TY - BOOK

T1 - Algebraic and Geometric Techniques for Optimization in Metric Vision

AU - Harker, Matthew

N1 - no embargo

PY - 2008

Y1 - 2008

N2 - This thesis is a collection of papers pertaining to the general topic of Metric Vision. The specific topics covered by the thesis are: Direct estimation via Grassmannian manifolds; the computation of distances between geometric objects; the calibration of optical measurement devices; and finally the automatic processing of surface data. The direct estimation of geometric objects is achieved by describing geometric objects by their Grassmannian coordinates and applying Lagrange multipliers to solve quadratic constrained least squares problems. Direct estimation of specific types of conics, coupled geometric objects, as well as multi-view relations is achieved. With respect to distance computation, it is shown that there exists a polynomial whose roots are the extremal distances between geometric objects. Moreover, a first order approximation to the distance may be made, which sacrifices accuracy in the name of computational cost. Further, the steps of the full calibration of a measurement device are outlined. From the nonlinear optimization of calibration parameters, follows the first order propagation of the covariance of the estimated parameters, and finally to describing the confidence envelopes about measured values. Finally, numerically perfectly conditioned polynomial moments for least squares surface representation are derived. While efficiently performing linear filtering to noisy data, they also enable the separation of overall surface structure from the surface relief. Finally, the least squares reconstruction of a surface from its gradient field is derived.

AB - This thesis is a collection of papers pertaining to the general topic of Metric Vision. The specific topics covered by the thesis are: Direct estimation via Grassmannian manifolds; the computation of distances between geometric objects; the calibration of optical measurement devices; and finally the automatic processing of surface data. The direct estimation of geometric objects is achieved by describing geometric objects by their Grassmannian coordinates and applying Lagrange multipliers to solve quadratic constrained least squares problems. Direct estimation of specific types of conics, coupled geometric objects, as well as multi-view relations is achieved. With respect to distance computation, it is shown that there exists a polynomial whose roots are the extremal distances between geometric objects. Moreover, a first order approximation to the distance may be made, which sacrifices accuracy in the name of computational cost. Further, the steps of the full calibration of a measurement device are outlined. From the nonlinear optimization of calibration parameters, follows the first order propagation of the covariance of the estimated parameters, and finally to describing the confidence envelopes about measured values. Finally, numerically perfectly conditioned polynomial moments for least squares surface representation are derived. While efficiently performing linear filtering to noisy data, they also enable the separation of overall surface structure from the surface relief. Finally, the least squares reconstruction of a surface from its gradient field is derived.

KW - Messtechnik

KW - Messunsicherheit

KW - Fehlerfortpflanzung

KW - Oberflächenrekonstruktion

KW - Optimierung

KW - polynomiale Momente

KW - Datenmodellierung

KW - metrology

KW - measurement uncertainty

KW - error propagation

KW - surface reconstruction

KW - optimization

KW - polynomial moments

KW - curve fitting

M3 - Doctoral Thesis

ER -