Question

Question 1: Color Theory (15 points)

In a three dimensional color space such as XYZ, any

color C with coordinates (X, Y, Z) can be expressed as a linear combination of the primaries P1, P2, P3 with coordinates (X1, Y1, Z1), (X2, Y2, Z2) and (X3, Y3, Z3) respectively. This may be expressed as

C (X, Y, Z) = α1*P1 (X1, Y1, Z1) + α2*P2 (X2, Y2, Z2) + α3*P3(X3, Y3, Z3)

In this question you are asked to show that similarly, the normalized chromaticity coordinates of C can also be expressed as a linear combination of the normalized chromaticity coordinates of P1, P2, P3. Proceed by answering the following:

 Find the normalized chromaticity coordinates of P1, P2, and P3 in terms of given known quantities (3 points)

 Express the normalized chromaticity coordinates of the color C in terms of the chromaticity coordinates of P1, P2, and P3 (6 points)

 Hence prove that the chromaticity coordinates of any color C (which is a linear combination of primaries P1, P2, and P3 in XYZ color space) can be represented also as a linear combination of the chromaticity coordinates of the respective primaries. (6 points)

Question 2: Generic Compression (10 points)

Consider a source that emits an alphabet with three symbols A, B, C having probability distributions as follows - P(A) = 0.625, P(B) = 0.125, P(C) = 0.25

 Construct a Huffman code and calculate its average code length. (2 points)

 For this three-symbol vocabulary, how many Huffman codes are possible. What are they? (2 points)

 Is the code you have defined optimal - give reasons! If not, what can you do to improve upon this. Show a solution by an example computing the avg. code length. (6 points)