Fabio Crosilla

Use of generalised Procrustes analysis for the photogrammetric block adjustment by independent models

Abstract The paper reviews at first some aspects of the generalised Procrustes analysis (GP) and outlines the analogies with the block adjustment by independent models. On this basis, an innovative solution of the block adjustment problem by Procrustes algorithms and the related computer program implementation are presented and discussed. The main advantage of the new proposed method is that it avoids the conventional least squares solution. For this reason, linearisation algorithms and the knowledge of a priori approximate values for the unknown parameters are not required. Once the model coordinates of the tie points are available and at least three control points are known, the Procrustes algorithms can directly provide, without further information, the tie point ground coordinates and the exterior orientation parameters. Furthermore, some numerical block adjustment solutions obtained by the new method in different areas of North Italy are compared to the conventional solution. The very simple data input process, the less memory requirements, the low computing time and the same level of accuracy that characterise the new algorithm with respect to a conventional one are verified with these tests. A block adjustment of 11 models, with 44 tie points and 14 control points, takes just a few seconds on an Intel PIII 400 MHz computer, and the total data memory required is less than twice the allocated space for the input data. This is because most of the computations are carried out on data matrices of limited size, typically 3×3.

Solving the PnP Problem with Anisotropic Orthogonal Procrustes Analysis

In this paper we formulate the Perspective-n-Point (a.k.a. exterior orientation) problem in terms of an instance of the an isotropic orthogonal Procrustes problem, and derive its solution. Experiments with synthetic and real data demonstrate that our method reaches the best trade-off between speed and accuracy. The MATLAB code reported in the paper testifies that it is also exceedingly simple to implement.

Regression analysis with partially labelled regressors: carbon dating of the Shroud of Turin

The twelve results from the 1988 radio carbon dating of the Shroud of Turin show surprising heterogeneity. We try to explain this lack of homogeneity by regression on spatial coordinates. However, although the locations of the samples sent to the three laboratories involved are known, the locations of the 12 subsamples within these samples are not. We consider all 387,072 plausible spatial allocations and analyse the resulting distributions of statistics. Plots of robust regression residuals from the forward search indicate that some sets of allocations are implausible. We establish the existence of a trend in the results and suggest how better experimental design would have enabled stronger conclusions to have been drawn from this multi-centre experiment.

An automatic classification and robust segmentation procedure of spatial objects

This paper proposes a statistical procedure for the automatic volumetric primitives classification and segmentation of 3D objects surveyed with high density laser scanning range measurements. The procedure is carried out in three main phases: first, a Taylor’s expansion nonparametric model is applied to study the differential local properties of the surface so to classify and identify homogeneous point clusters. Classification is based on the study of the surface Gaussian and mean curvature, computed for each point from estimated differential parameters of the Taylor’s formula extended to second order terms. The geometrical primitives are classified into the following basic types: elliptic, hyperbolic, parabolic and planar. The last phase corresponds to a parametric regression applied to perform a robust segmentation of the various primitives. A Simultaneous AutoRegressive model is applied to define the trend surface for each geometric feature, and a Forward Search procedure puts in evidence outliers or clusters of non stationary data.

Procrustean Point-Line Registration and the NPnP Problem

In this paper we formulate the point-line registration problem, which generalizes absolute orientation to point-line matching, in terms of an instance of the orthogonal Procrustes problem, and derive its solution. The same formulation solves the Non-Perspective-n-Point camera pose problem, which in turn generalizes exterior orientation to non-central cameras, i.e., Generalized cameras where projection rays do not meet in a single point. Our Procrustean solution is very simple and compact, and copes also with scaling. Experiments with simulated data demonstrate that our method compares favourably with the state-of-the-art in terms of accuracy.

Unsupervised Classification of Raw Full-Waveform Airborne Lidar Data by Self Organizing Maps

The paper proposes a procedure based on Kohonen’s Self Organizing Maps (SOMs) to perform the unsupervised classification of raw full-waveform airborne LIDAR (Light Detection and Ranging) data, without the need of extracting features from them, that is without any preprocessing. The proposed algorithm allows the classification of points into three classes (“grass”, “trees” and “road”) in two subsequent stages. During the first one, all the raw data are given as input to a SOM and points belonging to the category “trees” are extracted on the basis of the number of peaks that characterize the waveforms. In the second stage, data not previously classified as “trees” are used to create a new SOM that, together with a hierarchical clustering algorithm, allows to distinguish between the classes “road” and “grass”. Experiments carried out show that raw full-waveform LIDAR data were classified with an overall accuracy of 93.9%, 92.5% and 92.9%, respectively.

A Forward Search Method for Robust Generalised Procrustes Analysis

One drawback of Procrustes Analysis is the lack of robustness. To overcome this limitation a procedure that applies the Generalised Procrustes methods, by way of a progressive sequence inspired to the “forward search”, was developed. Starting from an initial centroid, defined by the partial point configuration satisfying the LMS principle, this is extended by joining, at every step, a restricted subset of the remaining points. At every insertion, the updated centroid, redetermined by the new considered points, is compared with the previous by way of the common elements. If significant variations of the similarity transformation parameters occur, they reveal the presence of outliers or non stationary points among the new elements just inserted.

Procrustean Photogrammetry: From Exterior Orientation to Bundle Adjustment

This work reviews the anisotropic row-scaling variant of the Procrustes analysis algorithms applied to develop new analytical tools for solving classical photogrammetric Computer Vision problems. In Garro et al. (Solving the pnp problem with anisotropic orthogonal procrustes analysis, 2012) the anisotropic row-scaling Procrustes analysis was first applied to perform the exterior orientation of one image. Moreover Fusiello and Crosilla (ISPRS J Photogrammetry and Remote Sens 102:209–221, 2015) provided a Procrustean formulation of the photogrammetric bundle block adjustment problem. Procrustean methods do not require any linearization nor approximated values of the unknown parameters and the results obtained are comparable in terms of accuracy with those given by the state-of-the-art methods.

Errors-in-Variables Anisotropic Extended Orthogonal Procrustes Analysis

This letter presents a novel total least squares (TLS) solution of the anisotropic row-scaling Procrustes problem. The ordinary LS Procrustes approach finds the transformation parameters between origin and destination sets of observations minimizing errors affecting only the destination one. In this letter, we introduce the errors-in-variables model in the anisotropic Procrustes analysis problem and present a solution that can deal with the uncertainty affecting both sets of observations. The algorithm is applied to solve the image exterior orientation problem. Experiments show that the proposed TLS method leads to an accuracy in the parameters estimation that is higher than the one reached with the ordinary LS anisotropic Procrustes solution when the number of points, whose coordinates are known in both the image and the external systems, is small.