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# CS 576 Assignment 1 solution

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## Written Part: (20 points)

Each question has marks displayed
Q.1 Suppose a camera has 450 lines per frame, 520 pixels per line, and 25 Hz frame rate.
The color sub sampling scheme is 4:2:0, and the pixel aspect ratio is 16:9.

The camera
uses interlaced scanning, and each sample of Y, Cr, Cb is quantized with 8 bits
• What is the bit-rate produced by the camera? (2 points)
• Suppose we want to store the video signal on a hard disk, and, in order to save
space, re-quantize each chrominance (Cr, Cb) signals with only 6 bits each for
Y,Cr,Cb. What is the minimum size of the hard disk required to store 10 minutes
of video (3 points)

Q.2 The following sequence of real numbers has been obtained sampling an audio
signal: 1.8, 2.2, 2.2, 3.2, 3.3, 3.3, 2.5, 2.8, 2.8, 2.8, 1.5, 1.0, 1.2, 1.2, 1.8, 2.2, 2.2, 2.2, 1.9,
2.3, 1.2, 0.2, -1.2, -1.2, -1.7, -1.1, -2.2, -1.5, -1.5, -0.7, 0.1, 0.9 Quantize this sequence by
dividing the interval [-4, 4] into 32 uniformly distributed levels (place the level 0 at -3.75,
the level 1 at -3.5, and so on. This should simplify your calculations).
• Write down the quantized sequence. (4 points)
• How many bits do you need to transmit it? (1 points)

Q.3 Temporal aliasing can be observed when you attempt to record a rotating wheel with
a video camera. In this problem, you will analyze such effects. Assume there is a car
moving at 36 km/hr and you record the car using a film, which traditionally record at 24
frames per second. The tires have a diameter of 0.4244 meters. Each tire has a white
mark to gauge the speed of rotation.

• If you are watching this projected movie in a theatre, what do you perceive the
rate of tire rotation to be in rotations/sec? (3 points)
• If you use your camcorder to record the movie in the theater and your camcorder
is recording at one third film rate (ie 8 fps), at what rate (rotations/sec) does the
tire rotate in your video recording (4 points)

• If you use an NTSC camera with 30 fps, what is the maximum speed that the car
can go at so that you see no aliasing in the recording (3 points)

## Programming Part: (80 points)

This assignment will help you gain a practical understanding of Quantization and
Subsampling to analyze how it affects visual media types like images and video.
We have provided you with a Microsoft Visual C++ project and a java class to display
two images side by side (left – original and right – output of your program).

Currently
both left and right correspond to the same input image. You are free to use this display
program as a start, or write your own in a language of your choice (excluding scripting
language such as python, javascript, matlab). In addition, please do not use any external
libraries found on the Internet for the implementation.

Input to your program will be five parameters where
• The first parameter is the name of the image, which will be provided in an 8 bit
per channel RGB format (Total 24 bits per pixel). You may assume that all
images will be of the same size for this assignment (CIF size = 352×288), more
information on the image format will be placed on the class website

• The next three parameters control the subsampling of your Y U and V spaces
respectively. For sake of simplicity, we will follow the convention that
subsampling occurs only along the width dimension and not the height. Each of
these parameters can take on values from 1 to n for some n, 1 suggesting no sub
sampling and n suggesting a sub sampling by n

• The last parameter Q controls quantization of your R, G and B values. It is a
number that specifies how many different values each channel can have
To invoke your program we will compile it and run it at the command line as
YourProgram.exe C:/myDir/myImage.rgb Y U V Q
where Y U V Q are the parameters as described above. Example inputs are shown below
and this should give you a fair idea about what your input parameters do and how your
program will be tested.

1. YourProgram,exe image1.rgb 1 1 1 256
There are 256 values (8 bits) per R G and B, and no subsampling in the Y, U or V ->
which implies that the output is the same as the input
2. YourProgram,exe image1.rgb 1 1 1 64
There are 64 values (6 bits) per R G and B and no subsampling in Y, U or V.
3. YourProgram,exe image1.rgb 1 2 2 256

There are 256 values (8bits) per R, G and B (no additional color quantization), but the U
and V channels are subsampled by 2. No subsampling in the Y channels.
Now for the details – In order the display an image on a display device, the normal choice
is an RGB representation. This is what the format of the input image is.

However, for
YUV processing reasons, you will have to convert the image in YUV space, process your
subsampling and reconvert it back to RGB space to show the output to display. Here is
the dataflow pipeline that illustrates all the steps.
Image
Display Input
Image

2. Convert to
YUV space
3. Process YUV
subsampling
for display
5. Convert
back to RGB
space
Display Input
Image

This code is already provided to you,
if you choose to make use of it
The RGB to YUV with the conversion
matrix is given below
Sub sample Y U and V separately
according to the input parameters

6. Quantize
RGB channels
are lost, prior to conversion to RGB all
the channels have to of the same size
Apply the inverse matrix to get the RGB
data

Quantize the color channels according to
the input parameter and display
Conversion of RGB to YUV
Given R, G and B values the conversion from RGB to YUV is given by
Y 0.299 0.587 0.114 R
U = 0.596 -0.274 -0.322 G
V 0.211 -0.523 0.312` B

Remember that if RGB channels are represented by n bits each, then the YUV channels
are also represented by the same number of bits.
RGB values are positive, but YUV can take negative values!
Conversion of YUV to RGB
Given R, G and B values the conversion from RGB to YUV is given by
R 1.000 0.956 0.621 Y
G = 1.000 -0.272 -0.647 U
B 1.000 -1.106 1.703 V

Remember that if YUV channels are represented by n bits each, then the RGB channels
are also represented by the same number of bits.
YUV channel can have negative values, but RGB is always positive!
Sub sampling of YUV & processing
Sub sampling, as you know will reduce the number of samples for a channel.
Eg for the input parameters
YourProgram.exe image1.rgb 1 2 2 256

In this example, the YUV image is not subsampled in Y, but by 2 in U and by 2 in V
resulting in
When converting back to the RGB space, all the YUV channels have to be of the same
size. However the sampling throws away samples, which have to be filled in
appropriately by average the neighborhood values. For example, for the above case a
local image area would look like

Y11U11V11 Y12 Y13U13V13 Y14 . . . . . line 1
Y21U21V21 Y22 Y23U23V23 Y24 . . . . . line 2
Y31U31V31 Y32 Y33U33V33 Y34 . . . . . line 3
Y41U41V41 Y42 Y43U43V43 Y44 . . . . . line 4
The missing values may be filled in using filters. Here is an example
U12 = (U11 + U13)/2 V12 = (V11 + V13)/2
U14 = (U13 + U15)/2 V14 = (V13 + V15)/2
…. And so on, to get

Y11U11V11 Y12U12V12 Y13U13V13 Y14U14V14 . . . . . line1
Y21U21V21 Y22U22V22 Y23U23V23 Y24U24V24 . . . . . line 2
Y31U31V31 Y32U32V32 Y33U33V33 Y34U34V34 . . . . . line 3
Y41U41V41 Y42U42V42 Y43U43V43 Y44U44V44 . . . . . line 4
Note the samples that you take to fill in values will change depending on the subsampling
parameters. The YUV components can now be converted to RGB space.

RGB Color Quantization.
Assume that the quantization levels are uniformly distributed. Initially we have 8 bits per
pixel per channel to start with. So the Q input value to your program can take on values
from the range 255 – 0. For instance
• Q=256, implies 8 bits per channel or 256 possible values for each channel, so the
output number of bits is same as input.

• Q=8, implies 3 bits per channel or 8 possible values which may be 0, 31, 63, 95,
127, 159, 191, 223,
• Q=64, implies 6 bits per channel or 64 possible values which may be 0, 3, 7, 11,
… 239, 243, 247, 251
• Remember Q may not necessarily be a power of 2.
So design your quantization function accordingly.

### What should you submit ?

• Your source code, and your project file or makefile, if any, using the submit
program. Please do not submit any binaries. We will compile your program and
execute our tests accordingly.
• Along with the program, also submit an electronic document (word, pdf,
pagemaker etc format) using the submit program that answers the fore-mentioned
analysis questions. You may use any (or all) input images for this analysis.