A Method for Measuring Distances in Motion Blurred Images

Motion blurring causes contrast boundaries to "move" relative to each other in photographs in the sense that the distances between them are different than in clear frames. (see  The Effect of Motion Blurring on Contrast Boundaries )  This means that any measurement methodology must take these differences into account.  The simplest approach would be to choose references that move the same amount.  This is accomplished by applying a very simple rule.  Measure distances between boundaries with the same polarity.  The polarity of an boundary is determined by which side is bright and which side is dark.  Bright on the right has the opposite polarity as dark on the right.  Strictly speaking, the contrast across the boundaries should also match.  In practice, boundaries with high contrast should be chosen because the difference in movement is affected less by the difference in contrast.

shutter open    shutter close

The above is an idealized representation where white represents the bright side of a contrast boundary and black represents the dark.  The top strip represents an image at the start of the exposure and is the actual appearance of the image.  The middle strip is the image at the end of the exposure and the bottom strip is the resulting image on film.  The vertical lines represent the position of the far left contrast boundary at the beginning and the end of the exposure.  The arrow between them represents the movement  of the image during the exposure or the blur length.  The bottom strip is the resulting image applying the rule that all areas that are hit by bright during the exposure are bright.  Boundaries with dark on the right are moved the same amount so the distance between them remains the same.  Boundaries with bright on the right are not moved.  In both possible cases, the distance between boundaries with the same polarity remains constant.  Thus the error introduced by blurring is minimized when same polarity references are chosen.
There are two cases to consider when the polarity is different.  The bright (white) areas in the starting image represent the distance between opposite polarity images with dark on the outside.  This distance increases by the blur length.  The dark (black) areas represent the distance between boundaries with bright on the outside.  This distance decreases by the blur length.  Reversing the blurring direction reverses the sense of right and left but the distance between same polarity boundaries remains the same.  The white areas will still grow and the black areas will still shrink.  The results of the two possible cases for different polarity boundaries, bright on the outside or dark on the outside remain the same.  (To actually see this, you need to view your screen in a mirror.)  There is a difference, however, in the results of image movement to the right and to the left.  A boundary with bright on the left is recorded in its shutter close position when movement is to the right and its shutter open position when the movement is to the left.  The opposite is true of opposite polarity boundaries.
.  The boundary is at its shutter close position when the bright is moved toward dark.  The boundary is at its shutter open position when dark is moved toward bright.  If the bright side is on the left, for example, then its position is shutter open when movement is to the right and shutter close when movement is to the left.  There are two instances where this method fails.

Pathological Case 1

The image direction changes direction during  the exposure.  When this happens,  boundaries of one polarity or the other, depending on the direction of image movement, do not represent either a shutter open or a shutter close position.  This usually means that there is not a lot of net movement in the image and thus the error introduced should be small.  This is an error in the time of the measurement, not the magnitude.

There is another case that can arise if one of the boundaries is moving relative to the other.

Pathological Case 2

This occurs when one boundary is moving (across the film) in the opposite direction of the other.  Then one boundary is in its shutter open position and one is in its shutter close position.  This is exactly the same as if the boundary that is moving opposite the direction of the image had its polarity switched.  Because of mirror image symmetry, that is exactly what happens. The results above for opposite polarity boundaries then follow.  The solution requires replacing one of the references with one of opposite polarity which is a known (or estimated) distance from the reference being replaced for the image in which this case occurs.