Multispectral filter wheel cameras: modeling aberrations with filters in front of lens

Transcription

1 Lehrtuhl für Bildverarbeitung Intitute of Imaging & Computer Viion Multipectral filter wheel camera: modeling aberration with filter in front of len Julie Klein and Til Aach Intitute of Imaging and Computer Viion RWTH Aachen Univerity, 5056 Aachen, Germany tel: , fax: web: in: IS&T/SPIE Electronic Imaging: Digital Photography VIII. See alo BibT E X entry below. BibT E author = {Julie Klein and Til Aach}, title = {Multipectral filter wheel camera: modeling aberration with filter in front of len}, booktitle = {{IS\&T/SPIE} Electronic Imaging: Digital Photography {VIII}}, year = {0}, page = {to appear}, addre = {San Francico, CA, USA}, month = {January --6}, publiher = {SPIE}, volume = {899} } 0 Society of Photo-Optical Intrumentation Engineer. Thi paper wa publihed in IS&T/SPIE Electronic Imaging: Digital Photography VIII and i made available a an electronic reprint with permiion of SPIE. One print or electronic copy may be made for peronal ue only. Sytematic or multiple reproduction, ditribution to multiple location via electronic or other mean, duplication of any material in thi paper for a fee or for commercial purpoe, or modification of the content of the paper are prohibited. document created on: November 5, 0 created from file: Klein0.tex cover page automatically created with CoverPage.ty available at your favourite CTAN mirror)

2 Multipectral filter wheel camera: modeling aberration for filter in front of len Julie Klein and Til Aach Intitute of Imaging and Computer Viion, RWTH Aachen Univerity, D-5056 Aachen, Germany ABSTRACT Aberration occur in multipectral camera featuring filter wheel becaue of color filter with different optical propertie being preent in the ray path. In order to enure an exact compenation of thee aberration, a mathematical model of the ditortion ha to be developed and it parameter have to be calculated uing the meaured data. Such a model already exit for optical filter placed between the enor and the len, but not for bandpa filter placed in front of the len. For thi configuration, the ray are firt ditorted by the filter and then by the len. In thi paper, we derive a model for aberration caued by filter placed in front of the len in multipectral camera. We compare thi model with ditortion obtained with imulation a well a with ditortion meaured during real multipectral acquiition. In both cae, the difference between modeled and meaured aberration remain low, which corroborate the phyical model. Multipectral acquiition with filter placed between the enor and the len or in front of the len are compared: the latter exhibit maller ditortion and the aberration in both image can be compenated uing the ame algorithm. Keyword: Multipectral camera, tranveral aberration, color filter, bandpa filter. INTRODUCTION Multipectral camera are ued in many domain for accurate color reproduction, ince by dividing the viible electromagnetic pectrum into more than three pectral channel they allow fulfilling the Luther rule much better than common RGB camera. One particular type of multipectral camera feature a monochrome enor and five to thirteen optical bandpa filter. The filter are poitioned either between the len and the enor 4, 8, 3 or in front of the len, a hown in Fig. ; thee configuration will be referred to a configuration B and configuration F, repectively. The configuration B i often preferred, for intance becaue filter placed in front of the len mut be bigger and are thu more expenive) in order to avoid any vignetting effect. While the aberration have already been modeled for thi configuration, 4 no comparable model ha been calculated for configuration F a far a the author know. In thi paper, a mathematical model i etablihed for configuration F and parallel with the aberration and model for configuration B are drawn. filter wheel len filter wheel monochrome camera len monochrome camera a) b) Figure : Two poible poition for the filter: configuration B between the enor and the len a) or configuration F in front of the len b). The aberration modeled in thi work concern configuration F. Further author information: Send correpondence to Julie Klein, julie.klein@lfb.rwth-aachen.de, Telephone: +49 0)

3 In the next ection, we will expoe the model for the tranveral aberration in configuration F and for the additional aberration. We then give reult obtained by imulating a camera with filter placed in front of the len and meauring the aberration. We alo compare the aberration in a real multipectral acquiition with the imulation and the model, before we finih with concluion.. MODELING OF ABERRATIONS The tranveral aberration appearing in the multipectral camera in configuration F are firt caued by the color filter and then by the len, a explained in the following paragraph. Once thee two type of ditortion are modeled for an optical ytem featuring one given filter, it i poible to model the relative ditortion between two different filter.. Filter aberration The calculation of the ditortion caued by filter relie on the model depicted in Fig.. The objective i repreented by a thin len of focal length f and the filter i repreented by a planar parallel plate of thickne l and refraction index n placed in front of the len with a tilt angle ϕ. The tilt angle i practically mall, but, due to unavoidable production pread of the filter wheel, it i not equal to zero, and varie from filter to filter. The ray that i conidered ha an angle θ with the optical axi and pae the center O of the len when the filter i in the ray path. Without any filter, the image point of the conidered object point i X; the image point ditorted becaue of the optical filter i X f. To derive the filter aberration d = X f X appearing on the enor plane, the firt tep i the calculation of the ditortion d f caued by the filter and meaured on the filter urface. Snell law for the ray reaching the firt filter urface tate inα) = n inβ) ) with the angle α of the incoming ray relative to the filter normal, the angle β of the ray refracted by the firt urface of the filter, the filter refraction index n and the refraction index of air et to. The ditortion d f can be meaured uing the point I, A and B and d f = IB IA ) object plane filter len enor plane n d c X c image point with filter and len aberration X f image point with filter aberration d X image point without any aberration φ α a β I A d f B d l θ O C image center E eential point n object point n l focal length f e Y e X e Z enor ditance Figure : Model for the filter poitioned in front of the len.

4 The vector IB can be een a a linear combination of the unit vector a defining the direction of the incoming ray and of the unit vector n which i perpendicular to the filter firt urface IB = ξ a + ξ n 3) The two unknown coefficient ξ and ξ can be calculated uing the projection of IB onto the two unit vector n defining the filter firt urface and n perpendicular to thi urface, repectively IB n = l coβ) inβ) = ξ inα) IB n = l = ξ coα) + ξ 4) where repreent the calar product and l i the thickne of the filter. The value of the unknown are then given by l inβ) ξ = coβ) inα) = l n coβ) ξ = l ξ coα) = l coα) ) 5) n coβ) The vector IA i calculated uing the intercept theorem IA = a l coα) Uing Eq. 6) and Eq. 3) after inertion of Eq. 5), the equation giving the ditortion d f on the filter urface become l d f = n coβ) a + l coα) ) l n n coβ) coα) a = l coα) ) n ) 7) n coβ) coα) a Once the ditortion d f on the filter urface i known, the ditortion d l on the len plane i defined a the um of the ditortion d f and a vector having the ame direction a a d l = d f + ξ 3 a 8) The unknown coefficient ξ 3 can be calculated a previouly uing the projection of d l onto the optical axi thu yielding d l e z = 0 = l coα) ) n e z ) n coβ) coα) a e 9) z + ξ 3 a e z ξ 3 = l coα) ) n coβ) coα) n e ) z a e z ) By inerting Eq. 7) and 9) into Eq. 8), the ditortion in the len plane can be calculated uing d l = l coα) ) n n e ) z a n coβ) a e z The ditortion d along the enor plane i then given by d = f d l with the ditance between the len and the enor and f the focal length of the len: d = f l coα) ) n n e ) z a, ) n coβ) a e z 6) 0) )

5 where the term note κ = f l a a e z can be actually replaced by X f ) and rewrite Eq. ) coα) n coβ) uing the intercept theorem. To implify the equation, we d = κ n κ n e z X f 3) which mean that the ditortion caued by the filter can be modeled by an affine tranformation d = κ n ez 0 0 κ n e x 0 κ n ez 0 κ n e y 0 0 κ n ez κ n e z Xf The ditortion term along the optical axi e z i 0 ince X f = x f, y f, ) T and Eq. 4) can be implified to κ n ez 0 κ n e x x f x f d = 0 κ n ez κ n e y y f = M f y f 5) with the matrix M f R 3 3. For an object placed at infinite ditance, the enor plane coincide with the focu plane, i.e., = f, and the ditortion d thu become null. The ray coming from an object at infinite ditance are parallel when no color filter i in their path and they focu on one given poition in the focu plane. When a color filter i utilized, the ray are refracted twice at the urface of the filter but remain parallel and with the ame direction at the output of the filter: they then focu exactly on the ame poition in the focu plane.. Relative filter ditortion We now conider the relative ditortion d due to the inertion of a new filter in the ray path, i.e., the ditortion caued when a filter F with the parameter l, n, ϕ, n and conequently related to the angle α and β ) i replaced by a filter F with the parameter l, n, ϕ, n and conequently related to the angle α and β ). Of coure, the ray leaving the filter F doe not necearily pa through O. The ditortion on the enor plane for thee two filter are n e z d, = κ n κ X f, n e 6) z d, = κ n κ X f, ) where κ i = f l i coαi) n i coβ i) for i =,. X f, i ued in both equation becaue we replaced a a e z by X f,. The relative ditortion d = d, d, i then given by d = κ n κ n }{{} κ n e z κ n e z ) X f, }{{ } T T X f, ) ) 4). 7) The firt part of Eq. 7), T, decribe a global tranlation for all the image poition. and the econd part of thi equation, T X f, ), i an affine diplacement that depend on the poition X f, of the image point ditorted by the filter F. The relative filter ditortion for the poition X f, = x f,, y f,, ) T can thu be modeled by an affine tranformation with the matrix M rel f R 3 3 d = M rel f x f, y f, 8) The third row of the matrix M rel f i compoed of 0 becaue the relative ditortion i in the enor plane, a in Eq. 5). The poition X f, = X f, + d of the image point ditorted by the filter F thu i an affine tranformation of the image point ditorted by the filter F. Thi reult i comparable to the model of aberration for filter placed between the len and the enor developed by Brauer et al., 3

6 .3 Len aberration After the filter aberration, the ray are alo ditorted by the len. The len ditortion include the primary monochromatic aberration and the chromatic aberration. 4 Thee ditortion are function of the ray coordinate at the ytem aperture. The primary monochromatic aberration are not affine but rather third-order term. 5 In the cae of paraxial imaging like in our optical ytem, the monochromatic aberration are neglected and only the chromatic aberration are taken into account. In thi work, they are approximated by an affine model. The image point X f, and X f, are ditorted to the image point X c, and X c,, repectively, becaue of the len aberration. The ditortion due to the len are d c, = X c, X f, and d c, = X c, X f,. They are calculated with an affine model uing the matrice M c, R 3 3 and M c, R 3 3 according to x f, d c, = M c, y f, 9) x f, d c, = M c, y f, We now eek to find the relative ditortion between the two image point X c, and X c,, which are the only image point we have acce to in our image X c, X c, = X c, X f, ) X c, X f, ) + X f, X f, ) = d c, d c, + d 0) From Eq. 8) and 9), we know that d c, and d are affine function of the image point X f,. d c, i an affine function of the image point X f,, which i in turn an affine function of the image point X f, according to Eq. 8) and X f, = X f, d, + d,. Thi mean that X c, X c, i an affine function of the image point X f,, and thu alo an affine function of the image point X c,. The whole ditortion X c = X c, X c, that we can meaure on our image or our imulation i finally an affine function of the image point X c, = x c,, y c,, ) T that can be expreed a X c = M x c, y c, ) with the matrix M R MEASUREMENTS OF ABERRATIONS To evaluate the accuracy of the affine model we derived, we performed ditortion meaurement in optical ytem featuring a color filter placed in front of a len, utilizing imulation data a well a data acquired with a real imaging ytem. We imulated the len according to the data in [6] with the imulation oftware Zemax Zemax Development Corporation, Bellevue, WA, USA) and meaured the relative aberration caued by two different color filter. The two filter had the ame thickne, i.e., l = l, and the ame refraction index, i.e., n = n, but different tilt angle: the reference filter wa tilted by + around e x and the filter for which the ditortion were imulated by around e y. The object imaged by thi imulated optical ytem wa a grid of point. We compared the poition on the enor plane of thee object point ditorted by the two filter. For each filter, we conidered only ray from one given wavelength out of the 7 central wavelength of the color filter of our real ytem. Beide the imulation of a multipectral camera, we alo acquired real multipectral image to meaure the ditortion with thi real optical ytem. The 7 color filter we utilize have central wavelength going from 400 nm to 700 nm in tep of 50 nm and bandwidth of about 40 nm and are mounted in a motorized filter wheel. The

7 len i an apherical AF-S DX Nikkor 8-70 nm, which i imilar to the len ued for the imulation, and the monochrome camera i a IDS ueye 40 CCD camera. In thi cae, the ditortion were not meaured uing ome point of interet pread over the image like for the imulation. Intead, we divide the image into 90 region of interet and calculate for each of thee the diplacement due to the current filter relative to the reference filter uing mutual information a imilarity meaure. Thi reult in a field of diplacement vector a hown in Fig. 4. The random ample conenu RANSAC) algorithm i then utilized to remove tochatic error in the vector field. More detail about thi meaurement are provided by Brauer et al. 3 We alo directly compared two multipectral image acquired with the two configuration hown in Fig.. We ued the ame monochrome camera, filter wheel and len to perform the two acquiition. 4. RESULTS The aberration of configuration F meaured with the imulated optical ytem are hown in Fig. 3a. For thee reult, we utilized a reference color filter the filter with wavelength 700 nm, that i, we traced the ray only for the wavelength 700 nm, and meaured the ditortion for the color filter with wavelength 500 nm. The object point that were followed are the corner of a grid. The imulated ditortion and the reult of our model are hown in the figure with black and white vector, repectively. At the corner of the enor plane, the ditortion are about.8 pixel. The vector of the imulated and of the modeled relative ditortion are very cloe and the error are below pixel ee Fig. 3b): thi indicate that the affine model i well uited for multipectral camera with optical filter placed in front of the len Grid point a) b) Figure 3: Aberration imulated for the corner poition of a grid for the color channel 500 nm relative to the color channel 700 nm black vector) and ditortion calculated with our model white) a). The ioline correpond to the length of the modeled ditortion, in pixel. The pixel error between the imulated and the modeled ditortion are hown in b) for the grid point. Model error in pixel We alo compared the image point imulated for the filter with wavelength 450 nm to the image point for the 6 other filter. The reference filter wa tilted by + around e x and the other filter by around e y. The mean and maximum error of our model are ummarized in Tab. for all filter: the mean error lie motly below 0.0 pixel and the maximum error i 0.05 pixel. Thee low error corroborate our affine model. For the real acquiition data, we did not ued point of interet in the image to meaure the ditortion, but rather whole region, a explained in Sec. 3. A can be een in Fig. 4a, the aberration meaured on eparate block of the image black vector) and the aberration from the affine model white vector) are very cloe. Outlier are poible during the meaurement of the ditortion ee the black vector on the bottom of the image) and are eliminated uing the RANSAC algorithm. The ditortion meaured between the color channel 650 nm and 550 nm are larger than the imulated one, reaching 8 pixel intead of.8 pixel for the imulation.

8 Error Filter in pixel) 400 nm 450 nm 500 nm 550 nm 600 nm 650 nm 700 nm Maximum Mean Table : Maximum and mean value of the model error for the even filter baed on imulated data. The reference filter here i filter 550 nm and the value are calculated over the grid point. Thi can be explained by the filter parameter: the tilt angle were et to ± for the imulation, but the real value are almot certainly different. The thickne and refraction index of the real filter may alo be not exactly the ame a in the imulation. The error of the model plotted in Fig. 4b are only calculated for the region marked a inlier by the algorithm. They are higher than with the imulation data: the mean error for thi filter i pixel. The mean and maximum error for the other filter can alo be taken from Tab.. The error are 0 to 00 time higher than the error in the imulation data, but remain low with mean error of about 0. pixel for all the filter. Thi can be explained by the ditortion themelve that are much larger than during the imulation. Another explanation can be the bandwidth of the filter we ued: it i about 40 nm, wherea for the imulation we conidered narrowband filter by taking only ray with the filter central wavelength. Thi would mean that the model for the chromatic aberration we ued could be improved with one of the model explained by Klein et al. 7 for intance Model error in pixel Region inlier a) b) Figure 4: Ditortion meaured for color channel 650 nm relative to color channel 550 nm black vector) and model value reulting from thi meaurement white) a). The vignetting effect i viible on the background image from color channel 650 nm. The pixel error between the meaured and the modeled ditortion are hown in b) for the 50 region of interet that are not outlier of the RANSAC algorithm. Error Filter in pixel) 400 nm 450 nm 500 nm 550 nm 600 nm 650 nm 700 nm Maximum Mean Table : Maximum and mean value of the model error for the even filter baed on real data. The reference filter here i filter 550 nm and the value are calculated over the inlier of the RANSAC algorithm. The matrice for the affine model of the aberration are very imilar for the imulation and for the real data, a can be een in Tab. 3. The lat row term are 0, ince the aberration are in the enor plane, i.e., perpendicular to e z. The diagonal term are cloe to, the amplitude of the tranlation term third column) and of the hear term term at poition,) and,) in the matrix) are larger for the real image acquiition than for the imulation.

9 Simulation: Real acquiition: M = M = Table 3: Example matrice for the affine model of the ytem aberration from Eq. ) for our imulation and our acquiition of real image. The ditortion matrix for the imulation correpond to Fig. 3a and the ditortion matrix for the real acquiition correpond to Fig. 4a. a) b) c) d) Figure 5: Region of a multipectral acquiition with configuration B a) and with configuration F c). The correponding image with corrected aberration are b) and d), repectively. Matrice for the correction are given in Tab. 4. Each region repreent an area of 6 6 pixel. We alo directly compared the ditortion of two multipectral imaging ytem in configuration B and configuration F. Part of the reulting image are hown in Fig. 5. The image obtained with configuration F Fig. 5c) are almot a much ditorted a the image obtained with configuration B Fig. 5a). The ame algorithm 3 wa utilized to compenate the ditortion in the two filter configuration and the unditorted image in Fig. 5d and 5b do not exhibit any remaining ditortion uch a color fringe. Thi mean that the algorithm that ha been developed for configuration B 3 can alo be ued to capture and compenate the tranveral aberration for configuration F. Matrice of the affine model for the two configuration are compared in Tab. 4. The matrice for the real acquiition in the two different configuration are much cloer compared to the matrice for imulation and real data in Tab. 3. The amplitude of the tranlation term and of the hear term are comparable for the two configuration. M = configuration B: configuration F : M = Table 4: Example matrice for the affine model of the ytem aberration from Eq. ) for configuration B and configuration F. The ditortion matrix for configuration B correpond to Fig. 5a and the ditortion matrix for configuration F correpond to Fig. 5c. 5. CONCLUSIONS We have developed a model for the aberration in multipectral camera where the optical filter are placed in front of the len. Auming a perfect len without aberration, the ditortion are affine and the model i imilar to the one for camera with filter poitioned between the len and the enor. We have confirmed the model by both imulating multipectral imaging with optical filter in front of the len and acquiring real image with uch a camera. After compenation of the ditortion uing our affine model, no color fringe remain viible. The model could further be improved by conidering another model for the len aberration.

10 6. ACKNOWLEDGMENTS The author acknowledge gratefully funding by the German Reearch Foundation DFG, grant AA5/ ). REFERENCES [] Luther, R., Au dem Gebiet der Farbreizmetrik, Zeitchrift für techniche Phyik 8, ). [] Brauer, J., Schulte, N., and Aach, T., Modeling and compenation of geometric ditortion of multipectral camera with optical bandpa filter wheel, in [5th European Signal Proceing Conference], September 007). [3] Brauer, J., Schulte, N., and Aach, T., Multipectral filter-wheel camera: Geometric ditortion model and compenation algorithm, IEEE Tranaction on Image Proceing 7, December 008). [4] Brauer, J. and Aach, T., Geometric calibration of len and filter ditortion for multipectral filter-wheel camera, IEEE Tranaction on Image Proceing 0, February 0). [5] Burn, P. D. and Bern, R. S., Analyi multipectral image capture, in [Proc. IS&T/SID 4th Color Imaging Conference CIC)], 4, 9 November 996). [6] Haneihi, H., Iwanami, T., Honma, T., Tumura, N., and Miyake, Y., Goniopectral imaging of threedimenional object, Journal of Imaging Science and Technology 455), ). [7] Helling, S., Seidel, E., and Biehlig, W., Algorithm for pectral color timulu recontruction with a evenchannel multipectral camera, in [Proc. IS&T nd European Conference on Color in Graphic, Imaging, and Viion CGIV)],, April 004). [8] Imai, F. H., Quan, S., Roen, M. R., and Bern, R. S., Digital camera filter deign for colorimetric and pectral accuracy, in [Proceeding of the Third International Conference on Multipectral Color Science, IS&T], 3 6 June 00). [9] Manouri, A., Marzani, F. S., Hardeberg, J. Y., and Gouton, P., Optical calibration of a multipectral imaging ytem baed on interference filter, SPIE Optical Engineering 44, February 005). [0] Ribé, A., Schmitt, F., Pillay, R., and Lahanier, C., Calibration and pectral recontruction for CRISATEL: An art painting multipectral acquiition ytem, Journal of Imaging Science and Technology 49, November/December 005). [] Tominaga, S., Fukuda, T., and Kimachi, A., A high-reolution imaging ytem for omnidirectional illuminant etimation, Journal of Imaging Science and Technology 5, July/Augut 008). [] Vilaeca, M., Mercadal, R., Pujol, J., Arjona, M., de Laarte, M., Huerta, R., Melgoa, M., and Imai, F. H., Characterization of the human iri pectral reflectance with a multipectral imaging ytem, Applied Optic 47, October 008). [3] Hardeberg, J. Y., Recent advance in acquiition and reproduction of multipectral image, in [EURASIP 4th European Signal Proceing Conference], 006). [4] Gro, H., Zügge, H., Pechka, M., and Blechinger, F., [Handbook of Optical Sytem], vol. 3: Aberration Theory and Correction of Optical Sytem, Wiley VCH Verlag GmbH 007). [5] Smith, W. J., [Modern Optical Engineering], McGraw-Hill 000). [6] Hayakawa, S., Zoom len ytem. Patent March 005). US. Pat. 005/ A. [7] Klein, J., Brauer, J., and Aach, T., Spatio-pectral modeling and compenation of tranveral chromatic aberration in multipectral imaging, Journal of Imaging Science and Technology 0). Accepted for publication.