What Is a Signal?

What counts as a signal

A signal is anything that bears information. An electrocardiogram, an EEG trace, speech, the speed of the wind, the Dow Jones index, a seismic recording, and an image are all signals — they just carry very different kinds of information, at very different rates.

What makes image processing its own field is the shape that information comes in: a signal can be a function of one, two, or three independent variables.

Dimensionality: 1-D, 2-D, 3-D

• 1-D — speech, the Dow Jones index, interest rates. The independent variable is time: $x(t)$.
• 2-D — an image. The independent variables are spatial position $(x,y)$, whether that's a real scene captured by a camera or a slice from a tomography scan.
• 3-D — wind speed and temperature over a region, indexed by longitude, latitude, and elevation.

This course lives almost entirely in 1-D and 2-D: 1-D to build intuition, 2-D because that's what an image is.

Continuous vs. discrete, analog vs. digital

Signals split into two families along two independent questions: is the domain continuous or discrete, and is the amplitude continuous or discrete?

• Continuous-time signal — defined at a continuum of times, with an amplitude that can range over a continuous (or finite) range of values.
• Analog — a continuous-time signal whose amplitude can also assume a continuous range of values. A voltage waveform straight off a sensor is analog.
• Discrete-time signal — defined only at discrete instants of time, i.e. a sequence of numbers. Its amplitude can still assume a continuous or a discrete set of values.
• Digital — a discrete-time signal whose amplitude is also restricted to a discrete set of values. This is the only kind of signal a computer can actually store.

Quantization: how many gray levels?

An image with 8-bit resolution per pixel has $2^8=256$ gray scales, running from 0 (black) to 255 (white). Drop the bit depth and you drop the number of distinguishable shades — the image doesn't get dimmer, it gets banded.