{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:26:32Z","timestamp":1760239592864,"version":"build-2065373602"},"reference-count":42,"publisher":"MDPI AG","issue":"23","license":[{"start":{"date-parts":[[2020,12,2]],"date-time":"2020-12-02T00:00:00Z","timestamp":1606867200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/S028730\/1"],"award-info":[{"award-number":["EP\/S028730\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>By placing a color filter in front of a camera we make new spectral sensitivities. The Luther-condition optimization solves for a color filter so that the camera\u2019s filtered sensitivities are as close to being linearly related to the XYZ color matching functions (CMFs) as possible, that is, a filter is found that makes the camera more colorimetric. Arguably, the more general Vora-Value approach solves for the filter that best matches all possible target spectral sensitivity sets (e.g., any linear combination of the XYZ CMFs). A concern that we investigate here is that the filters found by the Luther and Vora-Value optimizations are different from one another. In this paper, we unify the Luther and Vora-Value approaches to prefilter design. We prove that if the target of the Luther-condition optimization is an orthonormal basis\u2014a special linear combination of the XYZ CMFs which are orthogonal and are in unit length\u2014the discovered Luther-filter is also the filter that maximizes the Vora-Value. A key advantage of using the Luther-condition formulation to maximize the Vora-Value is that it is both simpler to implement and converges to its optimal answer more quickly. Experiments validate our method.<\/jats:p>","DOI":"10.3390\/s20236882","type":"journal-article","created":{"date-parts":[[2020,12,2]],"date-time":"2020-12-02T07:49:54Z","timestamp":1606895394000},"page":"6882","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Mathematical Investigation into the Design of Prefilters That Make Cameras More Colorimetric"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7381-6607","authenticated-orcid":false,"given":"Yuteng","family":"Zhu","sequence":"first","affiliation":[{"name":"School of Computing Sciences, University of East Anglia, Norwich NR4 7TJ, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4040-0554","authenticated-orcid":false,"given":"Graham D.","family":"Finlayson","sequence":"additional","affiliation":[{"name":"School of Computing Sciences, University of East Anglia, Norwich NR4 7TJ, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Hunt, R.W.G., and Pointer, M.R. (2011). Measuring Colour, John Wiley & Sons. [4th ed.].","DOI":"10.1002\/9781119975595"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/0734-189X(84)90180-4","article-title":"Exact reproduction of colored images","volume":"26","author":"Horn","year":"1984","journal-title":"Comput. Vis. Graph. Image Process."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1016\/S0016-0032(15)90396-4","article-title":"The transformation of color-mixture equations from one system to another","volume":"180","author":"Ives","year":"1915","journal-title":"J. Frankl. Inst."},{"key":"ref_4","first-page":"540","article-title":"Aus dem Gebiet der Farbreizmetrik","volume":"8","author":"Luther","year":"1927","journal-title":"Z. Tech. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"23205","DOI":"10.3390\/s141223205","article-title":"Perceptual color characterization of cameras","volume":"14","author":"Connah","year":"2014","journal-title":"Sensors"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1109\/83.277897","article-title":"Filter considerations in color correction","volume":"3","author":"Vrhel","year":"1994","journal-title":"IEEE Trans. Image Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"901","DOI":"10.1109\/83.597268","article-title":"Digital color imaging","volume":"6","author":"Sharma","year":"1997","journal-title":"IEEE Trans. Image Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"990","DOI":"10.1109\/83.597274","article-title":"Figures of merit for color scanners","volume":"6","author":"Sharma","year":"1997","journal-title":"IEEE Trans. Image Process."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Finlayson, G.D., Zhu, Y., and Gong, H. (2018, January 12\u201316). Using a simple colour pre-filter to make cameras more colorimetric. Proceedings of the 26th Color and Imaging Conference Final Program and Proceedings, Vancouver, BC, Canada.","DOI":"10.2352\/ISSN.2169-2629.2018.26.182"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Finlayson, G.D., and Zhu, Y. (2020). Designing Color Filters that Make Cameras More Colorimetric. IEEE Trans. Image Process., forthcoming.","DOI":"10.1109\/TIP.2020.3038523"},{"key":"ref_11","unstructured":"International Telecommunication Union (2020, November 15). BT.709: Parameter Values for the HDTV Standards for Production and International Programme Exchange. Available online: https:\/\/www.itu.int\/dms_pubrec\/itu-r\/rec\/bt\/R-REC-BT.709-5-200204-S!!PDF-E.pdf."},{"key":"ref_12","unstructured":"Wandell, B.A. (1995). Foundations of Vision, Sinauer Associates."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1109\/83.913597","article-title":"Inner products and orthogonality in color recording filter design","volume":"10","author":"Vora","year":"2001","journal-title":"IEEE Trans. Image Process."},{"key":"ref_14","unstructured":"Finlayson, G.D., and Zhu, Y. (2020, January 16\u201317). Unifying Optimization Methods for Color Filter Design. Proceedings of the 10th Colour and Visual Computing Symposium, CEUR Workshop Proceedings, Gj\u00f8vik, Norway."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Zhu, Y., and Finlayson, G.D. (2020). Designing a Color Filter via Optimization of Vora-Value for Making a Camera more Colorimetric. arXiv.","DOI":"10.2352\/issn.2169-2629.2020.28.28"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Boyd, S., Boyd, S.P., and Vandenberghe, L. (2004). Convex Optimization, Cambridge University Press.","DOI":"10.1017\/CBO9780511804441"},{"key":"ref_17","unstructured":"Zhu, Y., and Finlayson, G. (2020). An improved optimization method for finding a color filter to make a camera more colorimetric. Soc. Imaging Sci. Technol., 2020."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1109\/83.551700","article-title":"Mathematical methods for the design of color scanning filters","volume":"6","author":"Vora","year":"1997","journal-title":"IEEE Trans. Image Process."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Roshan, E., and Funt, B. (2020). Color Sensor Accuracy Index Utilizing Metamer Mismatch Radii. Sensors, 20.","DOI":"10.3390\/s20154275"},{"key":"ref_20","unstructured":"Quan, S., Ohta, N., and Katoh, N. (2000, January 7\u201310). Optimization of camera spectral sensitivities. Proceedings of the Eighth Color Imaging Conference: Color Science and Engineering Systems, Technologies, Applications, Scottsdale, AZ, USA."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"485","DOI":"10.2352\/J.ImagingSci.Technol.2002.46.6.art00003","article-title":"Unified Measure of Goodness and Optimal Design of Spectral","volume":"46","author":"Quan","year":"2002","journal-title":"J. Imaging Sci. Technol."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"814","DOI":"10.1109\/83.388082","article-title":"Optimal color filters in the presence of noise","volume":"4","author":"Vrhel","year":"1995","journal-title":"IEEE Trans. Image Process."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1109\/83.650859","article-title":"Optimal nonnegative color scanning filters","volume":"7","author":"Sharma","year":"1998","journal-title":"IEEE Trans. Image Process."},{"key":"ref_24","unstructured":"Farrell, J.E., and Wandell, B.A. (1995). Method and Apparatus for Identifying the Color of an Image. (5,479,524), U.S. Patent."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"267","DOI":"10.2352\/J.ImagingSci.Technol.2000.44.4.art00002","article-title":"Imaging colorimetry using a digital camera","volume":"44","author":"Wu","year":"2000","journal-title":"J. Imaging Sci. Technol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"670","DOI":"10.1364\/JOSA.35.000670","article-title":"Colorimetric specifications of Wratten light filters","volume":"35","author":"MacAdam","year":"1945","journal-title":"JOSA"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"105","DOI":"10.2352\/J.ImagingSci.Technol.2004.48.2.art00005","article-title":"Filter selection for multispectral color image acquisition","volume":"48","author":"Hardeberg","year":"2004","journal-title":"J. Imaging Sci. Technol."},{"key":"ref_28","unstructured":"Imai, F.H., Quan, S., Rosen, M.R., and Berns, R.S. (2001). Digital Camera Filter Design for Colorimetric and Spectral Accuracy, University of Joensuu."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Arad, B., and Ben-Shahar, O. (2017, January 22\u201329). Filter selection for hyperspectral estimation. Proceedings of the IEEE International Conference on Computer Vision, Venice, Italy.","DOI":"10.1109\/ICCV.2017.342"},{"key":"ref_30","unstructured":"Li, S., and Zhang, L. (2016, January 26\u201329). Optimal sensitivity design of multispectral camera via broadband absorption filters based on compressed sensing. Proceedings of the 3rd International Symposium of Space Optical Instruments and Applications, Beijing, China."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Li, S.X. (2018). Filter Selection for Optimizing the Spectral Sensitivity of Broadband Multispectral Cameras Based on Maximum Linear Independence. Sensors, 18.","DOI":"10.3390\/s18051455"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"074102","DOI":"10.1117\/1.OE.55.7.074102","article-title":"Filter selection based on light source for multispectral imaging","volume":"55","author":"Xu","year":"2016","journal-title":"Opt. Eng."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Mart\u00ednez-Domingo, M.\u00c1., Melgosa, M., Okajima, K., Medina, V.J., and Collado-Montero, F.J. (2019). Spectral Image Processing for Museum Lighting Using CIE LED Illuminants. Sensors, 19.","DOI":"10.3390\/s19245400"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1109\/83.491323","article-title":"Optimization of sensor response functions for colorimetry of reflective and emissive objects","volume":"5","author":"Wolski","year":"1996","journal-title":"IEEE Trans. Image Process."},{"key":"ref_35","unstructured":"Golub, G.H., and Van Loan, C.F. (1996). Matrix Computations, Johns Hopkins University Press. [3rd ed.]."},{"key":"ref_36","unstructured":"Zhu, Y., and Finlayson, G.D. (2020). Mathematical derivation for Vora-Value based filter design method: Gradient and Hessian. arXiv."},{"key":"ref_37","unstructured":"Nocedal, J., and Wright, S. (2006). Numerical Optimization, Springer."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Luenberger, D.G., and Ye, Y. (2015). Linear and Nonlinear Programming, Springer. [4th ed.].","DOI":"10.1007\/978-3-319-18842-3"},{"key":"ref_39","unstructured":"Wyszecki, G., and Stiles, W.S. (1982). Color Science: Concepts and Methods, Quantitative Data and Formulae, Wiley. [2nd ed.]."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1002\/col.10049","article-title":"A data set for color research","volume":"27","author":"Barnard","year":"2002","journal-title":"Color Res. Appl."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1499","DOI":"10.1364\/JOSAA.10.001499","article-title":"Measure of goodness of a set of color-scanning filters","volume":"10","author":"Vora","year":"1993","journal-title":"J. Opt. Soc. Am. A"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Jiang, J., Liu, D., Gu, J., and S\u00fcsstrunk, S. (2013, January 15\u201317). What is the space of spectral sensitivity functions for digital color cameras?. Proceedings of the 2013 IEEE Workshop on Applications of Computer Vision (WACV), Tampa, FL, USA.","DOI":"10.1109\/WACV.2013.6475015"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/23\/6882\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:40:25Z","timestamp":1760179225000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/23\/6882"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,2]]},"references-count":42,"journal-issue":{"issue":"23","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["s20236882"],"URL":"https:\/\/doi.org\/10.3390\/s20236882","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2020,12,2]]}}}