# Function approximation for lighting property

To relight an item under different illumination circumstances, we have to model the reflectance property of its surface elements. Mathematically, the reflectance property of a surface element is actually a spherical function. Its input domain is the spherical surface and the output range is the group of real numbers. To aid the relighting process, we generally make use of a spherical basis, either in discrete or continuous form, to approximate the spherical functions. This dissertation deals with a number of problems in the approximation process. Initially, the spherical harmonic (SH) technique is the conventional means for representing the low frequency lighting effects. In this method, a spherical function is approximated as a weighted sum of SH basis functions….

Contents

1 Introduction
1.1 Thesis Contribution
1.1.1 Noise-resistant ﬁtting for SH coeﬃcients
1.1.2 Spherical wavelet for all-frequency lighting
1.1.3 Multiscale spherical radial basis function (MSRBF)
1.2 Thesis Structure
2 Background
2.1 Lighting Model
2.1.1 Plenoptic illumination function for IBR
2.1.2 Rendering from reference images
2.1.3 Plenoptic illumination function for 3D objects
2.2 Diﬀerent Lighting Conﬁgurations
2.2.1 Directional source lighting
2.2.2 Point source lighting
2.2.3 Distant environment lighting
2.3 Spherical Harmonic (SH) and Hemispherical Harmonic (hemi-SH) basis
2.4 Coeﬃcient Estimation
2.4.1 Rendering
3 Noise-resistant Fitting for SH Coeﬃcients – Part I
3.1 Stability Analysis of Coeﬃcient Estimation
3.2 Magnitude of SH coeﬃcients
3.3 Noise Model
3.4 Constrained Solution
3.4.1 The constraint
3.4.2 Constrained least square
3.4.3 Speedup
3.4.4 Extension to the scattered sampling
3.4.5 Constrained eigen basis
4 Noise-resistant Fitting for SH Coeﬃcients – Part II
4.1 Double SH Basis Projection Approach
4.2 Coeﬃcient Estimation for Double SH Basis Projection
4.2.1 Discussion on the noise model
4.2.2 Energy bounded constraint
5 Spherical Wavelet for All-frequency Lighting
5.1 Overview
5.1.1 IAIdataset
5.1.2 Time-varying distant environment
5.1.3 Rendering in time-varying distant environment
5.2 Spherical Wavelet (SW)
5.2.1 Scaling functions and wavelet functions
5.2.2 ISWonIAI
5.2.3 Dual SW environment coeficient vector
5.3 Important Coeﬃcient Rendering
5.4 Results
5.4.1 Discrepancy
5.4.2 Error in the environment approximation
5.4.3 Error in rendered images
5.4.4 Visual comparison
6 Multiscale Spherical Radial Basis Function (MSRBF)