Real-Time Imaging 10 (2004) 239–250 Morphometrical data analysis using wavelets C.M. Takemura a,Ã , R.M. Cesar- Jr. a , R.A.T. Arantes b , L. da F. Costa b , E. Hingst-Zaher c , V. Bonato d , S.F. dos Reis e a Departamento de Cieˆncia da Computac - a˜o, Instututo de Matema´tica e Estatı´stica da Universidade de Sa˜o Paulo (USP-IME), 1010 Rua do Mata˜o, Cidade Universita´ria,CEP 05508-090, Sa˜o Paulo, SP, Brazil b USP-IFSC, Av. Trabalhador Sa˜ocarlense, 400Caixa Postal 369, CEP 13560-970, Sa˜o Carlos, SP, Brazil c USP-MZUSP, Av. Nazare´, 481 Bairro do Ipiranga,CEP 04263-000, Sa˜o Paulo, SP, Brazil d CREUPI-ICB, Av. He´lio Vergueiro Leite s/n,CEP 13990-000, Espı´rito Santo do Pinhal, SP, Brazil e UNICAMP-IB, Cidade Universita´ria ‘‘Zeferino Vaz’’, CEP 13083-970, Campinas, SP, Brazil Available online 10 August 2004 Abstract In this paper, we present a new shape analysis approach using the well-known wavelet transform and exploring shape representation by landmarks. First, we describe the approach adopted to represent the landmarks data as parametric signals. Then, we show the relation of the derivatives of Gaussian wavelet transform applied to the signal-to-differential properties of the shape that it represents. We present experimental results using real data to show how it is possible to characterize shapes through multiscale and differential signal-processing techniques in order to relate morphological variables with phylogenetic signal, environmental factors and sexual dimorphism. The goal of this research is to develop an effective wavelet transform-based method to represent and classify multiple classes of shapes given by landmarks. r 2004 Elsevier Ltd. All rights reserved. 1. Introduction The relationship between phenotype and pre-deter- mined time–space or genetic conditions has been the focus of many works presented to the scientific community. This kind of analysis usually involves morphometry, which can be described as the study of biological shapes. Traditional morphometrics generally apply multivariate statistical methods to size or shape variables such as distances and angles [1–3]. On the other hand, geometric morphometrics deals with geo- metrical relationships between these measurements [4]. Note that morphometrics is a multidisciplinary area related to shape analysis [5,6]. Nevertheless, there are few works that explore well-known multiscale and differential methods in the shape analysis and image processing community, e.g. wavelets [7–9], a gap that is partially filled by the method introduced in this paper. Many problems in a wide variety of disciplines may be addressed in terms of computer vision concepts and methods and it is possible to reduce many of these problems to shape analysis [5], where visual information like color, texture and motion can be discarded. The wavelet transform is a particularly useful tool to analyze non-stationary signals presenting local events because of its local analysis property [10,11]. Further- more, its scaling properties have been extensively used to implement multiscale tools for signal analysis, thus explaining its popularity in several practical applications [12,13]. The goal of the present work is to review and generalize the work presented in [6] where 2D landmark data sets were analyzed through the wavelet transform, ARTICLE IN PRESS www.elsevier.com/locate/rti 1077-2014/$-see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.rti.2004.05.006 Ã Corresponding author. E-mail addresses: maki@vision.ime.usp.br (C.M. Takemura), cesar@vision.ime.usp.br (R.M. Cesar- Jr.). URL: http://www.ime.usp.br/~cesar/.