IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 22, NO. 2, FEBRUARY 2013 501 Generalized Inverse-Approach Model for Spectral-Signal Recovery Shahram Peyvandi, Seyed Hossein Amirshahi, Javier Hernández-Andrés, Juan Luis Nieves, and Javier Romero Abstract—We have studied the transformation system of a spectral signal to the response of the system as a linear mapping from higher to lower dimensional space in order to look more closely at inverse-approach models. The problem of spectral- signal recovery from the response of a transformation system is generally stated on the basis of the generalized inverse-approach theorem, which provides a modular model for generating a spec- tral signal from a given response value. The controlling criteria, including the robustness of the inverse model to perturbations of the response caused by noise, and the condition number for matrix inversion, are proposed, together with the mean square error, so as to create an efficient model for spectral-signal recovery. The spectral-reflectance recovery and color correction of natural surface color are numerically investigated to appraise different illuminant-observer transformation matrices based on the proposed controlling criteria both in the absence and the presence of noise. Index Terms— Color, inverse problem, spectral analysis, spectral-signal reconstruction. I. I NTRODUCTION T HE RECENT development of multispectral imaging sys- tems together with an urgent need for a hyper-spectral data-acquisition system have given rise to extensive research in imaging science aimed at finding efficient ways of acquiring multispectral signals [1], [2]. Within this context, the over- sampling of hyperspectral signals as well as the inevitable noise involved in recording images demand an efficient process for image storage, communication and restoration without any loss of information. On the contrary, typical color-image acquisition devices record under-sampling spectral signals for each pixel. The spectral reconstruction of pixels is highly desirable in order to retrieve the colorimetric information of the image under any illumination conditions [3]. Manuscript received November 26, 2011; revised July 25, 2012; accepted August 24, 2012. Date of publication September 13, 2012; date of current version January 8, 2013. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Oscar C. Au. S. Peyvandi was with the Department of Textile Engineering, Amirkabir University of Technology, Tehran 15914, Iran. He is now with the Department of Psychology, Rutgers, State University of New Jersey, Newark, NJ 07102 USA (e-mail: peyvandi@psychology.rutgers.edu). S. H. Amirshahi is with the Department of Textile Engineering, Amirkabir University of Technology, Tehran 15914, Iran (e-mail: hamirsha@aut.ac.ir). J. Hernandez-Andres, J. L. Nieves and J. Romero are with the Departamento de Optica, Facultad de Ciencias, Universidad de Granada, Granada 18071, Spain (e-mail: javierha@ugr.es; jnieves@ugr.es; jromero@ugr.es). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIP.2012.2218823 In 1964 Cohen analyzed the reflectance spectra of 433 chips in the Munsell Book and fitted a linear model to a set of spec- tra [4]. This considerable achievement formed the foundations of spectral reflectance recovery using linear models [5]–[7]. On the basis of linear models of basis functions of the spectral dataset [8], [9] a variety of methods were later developed for spectral recovery in the field of color science [10]–[14]. Spectral-based techniques have also been investigated with the intention of recovering the reflectance of surface colors from the sensor response of a digital camera [15], [16]. As far as imaging applications are concerned, Bayesian inference for inverse problems has been widely used as a far-reaching tool for image restoration [17], [18]. In the Bayesian inference model for inverse problems the posterior distribution is obtained by prior knowledge of the signal and noise. The Wiener filter restoration method, resulting from Bayes theorem, is a common technique for signal restora- tion, image reconstruction and communications problems [3], [17]–[20]. In 1995 Brainard suggested that Bayesian method might be suitable for spectral recovery from the RGB sensor responses [21]. Bayesian decision theory was also employed to deal with the problem of color constancy to compute the posterior distribution of the illuminants and recover the physical properties of the surfaces in the scene for a given set of sensor responses [22]. The Wiener filter-restoration approach as a solution for inverse problems in imaging science has been widely used for spectral-reflectance reconstruction from image-capturing sensor responses [23]–[28] and also for color correction methods [29]–[32]. To build a parametric inverse-approach model for spectral recovery or color correction, a modular model is created, the arguments of which can be set up to optimize the inverse model based on a preferred criterion. We first of all present here a theoretical background to introduce a transformation system together with the Moore-Penrose inverse approach and Bayes’ method for inverse problems. Subsequently, in Section III, we introduce a theorem as a generalized approach for inverse problems in spectral-signal recovery from the response of a transformation system. The proposed theorem provides a general parametric form to generate mathematically a set of spectra given an individual response. The proposed theorem is extended to include the presence of noise as a perturbation of response in a transformation system. Since the optimum lighting condition for practical spectral recovery has always been a matter of great concern in color and imaging technology, in Section IV we investigate spectral- 1057–7149/$31.00 © 2012 IEEE