Sampling and Quantization

Sampling and quantization are the 2 independent choices made when converting an analog image to a digital image. Sampling sets the grid size $m \times n$. Quantization sets the number of levels $L = 2^k$. Together they fix the storage size $m \times n \times k$ bits per channel.

Coarse choices in either one cause visible artifacts.

Sampling

Sampling fixes the spatial resolution. Resolution is the number of samples per unit length, set by the size of the image sensor element.

Coarse sampling means large grid cells, so a small $m \times n$. It has 3 visible effects: loss of information, pixelation, and aliasing.

Sampling Rate

Number of samples per unit distance, one per grid cell.

Loss of Information

Each cell keeps only one value for the region it covers, so any variation finer than a cell is averaged away and cannot be recovered. Worse with large cells and scenes carrying fine detail.

Pixelation

Also known as the checkerboard effect or blocking. Each grid cell becomes one flat square of intensity, so a smooth or diagonal edge is replaced by a staircase of these squares. Most visible on high-contrast boundaries such as text against a plain background.

Worked Example

Nyquist Sampling Rate

A stripe pattern has $120$ cycles per millimetre. Find the minimum sampling rate that prevents aliasing.

Aliasing

Aliasing happens when a pattern in the scene repeats faster than the sampling grid can follow. The samples then fit a different, lower-frequency pattern that was never in the scene. That false pattern is the alias.

• Capturing a repeating pattern needs at least 2 samples per cycle of that pattern. This is the Nyquist criterion.
• A pattern with more than 1 cycle per 2 cells is undersampled.
• The alias frequency is the true frequency folded about half the sampling rate. A higher scene frequency gives a lower alias frequency.
• On fine repeating textures the alias shows up as broad curved bands, called moire fringes.

Anti-aliasing removes detail finer than the grid can represent, by optical blur or a Gaussian low-pass filter, before the image is sampled. A Gaussian filter weights nearby pixels with smoothly decreasing coefficients. This costs high-frequency detail, which is less noticeable when sub-sampling.

Quantization

Quantization fixes the grey-level depth. Grey-level depth is the number of discrete intensity levels $L$, set by the resolution of the analog to digital converter.

Coarse quantization means a small $L$ and a low bit depth $k$. The gaps between levels are large, so nearby intensities collapse to the same level. This causes false contouring.

False Contouring

Also known as Banding. Across a smooth gradient, a band of neighbouring intensities collapses onto one level and renders as a flat region. The jump to the next level shows up as a sharp edge that is not in the scene. These false edges resemble contour lines, most visible in skies and other large smooth areas.