Image and Video Acquisition, Representation and Storage

Abstract

What the reader should know to understand this chapter $\bullet$ Elementary notions of optics and physics. $\bullet$ Basic notions of mathematics.

Problems

3.1

Show that in the XYZ model the white is represented by the triple (1,1,1).

3.2

Consider the YIQ model. Show that in a grayscale image, where R$=$G$=$B, the chrominance components I and Q are null.

3.3

Consider the HSV model. Show that in the simplest form of HSV transformation, the hue (H) become undefined when the saturation S is null.

3.4

Compute in HSV model, the coordinates of cyan, magenta and yellow.

3.5

Repeat Problem 3.4 for the HSB model.

3.6

Take a videocassette registered under the NTSC system. How will it be displayed by a PAL videocassette recorder (VCR)? Explain your answer.

3.7

Implement the Huffman coding algorithm. Test the software on the following example: consider a file formed by 10,000 A, 2,000 B, 25,000 C, 5,000 D, 40,000 E, 18,000 F. Compute how many bits are required to code the file.

3.8

Consider the file formed by 20,000 B, 2,500 C, 50,000 D, 4,000 E, 1,800 F. Compare, in terms of memory required, fix-length and Huffman coding. Does there exist a case where fix-length and Huffman coding require the same memory resources? Explain your answer.

3.9

How much memory is required to store the movie Casablanca in its uncompressed version? Assume that the movie is black/white, has 25 frame/sec (each frame is 640 $\times$ 480 pixels), its runtime is 102 min. For sake of simplicity, do not consider the memory required to store the audio of the movie.

3.10

Repeat the Problem 3.9 for the movie Titanic. Titanic is a color movie, has 30 frame/sec, and its runtime is 194 min.

3.11

Repeat the Problem 3.10 for the high definition version of the movie Titanic. Assume that each frame i is 1,920 $\times$ 1,240 pixels and that movie is visualized using PAL or Secam system.

3.12

Repeat Problem 3.4 for the HIS model.

3.13

Repeat Problem 3.4 for the YUV model.

3.14

Implement Hu’s moments. Test your implementation on an Image verifying that the moments are invariant w.r.t. rotation.

3.15

Write the mathematical expression of the two dimensional Wavelet transform.

3.16

Implement the Wavelet Transform using Haar scaling function as Mother Function.

3.17

Consider the second-order Hessian matrix, H, defined as follows:

$$\begin{aligned} H = \left[ \begin{array}{ll} D_{xx} \quad D_{xy} \\ D_{yx} \quad D_{yy} \end{array} \right] \end{aligned}$$

(3.72)

Let ${\textit{Tr}}(H)$ and ${\textit{Det}}(H)$ be the trace and the determinant of the matrix H, respectively. Prove that holds the following formula

$$\begin{aligned} \varGamma = \frac{{\textit{Tr}}(H)^2}{{\textit{Det}}(H)} = \frac{(r+1)^2}{r}, \end{aligned}$$

(3.73)

where r is the ratio between the larger and the smaller eigenvalue. Moreover, show that $\varGamma$ takes the minimum when r is equal to 1.