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Published online by Cambridge University Press: 05 May 2012
Digital images today play a vital role in science and technology, and also in many aspects of our daily life. This book seeks to advance the analysis of images, especially digitized ones, through the statistical analysis of shapes. Its focus is on the analysis of landmark-based shapes in which a k-ad, that is, a set of k labeled points or landmarks on an object or a scene, is observed in two or three dimensions, usually with expert help, for purposes of identification, discrimination, and diagnostics.
In general, consider the k-ad to lie in ℝm (usually, m = 2 or 3) and assume that not all the k points are the same. Then the appropriate shape of the object is taken to be the k-ad modulo a group of transformations.
For example, one may first center the k-ad, by subtracting the mean of the k-ad from each of the k landmarks, to remove the effect of location. The centered k-ad then lies in a hyperplane of dimension mk - m, because the sum of each of the m coordinates of the centered k points is zero. Next one may scale the centered k-ad to unit size to remove the effect of scale or size. The scaled, centered k-ad now lies on the unit sphere Sm(k-1)-1 in a Euclidean space (the hyperplane) of dimension m(k - 1) and is now called the preshape of the k-ad.
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