Signals: 1D, 2D, 3D, 4D

A 2D digital image is really just a 2D matrix of numbers, like a spreadsheet. Each number is a pixel value. But, a 2D image is just one kind of signal we might want to represent, others are:

A sound is 1D (it changes in time but not space), so it can be represented by a row or column of numbers (a vector).
A 3D image, like an MRI scan, is represented by a stack of 2D images.
There are also 4D images: repeated 3D images, such as an fMRI scan. Let's dig a little deeper.

1D

A 1D signal, like a sound wave, changes through time but not space. Resolution is defined as digital samples occurring at regular intervals (e.g., ~8,000 times per second for voice, but up to 44,000 times per second for music!) along the x-axis. Bit Depth, just like its counterpart in the image, is the number of bits devoted to representing each sample. For music, that is often 16 bits for each stereo channel. Just like images, digitized sounds are more similar to the source if the resolution and bit-depth are high.

2D

A 2D image is represented by a 2D matrix of numbers like a spreadsheet, as we have already seen.

3D

In a 3D MRI image, each sample is a voxel. Voxels (or volume elements) are 3D pixels. Certain kinds of color images are also considered 3-D because they have multiple channels (e.g., Red, Green, Blue). You'll see more about this later, but, essentially, a 3D volume is a stack of 2D images.

4D

A 4D MRI image is a stack of 3D volumes. Functional Magnetic Resonance, or fMRI images, are 4D: Each volume represents a different time point. With the addition of the temporal dimension, we can now characterize both temporal and spatial resolution for the 4D image. Spatial resolution still refers to the size of each voxel, whereas temporal resolution refers to how many volumes per second are in the sequence. Spatial and temporal resolution trade-off: If you increase the spatial resolution, it takes more time to acquire each volume. If you want better temporal resolution, you have to give up spatial resolution. In addition to fMRI, movies are also 4D! This is because the movie is a sequence of color images, each of which is 3D. You'll see more about this later.

Watch this ~7-minute video clip.

DSP 1 (10 pts): Matrix Math