Image Processing Applications of Probability Theory

In this project, groups of four students will collaborate to experiment with probabilistic meth$-$

ods of generating and analyzing images in the MATLAB clone Octave$-$online. You will be working

with $512512$ gray$-$scale images that use one real number per picture element $("$pixel$")$ to represent

shades of gray: the range of these pixel values is $0,1,$ where $0$ represents black and $1$ represents

white. To display an image that is stored in matrix A$,$ simply type: imshow$($A$).$

You are given the following functions in an Octave$-$online bucket $($found here$)$:

$H=$Entropy$(P)$

Calculates entropy $H,$ in bits, when given quantizer output level probabilities in vector $P.$

Note: This function normalizes $P$ so that all of its coefficients sum up to $1.$

output$\_$levels $=$ MMSE$\_$levels$($data$,$ output$\_$levels$)$

Performs one iteration of the Lloyd$-$Max algorithm to move quantizer levels, in array

output$\_$levels, toward their optimal values $($optimal $=$ those that minimize the mean

squared error between samples in array "data" and their rounded$-$to$-$nearest level quantized

versions$).$

qdata $=$ Quantize$($data$,$output$\_$levels$)$

Quantizes each coefficient in data by rounding it to the nearest one of the values specified by

array output$\_$levels. Output matrix qdata matches the dimensions of input matrix data.

qdata $=$ Quantize$\_$count$($data$,$output$\_$levels$)$

After quantizing each coefficient in matrix data to the nearest value in output$\_$levels,

function Quantize$\_$count returns a count of samples at each output level. Output array

count matches the dimensions of input array output$\_$levels. $($This function is used to

estimate the probabilities of quantizer output levels.$)$

Random$\_$samples $(x,p,$dummy$)$

Generates samples of a discrete random variable $x$ whose PMF is specified by equal$-$length

vectors $x$ and $p$ : the probability that $x=x(i)$ is $p(i).$ These samples are returned in array

data, which is of the same size as input array dummy.

Histogram$($A$)$

Display the histogram of a gray$-$level image $($range $0,1)$ that is stored in matrix A$.$

$b=$Huffman$(p)$

Array b returns the number of bits per variable$-$length codeword corresponding to probability

values in array $p,$ as found via Huffman coding. Probability values in $p$ may be scaled by

a positive real constant. For example, four equally$-$likely outcomes may be represented by

either $p=\left[\begin{array}{cccc}0\mathrm{.}25& 0\mathrm{.}25& 0\mathrm{.}25& 0\mathrm{.}25\end{array}\right],$ or by $p=\left[\begin{array}{cccc}1& 1& 1& 1\end{array}\right].$Part I.

a$)$ Graph the image that was provided $($type "load image"; image is returned in matrix A$).$

b$)$ Graph the histogram of the original image.

c$)$ Quantize the image using amplitude levels $\{0\mathrm{.}0,0\mathrm{.}2,0\mathrm{.}4,0\mathrm{.}6,0\mathrm{.}8,1\mathrm{.}0\}($rounding$-$to$-$nearest

method$).$ Graph the quantized image.

d$)$ Graph the histogram of the quantized image.

e$)$ Count the number of quantized image pixels at each quantization level. Use these counts

to estimate the PMF of a random variable, $x,$ whose samples have the same probabilities.

Write an expression for $P_x(x),$ specifying every amplitude level and its probability.

f$)$ Based on $P_x(x),$ calculate $E[x].$

g$)$ Calculate the average of all pixels values in the quantized image.

h$)$ Calculate the average of all pixels values in the original image.

i$)$ Based on $P_x(x),$ calculate $E[x^2].$

j$)$ Calculate the average of all pixel values$-$squared in the quantized image.

k$)$ Calculate the average of all pixel values$-$squared in the original image.

Based on $P_x(x),$ generate samples of random variable $x$ in a matrix equal in size to that of

the original image. Graph this random image.

m$)$ Graph the histogram of the random image.

n$)$ Calculate the average of all pixels values in the random image.

o$)$ Calculate the average of all pixels values$-$squared in the random image.

Summarize the numerical results from parts $($f$)-($k$),($n$)-($o$)$ in a table. Please Write full answers to all question