spectrometers for optimum recovery of spectroscopic and spatial information,” Appl. Opt. 39(13), 2210–2220 (2000).
    [CrossRef] [PubMed]
  4. R. O. Green, “Spectral calibration requirement for Earth-looking imaging spectrometers in the solar-reflected spectrum,” Appl. Opt. 37(4), 683–690 (1998).
    [CrossRef] [PubMed]
  5. P. Mouroulis and M. M. McKerns, “Pushbroom imaging spectrometer with high spectroscopic data fidelity: experimental demonstration,” Opt. Eng. 39(3), 808–816 (2000).
    [CrossRef]
  6. R. A. Neville and L. Sun, “Karl Staenz, “Detection of spectral line curvature in imaging spectrometer data,” Proc. SPIE 5093, 144–154 (2003).
    [CrossRef]
  7. R. A. Neville, L. Sun, and K. Staenz, “Detection of keystone in imaging spectrometer data,” Proc. SPIE 5425, 208–217 (2004).
    [CrossRef]
  8. J. Zadnik, D. Guerin, R. Moss, A. Orbeta, R. Dixon, C. Simi, S. Dunbar, and A. Hill, “Calibration procedures and measurements for the COMPASS hyperspectral imager,” Proc. SPIE 5425, 182–188 (2004).
    [CrossRef]
  9. D. Schläpfer, J. Nieke, and K. I. Itten, “Spatial PSF Non-uniformity Effects In Airborne Pushbroom Imaging Spectrometry Data,” IEEE Trans. Geosci. Rem. Sens. 45(2), 458–468 (2007).
    [CrossRef]
  10. F. Dell’Endice, J. Nieke, D. Schläpfer, and K. I. Itten, “Scene-based method for spatial misregistration detection in hyperspectral imagery,” Appl. Opt. 46(15), 2803–2816 (2007).
    [CrossRef] [PubMed]
  11. P. Mouroulis and R. O. Green, “Spectral response evaluation and computation for pushbroom imaging spectrometers,” Proc. SPIE 6667, 66670G (2007).
    [CrossRef]
  12. J. T. Casey and J. P. Kerekes, “Misregistration impacts on hyperspectral target detection,” J. Appl. Remote Sens. 3(1), 033513 (2009).
    [CrossRef]
  13. T. Skauli, “Quantifying coregistration errors in spectral imaging,” Proc. SPIE 8158, 81580A, 81580A-8 (2011).
    [CrossRef]
  14. G. Lin, R. E. Wolfe, and M. Nishihama, “NPP VIIRS geometric performance status,” Proc. SPIE 8153, 81531V, 81531V-14 (2011).
    [CrossRef]
  15. T. Skauli, “Sensor noise informed representation of hyperspectral data, with benefits for image storage and processing,” Opt. Express 19(14), 13031–13046 (2011).
    [CrossRef] [PubMed]
  16. C. D. Claxton and R. C. Staunton, “Measurement of the point-spread function of a noisy imaging system,” J. Opt. Soc. Am. A 25(1), 159–170 (2008).
    [CrossRef] [PubMed]
  17. H. Du and K. J. Voss, “Effects of point-spread function on calibration and radiometric accuracy of CCD camera,” Appl. Opt. 43(3), 665–670 (2004).
    [CrossRef] [PubMed]
  18. S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
  19. H. Hovland, “Tomographic scanning imager,” Opt. Express 17(14), 11371–11387 (2009).
    [CrossRef] [PubMed]

2011 (3)

T. Skauli, “Quantifying coregistration errors in spectral imaging,” Proc. SPIE 8158, 81580A, 81580A-8 (2011).
[CrossRef]

G. Lin, R. E. Wolfe, and M. Nishihama, “NPP VIIRS geometric performance status,” Proc. SPIE 8153, 81531V, 81531V-14 (2011).
[CrossRef]

T. Skauli, “Sensor noise informed representation of hyperspectral data, with benefits for image storage and processing,” Opt. Express 19(14), 13031–13046 (2011).
[CrossRef] [PubMed]

2009 (2)

J. T. Casey and J. P. Kerekes, “Misregistration impacts on hyperspectral target detection,” J. Appl. Remote Sens. 3(1), 033513 (2009).
[CrossRef]

H. Hovland, “Tomographic scanning imager,” Opt. Express 17(14), 11371–11387 (2009).
[CrossRef] [PubMed]

2008 (1)

2007 (3)

F. Dell’Endice, J. Nieke, D. Schläpfer, and K. I. Itten, “Scene-based method for spatial misregistration detection in hyperspectral imagery,” Appl. Opt. 46(15), 2803–2816 (2007).
[CrossRef] [PubMed]

D. Schläpfer, J. Nieke, and K. I. Itten, “Spatial PSF Non-uniformity Effects In Airborne Pushbroom Imaging Spectrometry Data,” IEEE Trans. Geosci. Rem. Sens. 45(2), 458–468 (2007).
[CrossRef]

P. Mouroulis and R. O. Green, “Spectral response evaluation and computation for pushbroom imaging spectrometers,” Proc. SPIE 6667, 66670G (2007).
[CrossRef]

2004 (3)

R. A. Neville, L. Sun, and K. Staenz, “Detection of keystone in imaging spectrometer data,” Proc. SPIE 5425, 208–217 (2004).
[CrossRef]

J. Zadnik, D. Guerin, R. Moss, A. Orbeta, R. Dixon, C. Simi, S. Dunbar, and A. Hill, “Calibration procedures and measurements for the COMPASS hyperspectral imager,” Proc. SPIE 5425, 182–188 (2004).
[CrossRef]

H. Du and K. J. Voss, “Effects of point-spread function on calibration and radiometric accuracy of CCD camera,” Appl. Opt. 43(3), 665–670 (2004).
[CrossRef] [PubMed]

2003 (1)

R. A. Neville and L. Sun, “Karl Staenz, “Detection of spectral line curvature in imaging spectrometer data,” Proc. SPIE 5093, 144–154 (2003).
[CrossRef]

2001 (1)

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

2000 (2)

P. Mouroulis, R. O. Green, and T. G. Chrien, “Design of pushbroom imaging spectrometers for optimum recovery of spectroscopic and spatial information,” Appl. Opt. 39(13), 2210–2220 (2000).
[CrossRef] [PubMed]

P. Mouroulis and M. M. McKerns, “Pushbroom imaging spectrometer with high spectroscopic data fidelity: experimental demonstration,” Opt. Eng. 39(3), 808–816 (2000).
[CrossRef]

1999 (1)

P. Mouroulis, “Spectral and spatial uniformity in pushbroom imaging spectrometers,” Proc. SPIE 3753, 133–141 (1999).
[CrossRef]

1998 (1)

Casey, J. T.

J. T. Casey and J. P. Kerekes, “Misregistration impacts on hyperspectral target detection,” J. Appl. Remote Sens. 3(1), 033513 (2009).
[CrossRef]

Chrien, T. G.

Claxton, C. D.

Dell’Endice, F.

Dixon, R.

J. Zadnik, D. Guerin, R. Moss, A. Orbeta, R. Dixon, C. Simi, S. Dunbar, and A. Hill, “Calibration procedures and measurements for the COMPASS hyperspectral imager,” Proc. SPIE 5425, 182–188 (2004).
[CrossRef]

Dorn, R.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Du, H.

Dunbar, S.

J. Zadnik, D. Guerin, R. Moss, A. Orbeta, R. Dixon, C. Simi, S. Dunbar, and A. Hill, “Calibration procedures and measurements for the COMPASS hyperspectral imager,” Proc. SPIE 5425, 182–188 (2004).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Glockl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Green, R. O.

Guerin, D.

J. Zadnik, D. Guerin, R. Moss, A. Orbeta, R. Dixon, C. Simi, S. Dunbar, and A. Hill, “Calibration procedures and measurements for the COMPASS hyperspectral imager,” Proc. SPIE 5425, 182–188 (2004).
[CrossRef]

Hill, A.

J. Zadnik, D. Guerin, R. Moss, A. Orbeta, R. Dixon, C. Simi, S. Dunbar, and A. Hill, “Calibration procedures and measurements for the COMPASS hyperspectral imager,” Proc. SPIE 5425, 182–188 (2004).
[CrossRef]

Hovland, H.

Itten, K. I.

D. Schläpfer, J. Nieke, and K. I. Itten, “Spatial PSF Non-uniformity Effects In Airborne Pushbroom Imaging Spectrometry Data,” IEEE Trans. Geosci. Rem. Sens. 45(2), 458–468 (2007).
[CrossRef]

F. Dell’Endice, J. Nieke, D. Schläpfer, and K. I. Itten, “Scene-based method for spatial misregistration detection in hyperspectral imagery,” Appl. Opt. 46(15), 2803–2816 (2007).
[CrossRef] [PubMed]

Kerekes, J. P.

J. T. Casey and J. P. Kerekes, “Misregistration impacts on hyperspectral target detection,” J. Appl. Remote Sens. 3(1), 033513 (2009).
[CrossRef]

Leuchs, G.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Lin, G.

G. Lin, R. E. Wolfe, and M. Nishihama, “NPP VIIRS geometric performance status,” Proc. SPIE 8153, 81531V, 81531V-14 (2011).
[CrossRef]

McKerns, M. M.

P. Mouroulis and M. M. McKerns, “Pushbroom imaging spectrometer with high spectroscopic data fidelity: experimental demonstration,” Opt. Eng. 39(3), 808–816 (2000).
[CrossRef]

Moss, R.

J. Zadnik, D. Guerin, R. Moss, A. Orbeta, R. Dixon, C. Simi, S. Dunbar, and A. Hill, “Calibration procedures and measurements for the COMPASS hyperspectral imager,” Proc. SPIE 5425, 182–188 (2004).
[CrossRef]

Mouroulis, P.

P. Mouroulis and R. O. Green, “Spectral response evaluation and computation for pushbroom imaging spectrometers,” Proc. SPIE 6667, 66670G (2007).
[CrossRef]

P. Mouroulis and M. M. McKerns, “Pushbroom imaging spectrometer with high spectroscopic data fidelity: experimental demonstration,” Opt. Eng. 39(3), 808–816 (2000).
[CrossRef]

P. Mouroulis, R. O. Green, and T. G. Chrien, “Design of pushbroom imaging spectrometers for optimum recovery of spectroscopic and spatial information,” Appl. Opt. 39(13), 2210–2220 (2000).
[CrossRef] [PubMed]

P. Mouroulis, “Spectral and spatial uniformity in pushbroom imaging spectrometers,” Proc. SPIE 3753, 133–141 (1999).
[CrossRef]

Neville, R. A.

R. A. Neville, L. Sun, and K. Staenz, “Detection of keystone in imaging spectrometer data,” Proc. SPIE 5425, 208–217 (2004).
[CrossRef]

R. A. Neville and L. Sun, “Karl Staenz, “Detection of spectral line curvature in imaging spectrometer data,” Proc. SPIE 5093, 144–154 (2003).
[CrossRef]

Nieke, J.

F. Dell’Endice, J. Nieke, D. Schläpfer, and K. I. Itten, “Scene-based method for spatial misregistration detection in hyperspectral imagery,” Appl. Opt. 46(15), 2803–2816 (2007).
[CrossRef] [PubMed]

D. Schläpfer, J. Nieke, and K. I. Itten, “Spatial PSF Non-uniformity Effects In Airborne Pushbroom Imaging Spectrometry Data,” IEEE Trans. Geosci. Rem. Sens. 45(2), 458–468 (2007).
[CrossRef]

Nishihama, M.

G. Lin, R. E. Wolfe, and M. Nishihama, “NPP VIIRS geometric performance status,” Proc. SPIE 8153, 81531V, 81531V-14 (2011).
[CrossRef]

Orbeta, A.

J. Zadnik, D. Guerin, R. Moss, A. Orbeta, R. Dixon, C. Simi, S. Dunbar, and A. Hill, “Calibration procedures and measurements for the COMPASS hyperspectral imager,” Proc. SPIE 5425, 182–188 (2004).
[CrossRef]

Quabis, S.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Schläpfer, D.

D. Schläpfer, J. Nieke, and K. I. Itten, “Spatial PSF Non-uniformity Effects In Airborne Pushbroom Imaging Spectrometry Data,” IEEE Trans. Geosci. Rem. Sens. 45(2), 458–468 (2007).
[CrossRef]

F. Dell’Endice, J. Nieke, D. Schläpfer, and K. I. Itten, “Scene-based method for spatial misregistration detection in hyperspectral imagery,” Appl. Opt. 46(15), 2803–2816 (2007).
[CrossRef] [PubMed]

Simi, C.

J. Zadnik, D. Guerin, R. Moss, A. Orbeta, R. Dixon, C. Simi, S. Dunbar, and A. Hill, “Calibration procedures and measurements for the COMPASS hyperspectral imager,” Proc. SPIE 5425, 182–188 (2004).
[CrossRef]

Skauli, T.

Staenz, K.

R. A. Neville, L. Sun, and K. Staenz, “Detection of keystone in imaging spectrometer data,” Proc. SPIE 5425, 208–217 (2004).
[CrossRef]

Staunton, R. C.

Sun, L.

R. A. Neville, L. Sun, and K. Staenz, “Detection of keystone in imaging spectrometer data,” Proc. SPIE 5425, 208–217 (2004).
[CrossRef]

R. A. Neville and L. Sun, “Karl Staenz, “Detection of spectral line curvature in imaging spectrometer data,” Proc. SPIE 5093, 144–154 (2003).
[CrossRef]

Voss, K. J.

Wolfe, R. E.

G. Lin, R. E. Wolfe, and M. Nishihama, “NPP VIIRS geometric performance status,” Proc. SPIE 8153, 81531V, 81531V-14 (2011).
[CrossRef]

Zadnik, J.

J. Zadnik, D. Guerin, R. Moss, A. Orbeta, R. Dixon, C. Simi, S. Dunbar, and A. Hill, “Calibration procedures and measurements for the COMPASS hyperspectral imager,” Proc. SPIE 5425, 182–188 (2004).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. B (1)

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

IEEE Trans. Geosci. Rem. Sens. (1)

D. Schläpfer, J. Nieke, and K. I. Itten, “Spatial PSF Non-uniformity Effects In Airborne Pushbroom Imaging Spectrometry Data,” IEEE Trans. Geosci. Rem. Sens. 45(2), 458–468 (2007).
[CrossRef]

J. Appl. Remote Sens. (1)

J. T. Casey and J. P. Kerekes, “Misregistration impacts on hyperspectral target detection,” J. Appl. Remote Sens. 3(1), 033513 (2009).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

P. Mouroulis and M. M. McKerns, “Pushbroom imaging spectrometer with high spectroscopic data fidelity: experimental demonstration,” Opt. Eng. 39(3), 808–816 (2000).
[CrossRef]

Opt. Express (2)

Proc. SPIE (7)

P. Mouroulis and R. O. Green, “Spectral response evaluation and computation for pushbroom imaging spectrometers,” Proc. SPIE 6667, 66670G (2007).
[CrossRef]

P. Mouroulis, “Spectral and spatial uniformity in pushbroom imaging spectrometers,” Proc. SPIE 3753, 133–141 (1999).
[CrossRef]

R. A. Neville and L. Sun, “Karl Staenz, “Detection of spectral line curvature in imaging spectrometer data,” Proc. SPIE 5093, 144–154 (2003).
[CrossRef]

R. A. Neville, L. Sun, and K. Staenz, “Detection of keystone in imaging spectrometer data,” Proc. SPIE 5425, 208–217 (2004).
[CrossRef]

J. Zadnik, D. Guerin, R. Moss, A. Orbeta, R. Dixon, C. Simi, S. Dunbar, and A. Hill, “Calibration procedures and measurements for the COMPASS hyperspectral imager,” Proc. SPIE 5425, 182–188 (2004).
[CrossRef]

T. Skauli, “Quantifying coregistration errors in spectral imaging,” Proc. SPIE 8158, 81580A, 81580A-8 (2011).
[CrossRef]

G. Lin, R. E. Wolfe, and M. Nishihama, “NPP VIIRS geometric performance status,” Proc. SPIE 8153, 81531V, 81531V-14 (2011).
[CrossRef]

Other (1)

P. Mouroulis, D. A. Thomas, T. G. Chrien, V. Duval, R. O. Green, J. J. Simmonds, and A. H. Vaughan, Trade Studies in Multi/hyperspectral Imaging Systems—Final Report (NASA Jet Propulsion Laboratory, 1998).

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Figures (7)

Fig. 1
Fig. 1

Conceptual illustration of spatial coregistration error. Black lines represent nominal image pixel boundaries. The red and yellow lines represent the actual spatial responses for two bands in one image pixel. Coregistration errors result from differences in the position, size or shape of the spatial response, formulated in this paper in terms of the “sampling point spread function” (SPSF).

Fig. 2
Fig. 2

Illustration of two different cases of coregistration error. a) Sampling point spread function (SPSF) for one band, taken to be a Gaussian centered within the nominal pixel area indicated by black lines. b) SPSF for another band exhibiting coregistration error in the form of a 0.25 pixel shift with respect to the SPSF in a). c) An SPSF with a different coregistration error where the offset is the same as in b) and the peak is narrower. In d) and e), the SPSF in a) is plotted together with that of b) and c) respectively. In both cases, the boundary line where the SPSFs intersect defines the scene geometries which are most sensitive to the coregistration error, as discussed in the text. The graphs in f) and g) show cross sections of d) and e) along a line through the two peaks. Shading indicates the volume between SPSFs.

Fig. 3
Fig. 3

Weighting change resulting from an offset-type coregistration error. The graphs illustrate the case of two SPSFs offset by a small amount at a pixel center x0 = 0. The SPSFs are projected onto the direction of the offset, as discussed in the text. The maximum weighting error occurs for the case of a boundary in the scene at x = 0. The weighting error then corresponds to the area of the shaded rectangle. From the figure, it is clear that this is the same as the area between the SPSFs on one side of their intersection. This is the value of the coregistration metric (6) and approximates the conventional keystone measure, as discussed in the text.

Fig. 4
Fig. 4

Effect of coregistration error on the distribution of image pixels in spectral space for the case of two spectral bands i and j. The scene is assumed to consist of two randomly distributed materials A and B with spectra SA and SB respectively. In principle, all pixel spectra will fall on a line between the two endmember spectra. Even for an ideal sensor, the distribution is blurred by noise, as illustrated by the blue region, which represents a noise level of 1.5% of full scale. The red region shows the much larger broadening that may result from a coregistration error varying up to ε s,max =0.15 , equivalent to 15% keystone. This illustrates the potentially serious effect of coregistration error on image processing.

Fig. 5
Fig. 5

Spectral coregistration error. The peaks (blue) represent the spectral response functions (SRF) in a given band, for two image pixels with a spectral coregistration error between them. The resulting signal error is largest for an input spectrum with steps at the wavelengths where the SRFs intersect (black).

Fig. 6
Fig. 6

Illustration of interdependence of the spectral and spatial responsivity distributions. For simplicity, only one spatial dimension is considered. Assume a sensor with a spectral-spatial responsivity distribution which is uniform within the blue parallelogram, and zero elsewhere in the x-λ plane. The resulting SPSF f (x) and SRF g (x) are shown in the insets. The scene consists of a monochromatic point source indicated by a red dot. In one case, the source is at x1. Then the source will produce a larger output signal than expected from the SRF. If instead the source is at x2, the signal will be zero. Thus, the spectral-spatial interdependence can cause signal errors.

Fig. 7
Fig. 7

Sketch of a procedure to measure the SPSF by tomographic reconstruction. Left: A back-illuminated reticle with multiple slits is projected by a collimator (not shown) onto the imaging sensor field of view (green, assumed here to be a two-dimensional pixel array). The reticle is scanned in subpixel steps, as indicated by the blue arrows, while recording image data. Each bar and slit in the reticle is wide relative to the sensor pixel size, and the reticle is scanned by about one bar period. The scan is repeated in several discrete directions as indicated by the red arrow and dial. Right: The scanning creates a knife-edge cross section through all sensor pixels (green squares) in the field of view, assumed here to be a two-dimensional array. The SPSF of all sensor pixels can then be obtained by tomographic reconstruction [18,19].

Equations (27)

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

x,y f i (x,y)dxdy =1
w A,i = A f i (x,y)dxdy
S i = w A,i S A,i +(1 w A,i ) S B,i
Δw= A ( f j (x,y) f i (x,y) )dxdy
0 w A +Δw1
S j =( w A,i +Δw) S A,j +(1 w A,i Δw) S B,j = S B,j +( w A,i +Δw)( S A,j S B,j )
Δ S j =Δw( S A,j S B,j )
w A,i + w B,i =1= w A,j + w B,j w A,i w A,j = w B,j w B,i
A ( f i (x,y) f j (x,y) )dxdy = B ( f j (x,y) f i (x,y) )dxdy
Δ w max = 1 2 x,y | f j (x,y) f i (x,y) |dxdy = def ε s,ij
ε ¯ s = 1 PB(B1) p=1 P i ji ε ijp
ε s,max = max i,j,p ε s,ijp
ε ¯ s,i = 1 P(B1) ji p=1 P ε s,ijp .
P lim = P ε ¯ s .
f(x)= f(x,y)dy
Δ w max = ε s qf( x 0 )q/Δx
ε s qf( x 0 )q
Δ S i = ε ¯ s,i σ i
0 g p (λ)dλ =1.
ε λ,pq = def 1 2 0 | g p (λ) g q (λ) |dλ
f ip (x,y)= 0 F ip (x,y,λ)dλ   and   g ip (λ)= x,y F ip (x,y,λ)dxdy
w a = a F(x,y,λ)dxdydλ
w a,ideal = a f(x,y)g(λ)dxdydλ
w a = a ( F(x,y,λ)f(x,y)g(λ) )dxdydλ .
F(x,y,λ)f(x,y)g(λ)=0
ε λs,ip = def 1 2 xyλ | F ip (x,y,λ) f ip (x,y) g ip (λ) |dxdydλ
ε total = a s ε ¯ s + a λ ε ¯ λ + a λs ε ¯ λs

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