Demosaicking by Alternating Projections: Theory and Fast One-Step Implementation

Color image demosaicking is a key process in the digital imaging pipeline. In this paper, we study a well-known and influential demosaicking algorithm based upon alternating projections (AP), proposed by Gunturk, Altunbasak and Mersereau in 2002. Since its publication, the AP algorithm has been widely cited and compared against in a series of more recent papers in the demosaicking literature. Despite good performances, a limitation of the AP algorithm is its high computational complexity. We provide three main contributions in this paper. First, we present a rigorous analysis of the convergence property of the AP demosaicking algorithm, showing that it is a contraction mapping, with a unique fixed point. Second, we show that this fixed point is in fact the solution to a constrained quadratic minimization problem, thus, establishing the optimality of the AP algorithm. Finally, using the tool of polyphase representation, we show how to obtain the results of the AP algorithm in a single step, implemented as linear filtering in the polyphase domain. Replacing the original iterative procedure by the proposed one-step solution leads to substantial computational savings, by about an order of magnitude in our experiments.

[1]  W. Cheney,et al.  Proximity maps for convex sets , 1959 .

[2]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[3]  Arch W. Naylor,et al.  Linear Operator Theory in Engineering and Science , 1971 .

[4]  B. A. D. H. Brandwood.  A complex gradient operator and its applica-tion in adaptive array theory , 1983 .

[5]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[6]  Andreas Antoniou,et al.  Two-Dimensional Digital Filters , 2020 .

[7]  Jelena Kovacevic,et al.  Wavelets and Subband Coding , 2013, Prentice Hall Signal Processing Series.

[8]  Jr. James E. Adams,et al.  Design of practical color filter array interpolation algorithms for digital cameras , 1997, Electronic Imaging.

[9]  James E. Adams,et al.  Design of practical color filter array interpolation algorithms for digital cameras .2 , 1997, Proceedings 1998 International Conference on Image Processing. ICIP98 (Cat. No.98CB36269).

[10]  F. Deutsch.  Best approximation in inner product spaces , 2001 .

[11]  Robert Bregovic,et al.  Multirate Systems and Filter Banks , 2002 .

[12]  Wesley E. Snyder,et al.  Demosaicking methods for Bayer color arrays , 2002, J. Electronic Imaging.

[13]  H. Joel Trussell,et al.  Mathematics for demosaicking , 2002, IEEE Trans. Image Process..

[14]  Yücel Altunbasak,et al.  Color plane interpolation using alternating projections , 2002, 2002 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[15]  Thomas W. Parks,et al.  Adaptive homogeneity-directed demosaicing algorithm , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

[16]  Ning Zhang,et al.  Primary-consistent soft-decision color demosaic for digital cameras , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

[17]  Ning Zhang,et al.  Primary-consistent soft-decision color demosaicking for digital cameras (patent pending) , 2004, IEEE Transactions on Image Processing.

[18]  Rastislav Lukac,et al.  Demosaicked image postprocessing using local color ratios , 2004, IEEE Transactions on Circuits and Systems for Video Technology.

[19]  Xin Li,et al.  Demosaicing by successive approximation , 2005, IEEE Transactions on Image Processing.

[20]  R.W. Schafer,et al.  Demosaicking: color filter array interpolation , 2005, IEEE Signal Processing Magazine.

[21]  Sabine Süsstrunk,et al.  Linear demosaicing inspired by the human visual system , 2005, IEEE Transactions on Image Processing.

[22]  Lei Zhang,et al.  Color demosaicking via directional linear minimum mean square-error estimation , 2005, IEEE Transactions on Image Processing.

[23]  V. Katkovnik,et al.  Spatially adaptive color filter array interpolation for noiseless and noisy data: Articles , 2007 .

[24]  Jeanny Hérault,et al.  Practical implementation of LMMSE demosaicing using luminance and chrominance spaces , 2007, Comput. Vis. Image Underst..

[25]  Yap-Peng Tan,et al.  Adaptive Filtering for Color Filter Array Demosaicking , 2007, IEEE Transactions on Image Processing.

[26]  Karen O. Egiazarian,et al.  Spatially adaptive color filter array interpolation for noiseless and noisy data , 2007, Int. J. Imaging Syst. Technol..

[27]  David Alleysson,et al.  Frequency selection demosaicking: a review and a look ahead , 2008, Electronic Imaging.

[28]  Lei Zhang,et al.  Image demosaicing: a systematic survey , 2008, Electronic Imaging.

[29]  Laurent Condat,et al.  A generic variational approach for demosaicking from an arbitrary color filter array , 2009, 2009 16th IEEE International Conference on Image Processing (ICIP).

[30]  Giancarlo Calvagno,et al.  Regularization Approaches to Demosaicking , 2009, IEEE Transactions on Image Processing.