The Structure and Properties of Color Spaces and the Representation of Color Images (Synthesis Lectures on Image, Video, and Multimedia Processing)

88CHAPTER 6. SIGNALS AND SYSTEMS THEORYIn this process, three new signals have been deed in Eq. (6.67), which are a linear transformation of the original tristimulus values. Thus, these new values can be considered to be tristimulus values with respect to a new basis. As per Table 3.1, this dees the new primaries forming the basis LCC = {[L], [CH1 ], [CH2 ]}, 鈳も帯 鈳�鈳�鈳�鈳�1 1 1 [L] [P1 ] 鈳CH1 ]鈳�= 鈳ｂ1 1 �鈳�鈳P2 ]鈳�. (6.70) � 0 2 [CH2 ] [P3 ] ARGB鈫扡CCWe observe that the basis vector [L] = [P1 ] + [P2 ] + [P3 ] is in fact reference white, and that for any gray-scale image in which C1 (x) = C2 (x) = C3 (x), the two components CCH 1 (x) and CCH 2 (x) are zero. Applying the transformation of Eq. (6.67) to the color signals [F1 ](x), [F2 ](x) and [F3 ](x) deed in Eq. (6.62) and Eq. (6.63), we obtain 鈳�鈳�鈳�鈳�1 1 1 CL (x) 4 C1 (x) + 2 C2 (x) + 4 C3 (x) 1 1 1 F1,LCC (x) = 鈳ｂ 4 C1 (x) + 2 C2 (x) �4 C3 (x)鈳�= 鈳CH 1 (x)鈳�1 1 CCH 2 (x) �4 C1 (x) + 4 C3 (x) 鈳�鈳�鈳�鈳�1 1 1 �4 C1 (x) + 2 C2 (x) �4 C3 (x) CCH 1 (x) 1 1 1 F2,LCC (x) = 鈳�4 C1 (x) + 2 C2 (x) + 4 C3 (x) 鈳�= 鈳�CL (x) 鈳�(6.71) 1 1 扖CH 2 (x) C1 (x) �4 C3 (x) 鈳�鈳�鈳�1 4 鈳�1 CCH 2 (x) �4 C1 (x) + 4 C3 (x) 1 1 鈳�= 扚4,LCC (x) 扖CH 2 (x) F3,LCC (x) = 鈳�4 C1 (x) �4 C3 (x) 鈳�= 鈳�1 1 1 2 (CL (x) �CCH 1 (x)) 4 C1 (x) + 4 C3 (x) In this basis, [Cd ](x) in Eq. (6.57) can be written out explicitly as 鈳�鈳�CL (x) + CCH 1 (x) exp(j 2 x d2 ) + CCH 2 (x) 鈳�鈳�(exp(j 2 x d3 ) �exp(j 2x d4 )) 鈳�鈳�鈳�鈳�鈳�鈳�CCH 1 (x) + CL (x) exp(j 2 x d2 ) �CCH 2 (x) Cd,LCC (x) = 鈳�鈳�(exp(j 2 x d3 ) �exp(j 2x d4 )) 鈳�鈳�鈳�鈳�1 鈳CH 2 (x) �CCH 2 (x) exp(j 2 x d2 ) + 2 (CL (x) �CCH 1 (x))鈳�(exp(j 2x d3 ) �exp(j 2x d4 ))(6.72)In this basis, the [L] component is the conventional CFA signal. Given the deition of the di for the Bayer structure, it is straightforward to show that Cd,CH 1 (x) = Cd,L (x) exp(2(x d2 )), and so the [CH1] component provides the same information as the [L] component. However, the [CH2] component depends only on C1 (x) and C3 (x) and can perhaps provide additional information about the original components and thereby give improved demosaicking performance compared to methods using only the [L] component (i.e., the conventional CFA signal).This remains to be seen.