Image Model

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

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 sampling and quantization before a computer can store or process it.

Examples:

• Photographic film
• Image projected on the retina
• Analog video signal

Digital Image Model

A digital image is obtained from an analog image by 2 discretization steps.

• Sampling
  The spatial domain is divided into an $m \times n$ grid. Each cell $(i,j)$ covers a small finite region.
• Quantization
  The intensity of each cell is mapped to one of $L$ discrete levels.

$I(i,j)$ is the average intensity over cell $(i,j)$ after quantization. The digital image is the matrix of these values. Each element is a pixel.

Here:

• $m$: number of rows
• $n$: number of columns
• $i$: row index, $0 \le i \le m-1$
• $j$: column index, $0 \le j \le n-1$
• $L$: number of intensity levels, usually $L = 2^k$ for a bit depth $k$

$m \times n$ is the spatial resolution. $k$ is the bit depth. 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.

Coarse Sampling

Large grid cells, so a small $m \times n$ and low spatial resolution. Too sparse to record fine spatial variation.

• Small features fall between sample points and disappear.
• Edges and text look blocky. This is pixelation.
• Fast repeating patterns reappear as a false slower pattern. This is aliasing.

Coarse Quantization

Few intensity levels, so a small $L$ and low bit depth. The gaps between levels are large.

• Nearby intensities collapse to the same level.
• A slow gradient becomes a set of flat steps with visible jumps between them. This is banding, also called contouring.
• Most visible in skies and other large smooth regions.