摘 要

在這篇論文中，我們提出了一個系統，利用統計出來的特徵資料以及影像上的人臉特徵重建三維模型。此系統包含了兩個部分，事前的統計過程以及重建三維人臉的應用階段。在第一階段，我們提出了一個新的三維人臉校準的方法，是將三維人臉資料做柱狀座標的轉換遞迴地找出精確的校準關係，並在實驗中也證明此方法的確能夠降低1/7左右的誤差。利用這校準好的資料加上Radial Basis Function的技術取得合理的資料對應，並應用PCA得到在三維空間中人的臉部特徵變化。在實驗中也證實此統計方式能夠有效利用數個具代表性的人臉特徵線性組合出各種不同的三維人臉模型。

我們將事先統計好的人臉特徵資料應用於單張人臉影像的重建。在此提出的方法是利用Levenberg-Marquardt 的最佳化演算法，找到最佳人臉特徵的線性組合能夠將三維人臉與二維影像上的臉部特徵及輪廓表現上的差異降到最低。我們將三維上的人臉特徵及輪廓資訊以Weak-perspective的模式投影到二維空間中，並在此空間中來估測彼此間的誤差值。在人臉特徵間的誤差是以Euclidean distance來衡量，而在輪廓表現上的誤差，我們以partially one-way Haussdorf distance 來估測，同時考量這兩種誤差值，並取得一個最佳的權衡點以重建逼真的三維人臉模型。

在實驗結果中，我們將所有誤差量化以觀查其準確性，除了估測在二維空間中的誤差外，在模擬的資料中也直接的比較三度空間中的誤差，從數據中可以看出其誤差比例是很小的，如此可以顯示其準確性。我們也將從正面影像重建出的三維人臉中，比較其真正的側面以及重建的人臉側面，由此得知此三維重建演算法是合理也是相當可行的方法。

Abstract
In this thesis, we propose a new technique for reconstructing 3D head model from a single 2D image based on using a 3D eigenhead model. This system is composed of two components, offline training of the eigenhead model and online reconstruction of a 3D head model. For the first part, we propose a new 3D head alignment algorithm based on establishing dense point correspondences between 3D head model in the cylindrical coordinate to align the 3D head models in the training data set. With the proving of the experimental results, we reduce about 1/7 of the average error. We apply the Radial Basis Function technique to establish dense correspondences between each 3D face model and a reference face model, followed by the principal component analysis technique to compute the statistical eigenhead model.

For the 3D face reconstruction from a single image, the proposed algorithm finds the best linear combination of the eigenhead bases that minimizes an energy function composed of distances between the corresponding facial feature points and a one-way partial Haussdorf distance between the facial contours in the image domain. This energy minimization is accomplished by the iterative Levenberg-Marquardt algorithm with the initial guess determined by solving a linear system derived from the image projection constraints for the corresponding facial feature points. Experimental results are given to demonstrate the performance of the proposed algorithm

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