Newsgroups: sci.image.processing
Path: cantaloupe.srv.cs.cmu.edu!europa.chnt.gtegsc.com!howland.reston.ans.net!Germany.EU.net!Munich.Germany.EU.net!eso.org!news
From: ndevilla@eso.org (Nicolas Devillard)
Subject: Re : subpixel accuracy in grey scale images
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Message-ID: <1995Jun27.092318.16654@eso.org>
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Organization: ESO - European Southern Observatory, Garching by Munich
Date: Tue, 27 Jun 1995 09:23:18 GMT
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In his latest article about the subject of subpixel accuracy in grey scale images, John C. Schultz described the general way to correlate (or rather convolve in this method) two images. I would like to introduce the notion of subpixel localization more precisely, but first, let us stay at an upper level :

A correlation-based technique looks for the point where two images coincide most. You can find this point, either by finding a minimum of differences (Sum of Squared Differencies), or finding a maximum of coincidence (Convolution). Of course, both methods will give their best result as the most probable pixel.

Now let us have a closer look on the method :

Provided you assume the optical flow condition, one can prove the equation of a cross-correlation matrix is, locally to the extremum, a quadrique of equation :

z = a.x^2 + b.y^2 + c.xy + d.x + e.y + f

If you get a little further in the calculus, you will find c and d to be approximately zero in any case, that proves the problem to be separable in x and y.
Thus the equation of the cross-correlation matrix could be separated in 2 additionnal 1-D parabolic functions. Therefore the way to find the most coincidental point between two images is to fit a parabola in the 3 extremum values of the cross correlation matrix (in x and in y). The extremum of this parabola will be the real value of the most coincindental point.

This method proves to have a 1/96th theorical subpixel accuracy on 8bit gray level images, and an observed 1/10th subpixel accuracy in "normal" CCIR-601 video images, where the optical flow equation appears to be most significant.

Correlation-based techniques and their problems :

First of all, correlations are sensitive to gain and offset. As described by John C. Schultz, the best way to get beyond that is to normalize both your inputs (one is enough, though the demonstration is left to the reader !), if you do not have a mean to evaluate yourself gain and offset from other sources.

Correlations always return a result of best-correlated point, but that is "often" a wrong matching. Moreover, the risk of wrong matchings increases with the size of the research domain. Recognizing someone you only saw a photograph of, is easy in a group of 6 persons, is more difficult with a group of 36 (the chance of error is augmented in a linear way with the image surface in our case). So-called AI-techniques are a way to localize the search and thus minimize the chance of error. Many methods have prov
ed to be useful in such and such case, but the term AI only seems to...
"make good marketing"  ! Refer to hyperplane selection, dichotomy, temporal coherence in a sequence of images, machine learning, etc. to have more information about these techniques.

Hope I did not forget anything,

_______________________________________________________________________________

Nicolas Devillard . European Southern Observatory - Munich, Germany.


