Task $5$

Five different versions of the same fractal image were provided with the homework. These images have varying image resolutions from $128$ to $1024.$ Run the functions you have defined in Tasks $1$ and $2$ at least $25$ times for each fractal image with compression factor $($N$)$ of $8.$ Plot average run time vs image resolution. How does the algorithm scale with resolution $($Which of these: logarithmic, exponential, quadratic, or linear$)?$ In this homework, you will implement a basic compression method for images that will reduce the number of values in color scales

$($RGB$)$ to a compression factor.

In MATLAB, images can be represented with $3-$dimensional arrays $(MxN3)$ where the first dimension represents height, the second

dimension represents width, and the third dimension represents the color components $($RGB$).$ The values in this array can be

between $0-255$ with $0$ representing the lowest intensity for the color. The compression method you will implement in this homework

will reduce the number of values that the color intensities can take to the compression factor $(N)$ such that instead of using $256$ values

to represent the color intensity, the compressed image will use only $N$ distinct intensity values $(N$ should be a power of $2).$ In

implementation, the original intensity range $(0-255)$ will be divided into $N$ sub$-$ranges, and the values in each sub$-$range will be

mapped to a new value. As an example, Table $1$ shows the original intensity ranges and mapped values for $N=8.$ Use the following

steps to implement the algorithm for any value of $N(N$ should be a positive number that is a power of $2$ and should be less than $256)$;

Define the lower and upper limits for each sub$-$range $($Hint: For the first sub$-$range lower limit is $0,$ and upper limit is $(256/$N$)-1)$

Define the new values to which the original values will be mapped $($Hint: Pixels having values in the kth subrange will be

mapped to a single value that is $\{$:$\frac{(k-1)*256}{N}+\frac{256}{2*N})$

Map the pixel values in the original image to the new values.

Table $1.$ Example values for $N=8$