Question

The color of light can be represented in an RGB vector [R] [G] [B] where R, G and B represent the amounts of red, green and blue, respectively. The human eye and the brain transform the incoming RGB signal into the vision signal [R + G + B] [ ———— ] [I] [ 3 ] [L] = [ R - G ] [S] [ R + G ] [B - ———— ] [ 2 ] where I, L and S represent the intensity, long-wave signal, and short-wave signal, respectively. (a) Write down a matrix A such that A [R] = [I]. [Hint: This part is meant to be easy, as you [G] [L] [B] [S] should be able to read off A directly from the formula above of [I] in terms of [R].] [L] [G] [S] [B] (b) A pair of sunglasses reduces the perceived intensity of the incoming light by half. Write down a matrix P such that, if the incoming light is perceived as a vision signal [I] by the [L] [S] naked eyes, then P [I] would be the perceived vision signal if the incoming light first [L] [S] went through the sunglasses. [Hint: This part is also meant to be easy; the matrix P differs from the identity matrix in only one entry.] (c) Now write down a matrix S representing the effect the sunglasses have on the incoming light on the RGB level. That is, if the incoming light has RGB signal [R], then after it [G] [B] passes through the sunglasses it has RGB signal S [R]. [Hint: This is the most challenging [G] [B] part of this problem. Use your work in (a) and (b) to write down a matrix equation involving A, P and S, from which you should be able to solve for S using matrix multiplication and/or inverse.]

          The color of light can be represented in an RGB vector
																		[R]
																		[G]
																		[B]
where R, G and B represent the amounts of red, green and blue, respectively. The human eye
and the brain transform the incoming RGB signal into the vision signal
														[R + G + B]
														[  ————  ]
							[I]			[	  3	  ]
							[L] = 	[  R - G   ]
							[S]			[	 R + G   ]
														[B - ————  ]
														[	   2	  ]
where I, L and S represent the intensity, long-wave signal, and short-wave signal, respectively.

(a) Write down a matrix A such that A [R] = [I]. [Hint: This part is meant to be easy, as you
																						[G]	 [L]
																						[B]	 [S]
	should be able to read off A directly from the formula above of [I] in terms of [R].]
																																[L]	 			 [G]
																																[S]				 [B]
(b) A pair of sunglasses reduces the perceived intensity of the incoming light by half. Write
down a matrix P such that, if the incoming light is perceived as a vision signal [I] by the
																																									[L]
																																									[S]
naked eyes, then P [I] would be the perceived vision signal if the incoming light first
													[L]
													[S]
went through the sunglasses. [Hint: This part is also meant to be easy; the matrix P
differs from the identity matrix in only one entry.]
(c) Now write down a matrix S representing the effect the sunglasses have on the incoming
light on the RGB level. That is, if the incoming light has RGB signal [R], then after it
																																			[G]
																																			[B]
passes through the sunglasses it has RGB signal S [R]. [Hint: This is the most challenging
																								[G]
																								[B]
part of this problem. Use your work in (a) and (b) to write down a matrix equation
involving A, P and S, from which you should be able to solve for S using matrix
multiplication and/or inverse.]

The color of light can be represented in an RGB vector
[R]
[G]
[B]
where R, G and B represent the amounts of red, green and blue, respectively. The human eye
and the brain transform the incoming RGB signal into the vision signal
[R + G + B]
[ ———— ]
[I] [ 3 ]
[L] = [ R - G ]
[S] [ R + G ]
[B - ———— ]
[ 2 ]
where I, L and S represent the intensity, long-wave signal, and short-wave signal, respectively.

(a) Write down a matrix A such that A [R] = [I]. [Hint: This part is meant to be easy, as you
[G] [L]
[B] [S]
should be able to read off A directly from the formula above of [I] in terms of [R].]
[L] [G]
[S] [B]
(b) A pair of sunglasses reduces the perceived intensity of the incoming light by half. Write
down a matrix P such that, if the incoming light is perceived as a vision signal [I] by the
[L]
[S]
naked eyes, then P [I] would be the perceived vision signal if the incoming light first
[L]
[S]
went through the sunglasses. [Hint: This part is also meant to be easy; the matrix P
differs from the identity matrix in only one entry.]
(c) Now write down a matrix S representing the effect the sunglasses have on the incoming
light on the RGB level. That is, if the incoming light has RGB signal [R], then after it
[G]
[B]
passes through the sunglasses it has RGB signal S [R]. [Hint: This is the most challenging
[G]
[B]
part of this problem. Use your work in (a) and (b) to write down a matrix equation
involving A, P and S, from which you should be able to solve for S using matrix
multiplication and/or inverse.]

Added by Thomas W.

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