Estimation of Measurement/Process Variance

There are numerous sources of variation in this data: sensor noise, movement artifacts, expression changes, and potentially subtle processes such as heartbeat or breathing. For our purposes, we will lump all of these together into a single "noise" quantity. The goals of this subsection of the project are to estimate the levels of this noise (e.g., in terms of variance or standard deviation) and to estimate the minimum number of images that need to be acquired from a single subject to get a relatively robust/low noise estimate of the face shape.

Following registration, you will have a set of images in (close to) the same coordinate frame.[1] In principle, it's straightforward to measure variance among them. But there are a number of thorny issues.

First, what do we mean by "variance" here? There are many potential dimensions in which you could measure it. What we're really interested in measuring is "how much do face scan images vary with respect to each other?" There are certainly variations in all three spatial dimensions, as well as "within-image" variations (i.e., structural deformations due to puffing cheeks, closing eyes, etc.)

In practice, the "Z" dimension (depth) likely has the most scanner noise, as it is mathematically the most challenging to measure. So we'll focus on measuring variance in the Z coordinate. So we're looking for something like "how much does a set of images vary in the Z dimension"?

The second question is: what do we mean by "the variance"? The surfaces we're measuring are densely sampled continuous "depth fields" (i.e., 2D images with a depth assignment at every point). It's not clear what a single summary statistic might be. To tackle this, we'll measure two different things:

  1. A standard deviation field: an estimate of standard deviation in the Z direction at every (X,Y) point. This will be a fairly large amount of data, so it is best plotted rather than tabulated. Any plot style that makes clear the relationship of depth variance to location is sufficient, but I suggest either a contour plot or a 3D depth rendering. (If you provide the latter, please provide multiple views, so that it is clear what the full deviation field looks like.) Such a plot should allow you to see which parts of the face are subject to more or less variation. You may provide multiple plots, if it helps clarify the variation field.
  2. A summary statistic which reduces the entire deviation field to a single scalar. You should provide at least the mean deviation across the field, but you may provide other statistics if you like, as well.
[1] Technically, the alignment is only with respect to the landmarks -- you can expect the images to be fairly closely aligned in the vicinity of the defined landmarks, but they could differ almost arbitrarily outside those bounds (e.g., around the throat/collar bone, in the hair, etc.) [back]