Supposewe have measured correspondingpoints in a pair of -dimensionalimages,andwant to compute a similarity transformrelatingthem. Sucha situationoccursfor examplein Image-guidedSurgery, wherevery accuratealignmentsare required,and attainedthroughthe useof markers, or ultrasoundcalibrations[4]1. A similarity transformcombinesa scalinganda rigid bodytransformation.Theleastsquaressolutionis known, and theorthogonalrigid bodypartgaineda nameastheOrthogonal Procrustes Problem [6]. On theotherhand,the scalingfactorgetsmaybelessattentionthanit deserves,in particularits potentialbias. It is apparentlysimpler, but, comparedto rotations,thestandardleastsquaresolution[2] is not symmetric, in thesensethatcomputingin thereversedirectiongivesa differentanswer , this is currentlyan importantissuefor non linear registration,so it seemsimportantto clarify it for linearones.Also, thedistinctionbetweenfor examplearesizingof theobject,and a resizingof theimageoftendoesnot appearclearly.
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