2-D Signals and Systems

• a digital image is stored as a 2D array of pixels
• each pixel stores one intensity value
• image is a function $f(x,y)$ sampled on a rectangular grid
• spatial variables $x$ and $y$ indicates the spatial distances
• pixel are discrete
• pixels are influenced by its neighbours
• local operations use nearby pixel to compute a new values

2-D Impulse Signal

We can represent every pixel as a shifted and scaled impulse. For example, a pixel at $x=2$ and $y=3$ with pixel intensity half of the brightest impulse pixel will be:

$$f(x,y)=c\delta (x-i,y-j)=\{\begin{array}{ll}c,& x=i\text{ and }y=j\\ 0,& \text{otherwise}\end{array}$$

Where:

• $c$ represents the scaled pixel intensity, in our case above its $c=0.5$
• $i$ and $j$ is the shifted impulse, where $i=2$ and $j=3$

Building an Image from many Impulses

Any image $f(x,y)$ can be represented by the sum of a number of shifted and scaled impulses:

$$f(x,y)=\sum _{j=-\infty }^{\infty }\sum _{i=-\infty }^{\infty }f(i,j)\delta (x-i,y-j)$$

The above equation is for an infintely wide and long image. However, for a finite image:

$$f(x,y)=\sum _{j=-n}^n\sum _{i=-m}^{m}f(i,j)\delta (x-i,y-j)$$

where:

• the size of image are bounded by 2 variables, $[-n,n]$ and $[-m,m]$

Example

Question:
We have an input image $f(x,y)$. We wanna suppress the pixel random noise/smooth the image. We need to perform local average by taking the sum of the corresponding pixel with its 4 neighbour pixels in the input image (left, right, top, down).

Answer:
To express the sum of pixels:

$$g(x,y)=\frac{1}{5}[f(x,y)+f(x-1,y)+f(x+1,y)+f(x,y+1)+f(x,y-1)]$$

where:

• $f(x,y)$ represents the particular pixel we are trying to compute for, so its the center pixel
• $f(x+1,y)$ represents the pixel directly above the pixel

This direct approach where we sum the pixels and then divide by the number of pixels to find the local average, is a more intuitive, hand-written process.

Convolution can be expressed as:

$$g(x,y)=\sum _{j=-a}^a\sum _{i=-b}^{b}h(i,j)f(x-i,y-j)$$

where:

• $g(x,y)$ is the resulting pixel intensity in the output image at position $(x,y)$
• $f(x-i,y-j)$ represents the pixel values from the original image around position $(x,y)$
• $h(x,y)$ represents the specific multiplier (weight) assigned to each surrounding pixel

This is the general formula for Discrete 2D Spatial Convolution where we are convoluting between the input image $f$ and the sliding filter/kernel $h$.

This convolution approach is a more generalised framework that is being used in softwares.

Impulse response can be expressed as:

$$h(x,y)=\delta (x,y)+\delta (x-1,y)+\delta (x+1,y)+\delta (x,y-1)+\delta (x,y+1)$$

The above equation representing the impulse response can also be expressed as such:

2-D Image Systems

The output $g(x,y)$ of a 2-D system $T$ with an input $f(x,y)$ can be expressed as:

$$g(x,y)=T[f(x,y)]$$

Linear and Shift-Invariant Properties

Linearity:

$$T\{a{f}_1(x,y)+b{f}_2(x,y)\}=aT\{f_1\}+sbT\{f_2\}$$

3 factors required for Linearity:

• each input pixel appears only once
• each pixel are only to the first power
• each pixel are multiplied by a constant

Things that break Linearity

max, min, median, sorting, squaring, absolute value, thresholding, logarithm, gamma, histogram equalization, and clipping

Shift Invariance - the same processing applies to the whole image:

$$T\{f(x-x_0,y-y_0)\}=g(x-x_0,y-y_0)$$

1 Factor required for Shift-invariance:

• the weights are the same at every pixel positions

Things that break Shift-Invariance

weights that change with position, adaptive filters, coordinate changes such as rotation or resizing, and border rules that alter the window near the edges

Linear and Shift-Invariant (LSI) Systems

• Linear + Shift Invariant = LSI System
• An LSI system $T$ is only characterised by its impulse response, defined as:

$$h(x,y)=T[\delta (x,y)]$$

• For an LSI system, Convolution operation is defined as:

$$g(x,y)=\sum \sum f(m,n)\times h(x-m,y-n)=f(x,y)\ast h(x,y)$$