Due to scheduling conflicts, no recitation session on this topic was possible, however; to assist students in understanding Image Processing, this RHEA informational page was created. The material covered in this page covers topics presented in ECE438 lecture from 16-30 November. This page attempts to present a very general overview of the image filtering process, an understanding of boundary conditions, and examples of linear and non-linear filters.


Image Processing:

Image Filtering: process to sharpen or smooth image

f(m,n) → h(x,y) → g(m,n)

f(m,n) → input image h(x,y) → filter g(m,n) → filtered image


To calculate g(m,n) → h(m,n) ** f(m,n)

= ΣmΣn h(k,l) f(m-k, n-l)

= ΣkΣl h(m-k,n-l) f(k,l)

= ΣkΣl h_(k-m,l-n) f(k,l)


where

h_(x,y) = h(-m,-n)



Boundary Conditions:

Given a grid of pixels below, we are interested in determining a value for pixel(L) using the surrounding pixel values: A B C D E F G H I J K L M N O P

Several types of boundary conditions were discussed in lecture: FREE → filter image based only on pixels: {H, K, P}

TOROIDAL → filter image based on three surrounding pixels, and pixel from across image in the same row as L: {H, K, P, I}

REFLECTIVE → reflect pixel from inside image to outside boundary: {H, K, P, K}

SYMMETRIC → repeat edge row of pixels as boundary pixels: {H, K, P, L}


Examples of Image Filters:


Average filter: smoothes/blurs noise and edges.

h[k,l] = 1/16 * $ \begin{bmatrix} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 1 \end{bmatrix} $

Note: the averaging filter is separable:

h[k,l] = 1/16 * $ \begin{bmatrix} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 1 & 2 & 1 \end{bmatrix} = \begin{bmatrix} 1/4\\ 1/4\\ 1/2 \end{bmatrix} \begin{bmatrix} 1/4 & 1/2 & 1/4 \end{bmatrix} $


Since the filter is separable, H(u,v) = H(u) H(v)

If we Fourier Transform the filter, we discover that the filter is linear:

H(u) = Σ h(k)e-jku

= ¼ e-ju + ½ + ¼ e-ju

= ½(1 + cos(u))



Sobel Filter/Operator: detects edges in images. This operator returns a large response across edges, and no response when image is constant.

h[k,l] = 1/9 * $ \begin{bmatrix} -1 & -1 & -1 \\ -1 & 8 & -1 \\ -1 & -1 & -1 \end{bmatrix} $



Sharpening, using an Unsharpmask:


y[m,n] = f[m,n] + λ (f[m,n] - <f[m,n]>)


the component: f[m,n] - <f[m,n]> will mask the image with neighborhood average of pixels to extract variations in image.


f[m,n] + λ (variations): will add variations to sharpen original image.


How to calculate h[k,l]:

<f[m,n]> = ΣkΣl f[m+k, n+l] / (2M+1)2


where (2M+1) is the window size of the sample


g[m,n] = f[m,n] + λ (f[m,n] - <f[m,n]>)

= f[m,n] + λ ΣkΣl h’[k,l] f[m+k, n+l]

= ΣkΣl (δ[k,l] + λh’[k,l]) f[m+k, n+l]


the term: (δ[k,l] + λh’[k,l]) represents the filter – h_[k,l]


h_[k,l] = $ \begin{bmatrix} -lambda/N^2 & -lambda/N^2 & -lambda/N^2 \\ -lambda/N^2 & 1 + -lambda(1-1/N^2) & -lambda/N^2 \\ -lambda/N^2 & -lambda/N^2 & -lambda/N^2 \end{bmatrix} $



Non-Linear Image Filters:

We cannot convolve non-linear filters. An example of a non-linear filter is a ‘Median Filter’.


g(m,n) = median{f[m+k, n+l]}


If we look at the following set of pixels, we can easily determine the median, but the filter is not separable and therefore not linear:


$ \begin{bmatrix} 4 & 2 & 100 \\ 3 & 1 & 5 \\ 8 & 7 & 6 \end{bmatrix} $



Partial Differential Equation based image processing:


Consider an image that evolves with time (τ).


I(x,y,τ), where I(x,y,0) = fc(x,y)


according to the differential equation;


δI / δτ = δ2I / dx2 + δ2I / δy2


the solution for the evolved image is:


I(x,y,τ) = fc(x,y) ** (1/4πτ) e (–x2+y2/4τ)


and h(x,y) = (1/4πτ) e (–x2+y2/4τ)


further, we could discretize the equation to arrive at a linear time-invariant system



Perona-Malik Filter:


The Perona-Malik Filter attempts to remove image noise while not removing any significant parts of the image content:


a Perona-Malik Filter can be defined by the following function:


g[m,n] = f[m,n] + λ {ψ(f[m-1,n] – f[m,n]) + ψ(f[m+1,n] – f[m,n]) + ψ(f[m,n-1] – f[m,n]) + ψ(f[m,n+1] – f[m,n])}


examining the above equation, it is possible to determine that if no feature edge exists in the window then all Δf values will be small, and thus, ψΔf ~ Δf


in this case, the above equation can be rewritten as:


g[m,n] = f[m,n] + λ {f[m-1,n] + f[m+1,n] +f[m,n-1] + f[m,n+1] – f[m,n])}


which is simply an averaging filter.


however; if an edge is present, one (or more) Δf values will be large, and for large Δf, ψΔf ~ Δf .


in this case, will resemble the average filter, but with one of the edge values equal to 0.

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Seraj Dosenbach