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\noindent Math 155. Instructor: Luminita Vese. Teaching Assistant: Baichuan Yuan. 

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\noindent {\bf Homework $\#$ 3} {\bf Due on Friday, February 3}

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\noindent {\bf [1] } {\it Computational project: Spatial filtering} 
Consider the noisy X-ray image of circuit board corrupted by salt-and-pepper noise. Filter this image by applying a linear average filter with a $3\times 3$ mask (use the average mask with entries $w_{s,t}=\frac{1}{9}$, for all $s,t\in \{-1,0,1\}$). You can keep the border pixels unchanged. 

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\noindent {\bf [2] } Show that the Laplacian operation $\nabla^2 f=\frac{\partial ^2f}{\partial x^2}+\frac{\partial ^2f}{\partial y^2}$ is isotropic (invariant under rotations, or rotationally invariant). You will need the following equations relating
coordinates after axis rotation by an angle $\theta$: 
$$x=x'\cos\theta-y'\sin\theta$$
$$y=x'\sin\theta+y'\cos\theta$$
where $(x,y)$ are the unrotated and $(x',y')$ are the rotated coordinates. 

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\noindent {\bf [3] } Show that the continuous Laplacian is a linear operation, in other words show that the mapping $f\mapsto \nabla^2f$ is linear on the vector space of functions $f\in C^2$ in two dimensions (continuous and twice differentiable functions). 

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\noindent {\bf [4] }
 
(a) Show that the magnitude of the gradient $|\nabla f|=\sqrt{(\frac{\partial f}{\partial x})^2+(\frac{\partial f}{\partial y})^2}$ is an isotropic operation. 

(b) Show that the isotropic property is lost in general if 
the gradient magnitude is approximated by $|\nabla f|\approx |\frac{\partial f}{\partial x}|+|\frac{\partial f}{\partial y}|$. 



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