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Statistical Shape Analysis, 2nd Edition by Kanti V. Mardia, Ian L. Dryden

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8 2D Procrustes analysis using complex arithmetic

8.1 Introduction

This chapter provides an overview of the techniques introduced so far when a random sample of landmark configurations is available in two dimensions. In this chapter we make particular use of complex numbers which leads to simple methodology in this important case. Although most of the material for this chapter has been described using general matrix notation in previous chapters, it is often much simpler to use complex vectors in the 2D case. This chapter is designed to be largely self-contained for planar shape analysis using complex notation.

8.2 Shape distance and Procrustes matching

Consider two centred configurations y = (y1, …, yk)T and w = (w1, …, wk)T, both in , with y*1k = 0 = w*1k, where y* denotes the transpose of the complex conjugate of y. In order to compare the configurations in shape we need to establish a measure of distance between the two shapes.

A suitable procedure is to match w to y using the similarity transformations and the differences between the fitted and observed y indicate the magnitude of the difference in shape between w and y. Consider the complex regression equation

(8.1) numbered Display Equation

where A = (A1, A2)T = (a + ib, βeiθ)T are the 2 × 1 complex parameters with translation a + ib, scale ...

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