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\noindent Math 155: Instructor: Luminita VESE. Teaching Assistant: Siting LIU. 

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\noindent{\bf Homework $\#$ 2} \ \ {\bf Due on Friday, January 25} 

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\noindent{\bf [1]} Give a single intensity transformation function $T$ for spreading the intensities of an image so the lowest intensity is 0 and the highest is $L-1$. 

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\noindent{\bf [2]} (Histogram equalization in continuous variables) An image has the gray-level PDF 
$$p_r(r)=
\left\{
\begin{array}{ll}
\frac{6r+2}{3(L-1)^2+2(L-1)}\mbox{ if }0\leq r\leq L-1\\
0, \mbox{ otherwise }
\end{array}
\right.
$$
with $L-1>0$. 

(a) Verify some of the properties that a PDF has to satisfy: $p_r(r)\geq0$ for all $r\in (-\infty,\infty)$ and $\int_{-\infty}^{\infty}p_r(r)dr=1$. 

(b) Find the tranformation function $s=T(r)$ obtained through ``histogram equalization'' in continuous variables. 

(c) Verify that $p_s(s)$ is a uniform ``flat'' distribution for $s\in [0,1]$ (recall the formula $p_s(s)=p_r(r)\Big|\frac{dr}{ds}\Big|$). 

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\noindent{\bf [3]} (Histogram matching in continuous variables) An image has the gray-level PDF $p_r(r)=-2r+2$, with $0\leq r\leq 1$. 
It is desired to transform the gray levels of this image so that they will 
have the specified $p_z(z)=2z$, $0\leq z\leq 1$. Assume continuous quantities 
and find the transformation (in terms of $r$ and $z$) that will accomplish 
this (here $L-1=1$). 

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\noindent{\bf [4]} A linear spatial filter of size $(2a+1)\times (2b+1)$ defined by the transformation $H$, $g=H[f]$, is given by 
$$g(x,y)=\sum_{s=-a}^a\sum_{t=-b}^b w(s,t)f(x+s,y+t),$$
where $f(x,y)$ is a given input image, $a,b$ are positive integers, and $w(s,t)$ are weights for $-a\leq s\leq a$, $-b\leq t\leq b$.

(a) Give the definition of a linear transformation $H:V\rightarrow V$, where $V$ is a vector space. 

(b) Show that $H$ defined above is indeed a linear transformation (assume images defined on the entire plane, or ignore border effects). 

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