AN INCREMENTAL WEIGHTED LEAST SQUARES

APPROACH TO SURFACE LIGHTS FIELDS

Greg Coombe, Anselmo Lastra

Department of Computer Science, University of North Carolina at Chapel Hill

Keywords: Image-based rendering, surface light fields, appearance modelling, least-squares approximation.

Abstract: An Image-Based Rendering (IBR) approach to appearance modelling enables the capture of a wide variety

of real physical surfaces with complex reflectance behaviour. The challenges with this approach are

handling the large amount of data, rendering the data efficiently, and previewing the model as it is being

constructed. In this paper, we introduce the Incremental Weighted Least Squares approach to the

representation and rendering of spatially and directionally varying illumination. Each surface patch consists

of a set of Weighted Least Squares (WLS) node centers, which are low-degree polynomial representations

of the anisotropic exitant radiance. During rendering, the representations are combined in a non-linear

fashion to generate a full reconstruction of the exitant radiance. The rendering algorithm is fast, efficient,

and implemented entirely on the GPU. The construction algorithm is incremental, which means that images

are processed as they arrive instead of in the traditional batch fashion. This human-in-the-loop process

enables the user to preview the model as it is being constructed and to adapt to over-sampling and under-

sampling of the surface appearance.

1 INTRODUCTION

A Surface Light Field (SLF) (Wood, 2000) is a

parameterized representation of the exitant radiance

from the surface of a geometric model under a fixed

illumination. SLFs can model arbitrarily complex

surface appearance and can be rendered at real-time

rates. The challenge with SLFs is to find a compact

representation of surface appearance that maintains

the high visual fidelity and rendering rates.

One approach to this problem is to treat it as a

data approximation problem. The exitant radiance at

each surface patch is represented as a function, and

the input images are treated as samples from this

function. Since there are no restrictions imposed on

the geometry of the object or on the camera

locations, the input samples are located at arbitrary

positions. This is an example of a problem known as

scattered data approximation (Wendland, 2005).

(Coombe, 2005) introduced the notion of casual

capture of a SLF by moving a camera around an

object, tracking the camera pose using fiducials, and

incrementally updating the SLF. This enabled the

operator to see the result, add views where needed,

and stop when he or she was satisfied with the result.

However, one difficulty with the matrix factorization

approach is that it requires fully resampled matrices,

and so can be sensitive to missing data from

occlusions and meshing errors.

In this paper we present Incremental Weighted

Least Squares (IWLS), a fast, efficient, and

incremental algorithm for the representation and

rendering of surface light fields. It is a non-linear

polynomial approximation for multi-variate data,

based on the idea of Least Squares polynomial

approximation, which fits scattered data samples to a

set of polynomial basis functions. WLS is similar to

Figure 1: A model of a heart captured with our system

84

Coombe G. and Lastra A. (2006).

AN INCREMENTAL WEIGHTED LEAST SQUARES APPROACH TO SURFACE LIGHTS FIELDS.

In Proceedings of the First International Conference on Computer Graphics Theory and Applications, pages 84-91

DOI: 10.5220/0001358700840091

Copyright

c

SciTePress

piecewise polynomial approximation and splines,

except that the reconstruction is non-linear.

Weighted Least Squares (Wendland, 2005)

generalizes Least Squares by computing a set of

approximations with associated weighting terms.

These weighting terms can be either noise (in

statistics) or distance (in graphics and computational

geometry (Ohtake, 2003)). If we use distance, then

WLS becomes a local approximation method. This

local approximation is extended to a global

approximation by computing the Partition of Unity

(Ohtake, 2003; Shepard, 1968).

This paper offers the following contributions:

• We apply existing mathematical tools such as

the Weighted Least Squares approximation

technique by casting surface light fields as a

scattered data approximation problem.

• We introduce Incremental Weighted Least

Squares, an incremental approach to surface

light field construction that enables

interactive previewing of the reconstruction.

• Using the IWLS representation, we develop a

real-time surface light field rendering

algorithm, implemented on the GPU, which

provides direct feedback about the quality of

the surface light field.

The paper proceeds as follows. We first discuss

previous approaches to light field capture and

representation, including the difference between

batch processing and incremental processing. In

Section 3 we discuss Least Squares fitting and the

generalization to Weighted Least Squares. We then

introduce IWLS and describe how the WLS

representation can be incrementally constructed and

rendered. In Section 4 we discuss implementation

details of the capture and rendering system, and then

present results and conclusion.

2 BACKGROUND

A good overview of the state of the art in material

modelling by image acquisition is provided by the

recent Siggraph course on Material Modelling

(Ramamoorthi, 2002), and the Eurographics State of

the Art Report on Acquisition, Synthesis and

Rendering of Bidirectional Texture Functions

(Mueller, 2004). The choice of representation of this

captured data is crucial for interactive rendering.

(Lensch, 2001) uses the Lafortune representation

(Lafortune, 1997) and clustered samples from

acquired data in order to create spatially-varying

BRDFs. (Gardner, 2003) and (McAllister, 2002)

describe BDRF capture devices and methods for

BRDF representation.

Data-driven representation can be divided into

parametric and non-parametric approaches. A

parametric approach assumes a particular model for

the BRDF (such as the Lafortune model (Lafortune,

1997) used by (McAllister, 2002)). These models

have difficulty representing the wide variety of

objects that occur in real scenes, as observed by

Hawkins in (Yu, 1999).

A non-parametric approach uses the captured

data to estimate the underlying function and makes

few assumptions about the behavior of the

reflectance. Thus non-parametric models are capable

of representing a larger class of surfaces, which

accounts for their recent popularity in image-based

modelling (Chen, 2002; Furukawa, 2002; Matuzik,

2003; Zickler, 2005). Our approach uses a non-

parametric model to represent surface light fields.

2.1 Surface Light Fields

Surface light fields (Wood, 2000) parameterize the

exitant radiance directly on the surface of the model.

This results in a compact representation that enables

the capture and display of complex view-dependent

illumination of real-world objects. This category of

approaches includes view-dependent texture

mapping (Debevec, 1996; Debevec, 1998; Buehler,

2001), which can be implemented with very sparse

and scattered samples, as well as regular

parameterizations of radiance (Levoy, 1996; Gortler,

1996). (Wood, 2000) use a generalization of Vector

Quantization and Principal Component Analysis to

compress surface light fields, and introduce a 2-pass

rendering algorithm that displays compressed light

fields at interactive rates. These functions can be

constructed by using Principal Component Analysis

(Nishino, 1999; Chen, 2002; Coombe, 2005) or non-

linear optimization (Hillesland 2003). The function

parameters can be stored in texture maps and

rendered in real-time (Chen, 2002).

These approaches can suffer from difficulties

stemming from the inherent irregularity of the data.

If they require a complete and regularly sampled set

of data, an expensive resampling step is needed. To

avoid these problems, we treat surface light field

reconstruction as a scattered data approximation

problem (Wendland, 2005). Scattered data

approximation can be used to construct

representations of data values given samples at

arbitrary locations (such as camera locations on a

hemisphere or surface locations on a model).

AN INCREMENTAL WEIGHTED LEAST SQUARES APPROACH TO SURFACE LIGHTS FIELDS

85

A common scattered data approximation

technique uses Radial Basis Functions (RBFs)

(Moody, 1989). (Zickler, 2005) demonstrated the

ability of RBFs to accurately reconstruct sparse

reflectance data. Constructing this approximation

requires a global technique, since every point in the

reconstruction influences every other point. This is a

disadvantage for an incremental algorithm, as every

value must be recomputed when a new sample

arrives. It is also difficult to render efficiently on

graphics hardware, since many RBF algorithms rely

on Fast Multipole Methods to reduce the size of the

computation (Carr, 2001). We would like a method

that has the scattered data representation ability of

RBFs, but without the complex updating and

reconstruction.

2.2 Incremental Methods

Most of the research in image-based modelling has

focused on batch-processing systems. These systems

process the set of images over multiple passes, and

consequently require that the entire set of images be

available. Incorporating additional images into these

models requires recomputing the model from

scratch.

Formulating surface light field construction as an

incremental processing approach avoids these

problems by incrementally constructing the model as

the images become available. (Matusik, 2004) used

this approach with a kd-tree basis system to

progressively refine a radiance model from a fixed

viewpoint. (Schirmacher, 1999) adaptively meshed

the uv and st planes of a light field, and used an error

metric along the triangle edges to determine the

locations of new camera positions. (Coombe, 2005)

used an incremental PCA algorithm to construct

surface light fields in an online fashion, but required

resampling the data for matrix computation.

3 INCREMENTAL WEIGHTED

LEAST SQUARES

IWLS is a technique to incrementally calculate an

approximation to the exitant radiance at each surface

patch given a set of input images. The process is

divided into two parts: constructing the WLS

representation from the incoming images, and

rendering the result. In this section we review Least

Squares fitting and the generalization to Weighted

Least Squares. We then describe how these WLS

representations can be incrementally constructed and

rendered.

3.1 Least Squares Approximation

Least Squares methods are a set of linear

approximation techniques for scattered data. Given a

set of N scalar samples

ℜ∈

i

f

at points

d

i

x ℜ∈

,

we want a globally-defined function f(x) that best

approximates the samples. The goal is to generate

this function f(x) such that the distance between the

scalar data values f

i

and the function evaluated at the

data points f(x

i

) is as small as possible. This is

written as

min f (x

i

) − f

i

i

∑

(Note: this discussion follows the notation of

(Nealen, 2004)). Typically, f(x) is a polynomial of

degree m in d spatial dimensions. The coefficients of

the polynomial are determined by minimizing this

sum. Thus f(x) can be written as

f (x)

=

b(x)

T

c

where b(x) = [b

1

(x) b

2

(x) ... b

k

(x) ]

T

is the

polynomial basis vector and c = [c

1

c

2

... c

k

] is the

unknown coefficient vector. A set of basis functions

b(t) is chosen based on the properties of the data and

the dimensionality.

To determine the coefficient vector c, the

minimization problem is solved by setting the partial

derivatives to zero and solving the resulting system

of linear equations. After rearranging the terms, the

solution is:

[

]

∑∑

−

=

i

ii

i

T

ii

fxbxbxbc )()()(

1

For small matrices, this can be inverted directly.

For larger matrices, there are several common

matrix inversion packages such as BLAS

(Remington, 1996) and TGT. The size of the matrix

to be inverted depends upon the dimensionality d of

the data and the degree k of the polynomial basis.

3.2 Weighted Least Squares

One of the limitations of Least Squares fitting is that

the solution encompasses the entire domain. This

global complexity makes it difficult to handle large

data sets or data sets with local high frequencies. We

would prefer a method that considers samples that

are nearby as more important than samples that are

far away. This can be accomplished by adding a

GRAPP 2006 - COMPUTER GRAPHICS THEORY AND APPLICATIONS

86

distance-weighting term Θ(d) to the Least Squares

minimization. We are now trying to minimize the

function

∑

−−Θ

i

iii

fxfxx )()(min

A common choice for the distance-weighting

basis function Θ(d) is the Wendland function

(Wendland, 1995)

)1

4

()1()(

4

+−=Θ

h

d

h

d

d

which is 1.0 at d = 0, and falls off to zero at the

edges of the support radius h. This function has

compact support, so each sample only affects a small

neighborhood. There are many other weighting

functions that could be used, such as multiquadrics

and thin-plate splines, but some functions have

infinite support and thus must be solved globally.

Instead of evaluating a single global

approximation for all of the data samples, we create

a set of local approximations. These approximations

are associated with a set of points

x

, which are

known as centers. At each of these centers, a low-

degree polynomial approximation is computed using

the distance-weighted samples x

i

in the local

neighborhood.

[]

∑∑

−Θ−Θ=

−

i

iii

i

T

iii

fxbxxxbxbxxxc )()()()()()(

1

We now have a set of local approximations at

each center. During the reconstruction step, we need

to combine these local approximations to form a

global approximation. Since this global function is a

weighted combination of the basis functions, it has

the same continuity properties.

The first step is to determine the m nearby local

approximations that overlap this point and combine

them using a weight based on distance. However,

the functions cannot just be added together, since the

weights may not sum to 1.0. To get the proper

weighting of the local approximations, we use a

technique known as the Partition of Unity (Shepard,

1968), which allows us to extend the local

approximations to cover the entire domain. A new

set of weights Φ(j) are computed by considering all

of the m local approximations that overlap this point

∑

=

Θ

Θ

=Φ

m

i

i

j

j

x

x

x

1

)(

)(

)(

The global approximation of this function is

computed by summing the weighted local

approximations.

∑

=

Φ=

m

j

j

T

j

xcxbxxf

1

)()()()(

The WLS representation allows us to place the

centers

x

wherever we like, but the locations are

fixed. This is in contrast to the Radial Basis

Function method, which optimizes the location of

the centers as well as the coefficients. We discuss

several strategies for center placement in Section 4.

3.3 Incremental Construction

Surface light field representation can be treated as a

batch process by first collecting all of the images

and then constructing and rendering the WLS

representations. The advantage of batch processing

is that all of the sample points are known at the time

of construction, and the support radii and locations

of the centers can be globally optimized.

The disadvantage to batch processing is that it

provides very little feedback to the user capturing

the images. There is often no way to determine if the

surface appearance is adequately sampled, and

undersampled regions require recomputing the WLS

representation. A better approach is to update the

representation as it is being constructed, which

allows the user to preview the model and adjust the

sampling accordingly. In this section we describe

two approaches to incrementally update the WLS

approximation.

3.3.1 Adaptive Construction

The adaptive construction method starts with all of

the centers having the maximum support radius. As

new image samples are generated, they are tested

against the support radius of a center, and added to

that center’s neighborhood list. The WLS

Figure 2: A diagram of the weighted least squares

approach to function representation. Each cente

r

constructs a low-degree polynomial approximation base

d

on samples in their neighborhood. These local

approximations are then combined to form a global

approximation.

AN INCREMENTAL WEIGHTED LEAST SQUARES APPROACH TO SURFACE LIGHTS FIELDS

87

approximation is computed from the samples in the

neighborhood list. As each center’s list gets larger,

the support radius is decreased, and samples are

discarded if they no longer fall within the

neighborhood. In our implementation, this involves

several user-defined parameters; we typically

decrease the radius by 25% if the number of samples

is more than 4 times the rank of the approximation.

3.3.2 Hierarchical Construction

The adaptive approach has the disadvantage that a

single sample can cause the recomputation of

numerous WLS coefficients, particularly in the

initial phases when the support radii are large. The

hierarchical approach avoids this computation by

subdividing the domain as a quadtree. Initially, the

representation consists of a single WLS center with

a large support radius. When a new image sample

arrives, the hierarchy is traversed until a leaf node is

reached. The sample is deposited at the leaf node

and the WLS is recalculated. If the number of

samples in a leaf node is larger than a pre-

determined threshold, the leaf node is split into four

children. Each child decreases its support radius and

recomputes its WLS coefficients. Note that a sample

can be a member of more than one leaf node, since

the support radii of the nodes can overlap. There are

several user-defined parameters; we have had good

results splitting the nodes if the number of samples

exceeds 4 times the rank of the approximation, and

decreasing the area of the neighborhood by half

(decreasing the radius by 1/√2).

3.4 Rendering

Weighted Least Squares conforms well to the

stream-processing model of modern graphics

hardware. Each surface patch is independent and is

calculated using a series of simple mathematical

operations (polynomial reconstruction and Partition

of Unity). More importantly, the local support of the

WLS centers means that reconstruction only requires

a few texture lookups in a small neighborhood.

After the centers and the WLS coefficients have

been computed (using either the adaptive or the

hierarchical technique), they are stored in a texture

map for each surface patch. The coefficients are laid

out in a grid pattern for fast access by the texture

hardware. The adaptive centers are typically

arranged in a grid pattern, but the hierarchical

pattern must be fully expanded before it is saved to

texture. This is done by copying down any leaf

nodes that are not fully expanded.

During rendering, the viewpoint is projected

onto the UV basis of the surface patch and used to

index into the coefficient texture. The samples from

the neighborhood around this element comprise the

set of overlapping WLS approximations. Texture

lookups are used to collect the neighboring centers

and their coefficients. These polynomial coefficients

are evaluated and weighted by their distance. The

weights are computed using the Partition of Unity,

which generates the final color for this surface patch.

Once the color at each patch has been

determined, we need a method to interpolate the

colors across the model to smoothly blend between

surface patches. One approach is to simply

interpolate the colors directly. However this

approach is incorrect, as it interpolates the values

after the Partition of Unity normalization step. This

generates artifacts similar to those encountered when

linearly interpolating normal vectors across a

triangle. The correct approach is to perform the

normalization after the interpolation. For our system,

we can accomplish this by interpolating the weights

and colors independently, and using a fragment

program to normalize the weights at every pixel.

4 IMPLEMENTATION

The data structure and camera capture are managed

on the CPU and function reconstruction and

rendering is handled by the GPU. In this section we

discuss how input images are converted into surface

Figure 3: A diagram of the surface lightfield capture an

d

rendering system. Images are captured using a handhel

d

video camera, and passed to the system. Using the mesh

information, visibility is computed and the surface

locations are back-

p

rojected into the image. Each of these

samples are incorporated into the Incremental Weighte

d

Least Squares approximation, and sent to the card fo

r

rendering. The user can use this direct feedback to decide

where to move the video camera to capture more images.

GRAPP 2006 - COMPUTER GRAPHICS THEORY AND APPLICATIONS

88

patch samples, which involves camera capture, pose

estimation, and visibility computation. A diagram of

the system is shown in Figure 3.

In order to project the image samples onto the

geometry, the camera's position and orientation must

be known. Our system uses a tracked video camera

to capture images of the object. The camera was

calibrated with Bouguet’s Camera Calibration

Toolbox, and images are rectified using Intel`s Open

Source Computer Vision Library. To determine the

pose of the camera with respect to the object, a stage

was created with fiducials along the border. The 3D

positions of the fiducials are located in the camera's

coordinate system in real-time using the ARToolkit

Library. This library uses image segmentation,

corner extraction, and matching techniques for

tracking the fiducials. This system is similar to the

system presented in our earlier paper (Coombe,

2005).

Table 1: A description of the models and construction

methods used for timing data.

Once the camera pose has been estimated, the

visibility is computed by rendering the mesh from

the point of view of the camera. The depth buffer is

read back to the CPU, where it is compared against

the projected depth of each vertex. If the vertex

passes the depth test, it samples a color from the

projected position on the input image.

For rendering efficiency, the coefficient textures

are packed into a larger texture map. Each texture

stores the coefficients for one term of the

polynomial basis. For all of the examples in this

paper we use a 3-term polynomial basis. We found

that higher-order polynomial bases were susceptible

to over-fitting (Geman, 1992), which occurs when

the number of input samples is small enough that the

polynomial bases try to fit minor errors in the data,

rather than the overall shape. The consequence is

that reconstruction is very accurate at sample

positions, but oscillates wildly around the edges.

Using a lower-degree polynomial avoids this

problem.

The positions of the surface patches, which are

determined a priori, are represented as either

vertices or texels. For most of the models we use

vertices, and the renderer uses a vertex texture fetch

to associate surface patches with vertices.

4.1 Results

We have implemented this system on a 3.2 GHz

Intel Pentium processor with an Nvidia GeForce

7800. A graph of timing results from several

different models is shown in Figure 4, and the

parameters used for these timings are shown in

Table

1

. The rendering algorithm is compact, fast, and

efficient and can render all of the models in this

paper at over 200 frames per second. An image

Model # Vertices Construction # Centers

Bust A 31K Hierarchical 16

Heart A 4K Hierarchical 16

Pitcher A 29K Hierarchical 16

Bust B 14K Hierarchical 16

Bust C 14K Hierarchical 64

Heart B 4K Adaptive 16

Figure 4: Timing results (in seconds per image) for the

IWLS construction. We measured three quantities; the time

to compute the visibility and reproject the vertices into the

image, the Least Squares fitting times, and the time to

transfer the computed results to the graphics card fo

r

rendering.

0

0.5

1

1.5

2

2.5

bust A heart A pitcher

A

bust B bust C heart B

Transfer

WLS

Visibility/Reprojection

Figure 5: A side-by-side comparison of the WLS

reconstruction with an input image that was not included

in the training set.

AN INCREMENTAL WEIGHTED LEAST SQUARES APPROACH TO SURFACE LIGHTS FIELDS

89

generated with our system is shown in Figure 1, and

a side-by-side comparison is shown in Figure 5.

The hierarchical construction method is much

faster than the adaptive construction method due to

the fact that the adaptive construction method can

potentially cause the recomputation of a number of

coefficients. For the 4K-vertex heart model, the

adaptive construction generated about 5.1 Least

Squares fitting computations per image, while the

hierarchical construction only generated about 1.7.

As this is the most time-consuming aspect of the

process, reducing the number of Least Squares fits is

important to achieve good performance. This

performance gain enables higher resolution

reconstruction; note that the bust model with 64

centers is only 1.4 times slower than the 16 center

version, even though it has 4 times as many

coefficients. However, the increased number of

coefficients is reflected in the data transfer time,

which is close to 4 times longer.

A potential issue with the hierarchical

construction is that it could introduce error. We

conducted an experiment to compare the quality of

the reconstruction with a reference batch process

which has global knowledge. The results are shown

in Figure 6.

We have tried several center placement

strategies; a uniform grid over projected hemisphere

directions, a uniform disk using Shirley’s concentric

mapping (Shirley, 1997), and jittered versions of

each. A comparison is shown in

Table 2. For most of

the models in this paper we use the grid method due

to its ease of implementation.

Table 2: RMS Error values from reconstructing a WLS

approximation while varying the center layout. The error

was computed with a training set of 64 images and an

evaluation set of 74 images. The disk is a slight

improvement in terms of error compared to the grid, and it

has a large benefit in terms of reducing the variability of

the error.

Layout Mean RMS Error StD RMS Error

Uniform Grid 0.0519 0.0086

Jittered Grid 0.0608 0.0148

Uniform Disk 0.0497 0.0045

Jittered Disk 0.0498 0.0036

5 CONCLUSION

We have introduced Incremental Weighted Least

Squares (IWLS), a fast, efficient, and incremental

algorithm for the representation and rendering of

surface light fields. IWLS can be used to render high

quality images of surfaces with complex reflectance

properties. The incremental construction is useful for

visualizing the representation as it is being captured,

which can guide the user to collect more images in

undersampled regions of the model and minimize

redundant capture of sufficiently sampled regions.

The rendering algorithm, which is implemented on

the GPU for real-time performance, provides

immediate feedback to the user.

5.1 Future Work

There are several improvements to our system that

we are interested in pursuing. Currently, the surface

patches are computed independently and do not

share information. This choice was made in order to

allow the algorithm to reconstruct as broad a class of

surfaces as possible. However, many surfaces have

slowly-varying reflectance properties which could

be exploited for computational gain. Each surface

patch could collect WLS coefficients from itself as

well as its neighbors. This would involve adjusting

the distance weighting to reflect the distance along

the surface of the model. We could also use an

approach similar to Zickler (Zickler, 2005) to share

reflectance values across a surface.

This system was designed to construct and

render surface light fields, which allow arbitrary

Figure 6: The reconstruction error of the hierarchical

construction versus a batch construction for a single surface

p

atch of the bust model. Each method used only the inpu

t

samples available, and the error was measured against the

full set of samples. The hierarchical algorithm is initially

superior to the batch algorithm, and continues to be simila

r

in error behavior while also being much faster to compute.

GRAPP 2006 - COMPUTER GRAPHICS THEORY AND APPLICATIONS

90

viewpoints but a fixed lighting. We are interested in

applying IWLS to the problem of reconstructing

arbitrary lighting, but from a static viewpoint. This

is similar to Polynomial Texture Mapping

(Malzbender, 2001), which used Least Squares

fitting and custom hardware to render images with

varying lighting.

Mathematically, the Weighted Least Squares

approach generalizes easily to multiple dimensions

by simply modifying the polynomial basis.

However, the substantial increase in data would

require re-thinking the construction and rendering

components of our system.

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