Image Model | Sahithyan's Notes

An image is modeled as a function $I(x,y)$ returning the light intensity at position $(x,y)$ .

Here:

(x,y)

Here:

$x, y$: spatial coordinates of a point in the image
$I(x,y)$: light intensity at that point, bounded between a minimum and maximum

Analog Image Model

$x, y$ are continuous and range over a bounded region. $I$ is defined at every point of that region and varies continuously.

Properties:

Infinite spatial resolution
A value exists between any two points.
Continuous intensity
No fixed set of allowed values.
Not directly computable
Needs a discrete representation before a computer can store or process it.

Examples:

A digital image is a discrete representation of an image, with a finite number of positions and a fixed number of intensity values.

Finite grid of pixels
The spatial domain is an $m \times n$ grid. Each cell $(i,j)$ covers a small finite region.
Fixed number of levels
The intensity of each cell is one of $L$ discrete levels.

$I(i,j)$ is the intensity of cell $(i,j)$ , one of the $L$ levels. The digital image is the matrix of these values. Each element is a pixel.

\begin{bmatrix} I(0,0) & I(0,1) & \cdots & I(0,n-1) \\ I(1,0) & I(1,1) & \cdots & I(1,n-1) \\ \vdots & \vdots & \ddots & \vdots \\ I(m-1,0) & I(m-1,1) & \cdots & I(m-1,n-1) \end{bmatrix}

Here:

$m \times n$ is the spatial resolution. An 8-bit grayscale image has $L = 256$ levels, $0$ for black and $255$ for white.

Storage size in bits is $m \times n \times k$ for a single channel. A color image repeats this per channel.

In a computer program, an image is simply a matrix of integer values. Image processing means manipulating this matrix.