Sampling & quantization

Images as sampled signals

The light arriving at the sensor is a continuous function $f(x, y)$ of the position on the image plane, with one value for gray images or three for color. A digital image stores this function only on a regular grid, and only with a finite set of values:

Here $s$ is the distance between samples and $Q$ is the quantizer. The image is then a matrix of integers: row $j$, column $i$, with the origin in the top-left corner.

Sampling

The spatial resolution is the number of samples. Using a cell size $s$ keeps one value per $s \times s$ cell of the original, so the image shrinks to $\lfloor W/s \rfloor \times \lfloor H/s \rfloor$ pixels. Enlarged back to the original size, every sample becomes a visible block.

The value of a sample can be taken at the cell center (an ideal point sample), or averaged over the cell. The average is closer to a real sensor pixel, which collects all light falling onto its area. Detail finer than the grid cannot be represented either way: point samples turn it into false coarse patterns (aliasing, clearly visible in the zone plate), while averaging blurs it away first.

Quantization

With $b$ bits per value there are $L = 2^b$ levels. For values $v \in [0, 1]$:

In smooth gradients, neighboring pixels round to the same level for a while and then jump to the next one. This creates visible steps called banding or false contours. With 8 bits (256 levels) the steps are usually too small to see, which is why 8 bits per channel is the standard for images. Camera sensors often record 10 to 14 bits, which leaves room for later processing such as brightening dark areas.

Try this

• Set the bits to 1. Every channel is either 0 or 1, so a color image has only $2^3 = 8$ colors.
• Pick the gradients image and increase the bits one at a time. From which bit depth on do the bands in the dark ramp disappear for you?
• Click on the sky of the landscape at 4 bits and look at the row plot: the thick staircase only takes values on the grid lines.
• Switch on grayscale and compare with color at the same bit depth. Color images with 3 bits per channel still have 512 colors, gray images only 8 levels.
• Choose the zone plate with cell size 4 and “cell center”. Rings appear where there are none; switch to “cell average” and most of them fade to gray.
• Look at the fence of the landscape with a large cell size. The pickets merge, split or disappear depending on where the grid happens to hit them.
• Compare two settings with the same storage: 2 × 2 cells at 8 bits and 1 × 1 cells at 2 bits. Which loss is more visible?