MEASUREMENT NOISE IN PHOTOMETRIC STEREO BASED

SURFACE RECONSTRUCTION

Toni Kuparinen, Ville Kyrki

Department of Information Technology, Lappeenranta University of Technology, P.O. Box 20, 53851 Lappeenranta, Finland

Pekka Toivanen

Department of Computer Science, University of Kuopio, P.O.Box 1627, 70211 Kuopio, Finland

Keywords:

Surface reconstruction, photometric stereo, gradient ﬁelds, Fourier domain, Wiener ﬁlter, denoising, white

noise.

Abstract:

In this paper, we present a noise reduction method for photometric stereo based surface reconstruction of

surfaces with high frequency height variation. Such surfaces are important for many industrial settings, for

example, in paper and textile manufacturing. The paper presents the derivation of the effect of white image

noise to gradient ﬁelds. Based on the derivation, a denoising approach of the gradient ﬁelds using the Wiener

ﬁlter is proposed. Several known surface reconstruction methods with and without the proposed denoising

approach are evaluated experimentally, with respect to the effect of the noise, and the boundary conditions

of the reconstruction. The experimental results validate that the proposed approach improves the surface

reconstruction on surfaces with high frequency height variation.

1 INTRODUCTION

Surface topography is a highly important quality pa-

rameter in many industrial applications, such as paper

and textile manufacturing. Undesired surface topog-

raphy variations can reﬂect imperfections in manufac-

turing process, product operational efﬁciency, and life

expectancy. Depth recovery techniques, such as shape

from shading (SfS) (Horn, 1990), and photometric

stereo (PS) (Woodham, 1978), can provide surface

gradients in a fast and non-contact manner. In order

to obtain the surface topography, that is, the relative

height values of the surface, the differential surface

gradients have to be integrated. However, in practice

the surface gradients are corrupted by noise, resulting

from imaging and other measurement errors.

Several solutions have been proposed to integrate

the measured gradient ﬁelds. A traditional method

for integrating the surface height from gradient infor-

mation is the Frankot-Chellappa algorithm (Frankot

and Chellappa, 1988). Another popular method is the

Poisson solver (Simchony et al., 1990). Typically,

the performance of the surface reconstruction meth-

ods has been evaluated on surfaces with large, smooth

objects, such as ﬂower pots, faces, peaks, and ramps

with rather strong additive Gaussian noise (Noakes

and Kozera, 2003; Karacali and Snyder, 2004; Wei

and Klette, 2003; Agrawal et al., 2006).

However, monitoring of surface roughness and

texture, that is, higher frequency variations with lim-

ited noise levels, are frequently of interest in manu-

facturing processes. McGunnigle (McGunnigle and

Chantler, 2003) presented a framework for measure-

ment and modelling of rough surfaces. They evalu-

ated several surface description models, but did not

reconstruct surfaces with or without noise. Recently

Hansson (Hansson and Johansson, 2000) has studied

two-light photometric stereo in paper surface recon-

struction. Based on Hansson’s work, Kuparinen ex-

tended the Hansson’s two light method to four-light

photometric stereo using Symmetrical binary weights

in (Kuparinen et al., 2007). Kuparinen also showed

that the minimization based Frankot-Chellappa and

Poisson methods smooth the reconstructed surface re-

moving the inherent high-frequency containing sig-

nal (Kuparinen et al., 2007).

This paper studies the photometric surface recon-

struction of high-frequency varying surfaces. As par-

ticular contributions of the paper, a model of noise for

the gradient images is derived, and an evaluation of

the noise on reconstructed surfaces is presented. In

addition, a denoising method based on Wiener ﬁlter-

571

Kuparinen T., Kyrki V. and Toivanen P. (2008).

MEASUREMENT NOISE IN PHOTOMETRIC STEREO BASED SURFACE RECONSTRUCTION.

In Proceedings of the Third International Conference on Computer Vision Theory and Applications, pages 571-576

DOI: 10.5220/0001083805710576

Copyright

c

SciTePress

τ

σ

Camera

x−axis

Light

Figure 1: Geometry of the imaging setup. (Kuparinen et al.,

2007)

ing is presented and evaluated on both artiﬁcial and

real data. Drbohlav (Drbohlav and Chantler, 2005)

derived also models for additive noise in photomet-

ric stereo, but did not present experimental results on

noise reduction. Finally, the effect of the boundary

conditions of the reconstruction is studied.

2 PHOTOMETRIC STEREO

2.1 The Estimation of Gradient Field

In photometric stereo, the viewing direction is held

constant while the direction of the illumination be-

tween successive images is varied. Image radiance

values in successive views are used to determine the

surface orientation at each image point (Woodham,

1978). For Lambertian surfaces, the reﬂected inten-

sity is independent of the viewing direction. How-

ever, the intensity depends on the direction of the light

source.

Lambert’s Law (Lambert, 2001) represents the

image intensity i at the point (x,y)

i = ρλ(l

T

·n), (1)

where ρ is the surface albedo, λ is the in-

tensity of the light source, n = [n

1

,n

2

,n

3

]

T

=

[p,q,1]

T

√

p

2

+q

2

+1

is the unit normal to the surface and l =

[cos(τ)sin(σ),sin(τ)sin(σ),cos(σ)]

T

is the unit vec-

tor toward the light source. Elements p and q are

surface partial derivatives measured along the x and

y axes, respectively. The angles of illumination, the

tilt τ, and slant σ are illustrated in Fig. 1. Orthogonal

projection and constant illumination over the surface

are assumed in Lambert’s Law.

For a four-light case, if the same slant angle σ is

used for all light sources, and the tilt angles 0

◦

, 180

◦

,

90

◦

and 270

◦

are used for images i

1

through i

4

, re-

spectively, the surface gradient ﬁelds can be derived

using photometric stereo as follows:

p =

2

tan(σ)

i

1

−i

2

(i

1

+ i

2

+ i

3

+ i

4

)

(2)

and

q =

2

tan(σ)

i

3

−i

4

(i

1

+ i

2

+ i

3

+ i

4

)

, (3)

where i

1

and i

2

are intensity vectors of 0

◦

and 180

◦

tilt angles, and i

3

and i

4

intensity vectors of 90

◦

and

270

◦

tilt angles.

2.2 Reconstruction Methods

In order to obtain surface topography, the surface gra-

dients have to be integrated. However, in practice the

surface gradients contain noise, which can be derived

from imaging and other measurement errors.

If the gradient ﬁelds are samples from a larger sur-

face, e.g., textile or paper surface, the boundary con-

ditions of the gradient ﬁelds are of high importance.

Traditionally, the boundary conditions are omitted,

since the object to be reconstructed is extracted from

its surroundings, i.e., the edges of images are black,

and the boundary conditions are not then relevant.

In surface reconstruction, one of three different

boundary conditions is usually applied: Dirichlet,

Neumann, or periodic boundary condition. Dirichlet

boundary condition assumes that the height values of

the boundary are known. Usually, this accounts to set-

ting the height values at the boundary to zero. Neu-

mann boundary conditions assume that the directed

derivatives of the boundary are known, usually that

the directed gradients of the boundary can be set to

zero. In periodic boundary conditions, the surface

is assumed to continue periodically, i.e., the surface

continues identically to the other side, similar to the

discrete Fourier transform (Simchony et al., 1990).

Next, particular solutions to the problem of in-

tegrating the measured gradient ﬁelds are described

and then subsequently experimentally evaluated in the

next section.

Hansson and Johansson presented a two-light PS

method in (Hansson and Johansson, 2000). The imag-

ing was modeled as

s

′

p

(x,y) = s

p

(x,y) ∗PSF + n(x, y), (4)

where s

p

(x,y) is the directional derivative, n(x,y) is

the noise, and s

p

(x,y) ∗PSF represents convolution

of the signal by a point-spread function. They used a

Wiener ﬁlter for the computation of the surface height

from the directed derivatives. Boundary conditions in

Hansson method are twofold: in gradient ﬁeld esti-

mation of p-gradient, the sum of height values are as-

sumed to be zero, whereas in the integration phase,

the periodic boundary conditions are used as the dis-

crete Fourier transform is used. Thus, it is important

to note that the method is not isotropic.

In Symmetric, similar assumptions for boundary

conditions hold as for Hansson’s method.

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

572

The Frankot-Chellappa (Frankot and Chellappa,

1988) algorithm aims to minimize the least square re-

construction error given by

J(Z) =

Z Z

((Z

x

− p)

2

+ (Z

y

−q)

2

)dxdy, (5)

where Z is the surface to be obtained, {Z

x

,Z

y

}the gra-

dient ﬁeld of Z, and {p,q} the given non-integrable

gradient ﬁeld. The gradient ﬁeld of Z can be written

as {Z

x

,Z

y

}= {p,q}+{ε

x

,ε

y

}, where {ε

x

,ε

y

}denotes

the correction gradient ﬁeld, which makes the non-

integrable ﬁeld integrable (Agrawal et al., 2006). In

Frankot-Chellappa method, periodic boundary condi-

tions are applied, and the non-integrable gradient ﬁeld

is projected on to a set of integrable functions using

the Fourier basis functions.

Simchony presented Poisson solver for surface re-

construction in (Simchony et al., 1990). The approach

is similar to Frankot-Chellappa, such that the norm

of the correction gradient ﬁeld, Equation 5, is min-

imized. The second partial derivatives in Poisson

equation are approximated using central differencing

method. Simchony (Simchony et al., 1990) develops

two algorithms for Poisson solver: 1) ﬁnite difference

calculation in Fourier domain with periodic boundary

conditions, and 2) ﬁnite difference calculation in time

domain with Neumann boundary conditions.

2.3 Noise in Photometric Stereo

In practice, the surface gradients contain noise, which

can be derived from imaging and other measurement

errors. So far, the effect of the noise has not been

thoroughly examined. If the imaging conditions are

stable, the noise level of the sensors can be assumed

stationary and measurable, and the information can be

used in order to evaluate the effect of the noise on the

gradient ﬁeld and the ﬁnal reconstructed surface.

Next, we propose a derivation for the inﬂuence of

imaging noise to the gradient ﬁelds. We also propose

a method for the restoration of the degraded gradient

ﬁelds for photometric stereo. We make the assump-

tion that additive white noise, N(0,s

2

), affects all the

captured images: zero mean, Gaussian random varia-

tion with variance s

2

is added to images.

Taking into account the noise, Equation 2 can be

written as

p =

2

tan(σ)

(i

1

+ N

1

(0,s

2

)) −(i

2

+ N

2

(0,s

2

))

∑

4

k=1

(i

k

+ N

k

(0,s

2

))

=

2

tan(σ)

i

1

−i

2

+ N(0, 2s

2

)

∑

4

k=1

i

k

+ N(0, 4s

2

)

. (6)

Note that the variance of the sum of two indepen-

dent normal distributions is the sum of the distribution

variances.

In the denominator of Equation 6, the effect of

the noise can be excluded, when

∑

4

k=1

i

k

>> 0 and

N(0,4s

2

) is close to zero, that is, when the noise vari-

ance is small. This assumption is true for many indus-

trial applications where the imaging conditions can be

kept good and stable. Using the assumption and sep-

arating the gradient and the noise, the gradients are

then

p =

2

tan(σ)

i

1

−i

2

(

∑

4

k=1

i

k

)

+ n, (7)

and

q =

2

tan(σ)

i

3

−i

4

(

∑

4

k=1

i

k

)

+ n, (8)

with the noise n for both gradient ﬁelds

n =

2

tan(σ)

N(0,2s

2

)

(

∑

4

k=1

i

k

)

. (9)

Making a further assumption that the noise vari-

ance is constant over the whole gradient image, the

power spectrum of the noise ﬁeld, N(u,v), can be ap-

proximated as a constant ﬁeld by averaging the sum

of intensity images:

N(u,v) =

2

tan(σ)

2s

2

1

MN

∑

M

x=1

∑

N

y=1

i

a

(x,y)

, (10)

where i

a

=

∑

4

k=1

i

k

, and the image size is M ×N pix-

els. The effect of white noise can be derived in an

identical manner for other combinations of slant and

tilt angles for photometric stereo.

Wiener ﬁltering is an optimal approach for restor-

ing images degraded by noise, when the Signal-to-

Noise Ratio (SNR) and Point Spread Function (PSF)

are correctly known (Gonzalez and Woods, 2002).

We propose that Wiener ﬁltering is applied to gradi-

ent ﬁelds to restore them from the noise. In this work,

PSF is omitted, and the Wiener ﬁlter is utilized in the

Fourier domain as follows

H =

1

1+ SNR(u,v)

−1

, (11)

where SNR(u, v) = |G(u,v)|

2

/|N(u,v)|

2

is the signal-

to-noise ratio in the frequency domain. G(u,v) is the

power spectrum of the gradient ﬁeld from photomet-

ric stereo, and N(u,v) the power spectrum of the noise

calculated using Equation 10.

3 EXPERIMENTS

Surface reconstruction methods Hansson, Symmet-

ric, Frankot-Chellappa and Poisson solver are next

evaluated on gradient ﬁelds of textured surfaces with

high frequency variation and noise. The focus is to

MEASUREMENT NOISE IN PHOTOMETRIC STEREO BASED SURFACE RECONSTRUCTION

573

evaluate the effect of noise level, denoising, and the

boundary conditions for the integration methods in

surface reconstruction. Evaluation measure is the sur-

face reconstruction error. Two versions of the Poisson

solver are evaluated: 1) ﬁnite difference calculation

in time domain with Neumann boundary conditions,

and 2) ﬁnite difference calculation in Fourier domain

with periodic boundary conditions, denoted as Pois-

son, and Poissonf, respectively.

The evaluation is performed for both simulated

data and real application data. Gradient ﬁelds are cal-

culated using two approaches: 1) analytically for the

simulated surface, and 2) applying photometric stereo

to real image data from paper surface for the applica-

tion. Both surfaces contain high-frequency variation.

3.1 Simulated Data

A chirp-surface contains periodic variation of increas-

ing frequency with time, the amplitude of the surface

remaining constant, as shown in Figure 2. In the ex-

periments, the frequency was increased logarithmi-

cally given the following equation for the surface:

S = sin(

2π f

0

ln(k)

(k

t

−1)), (12)

where f

0

is the frequency at t = 0, k is the rate of ex-

ponential increase in frequency, and t is the time. For

a surface S(x,y), time t was deﬁned as t = 0.5(x+ y).

The chirp-function was differentiated analytically in

order to obtain the horizontal and vertical gradient

ﬁelds, which were degraded with two levels of addi-

tive white noise: 5 dB, and 10 dB. The noise levels

were then calculated using Equation 10 from the stan-

dard deviation of gradient ﬁelds. The degraded gra-

dient ﬁelds were Wiener ﬁltered, and surfaces were

integrated from the degraded and Wiener ﬁltered gra-

dient ﬁelds.

Figure 2 and Table 1 present the reconstructed sur-

faces and the reconstruction errors. In Fig. 2, the ef-

fects of boundary conditions are visible. For meth-

ods applying periodic boundary conditions , that is,

Hansson, Symmetric, Frankot-Chellappa, and Pois-

sonf, periodic reconstruction errors near the surface

boundaries are clearly observable. Poisson solver

with Neumann boundary conditions reconstructs the

boundaries without any noticeable error. For Hans-

son, also the horizontal surface reconstruction error

deteriorates the reconstruction result.

The reconstruction errors in Table 1 were cal-

culated from surfaces, which were scaled to the

zero mean and unit variance. Reconstruction er-

rors between the chirp-surface, and reconstructed sur-

faces were analyzed with two measures: 1) root

Chirp-surface Hansson

Symmetric Frankot-Chellappa

Poisson Poissonf

Figure 2: The original surface and the reconstructed sur-

faces from Wiener ﬁltered gradient ﬁelds of simulation data.

mean square error (RMSE), and 2) high-pass RMSE

(HPRMSE). The HPRMSE were calculated from sur-

faces, which were high-pass ﬁltered with cutoff at

10% of resolution. RMSE emphasizes reconstruc-

tion errors on longer wavelengths, HPRMSE errors

on higher frequencies.

In RMSE error, the Poisson solver with Neumann

boundary conditions outperforms clearly the other

methods in all the noise levels. At the same time,

the reconstruction error in longer wavelengths for

Hansson is signiﬁcantly larger. Wiener ﬁltering has

only a minor effect to RMSE in surface reconstruc-

tion. In high-frequency reconstruction error, Poissonf

outperforms the other methods, with Symmetric and

Frankot-Chellappa being comparably close. Hansson

and Poisson exhibit remarkably larger reconstruction

errors in HPRMSE. It is also evident that the proposed

Wiener ﬁltering enhances the reconstruction results in

high frequencies on 5 dB noise level.

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

574

Table 1: Surface reconstruction errors: RMSE, and High-pass RMSE calculated from 10% high pass ﬁltered surfaces. S, S

n

,

and S

wnr

are the reconstruction errors from original gradient ﬁelds, noise added gradient ﬁelds, and Wiener ﬁltered gradient

ﬁelds at 10 dB and 5dB noise levels. The smallest reconstruction errors for each experiment are bolded.

RMSE HPRMSE

S S

n

S

wnr

S

n

S

wnr

S S

n

S

wnr

S

n

S

wnr

Simulated data 10dB 10dB 5dB 5dB 10dB 10dB 5dB 5dB

Gradient ﬁeld 0.140 0.140 0.417 0.409 0.134 0.134 0.395 0.388

Hansson 0.417 0.513 0.513 0.861 0.852 0.136 0.292 0.292 0.622 0.615

Symmetric 0.252 0.260 0.260 0.306 0.306 0.022 0.030 0.030 0.070 0.068

Frankot-Chellappa 0.239 0.246 0.246 0.288 0.288 0.020 0.027 0.027 0.061 0.060

Poisson 0.161 0.168 0.168 0.223 0.223 0.095 0.097 0.097 0.119 0.118

Poissonf 0.239 0.245 0.245 0.286 0.285 0.013 0.019 0.019 0.048 0.047

Application data

Gradient ﬁeld 0.003 0.003 0.010 0.008 0.003 0.003 0.009 0.007

Hansson 0.035 0.035 0.108 0.105 0.023 0.023 0.073 0.069

Symmetric 0.034 0.034 0.106 0.101 0.028 0.028 0.089 0.082

Frankot-Chellappa 0.013 0.013 0.043 0.042 0.006 0.006 0.020 0.018

Poisson 0.014 0.014 0.044 0.042 0.007 0.007 0.022 0.019

Poissonf 0.013 0.013 0.041 0.040 0.005 0.005 0.016 0.015

3.2 Application Example

In the second experiment, paper surfaces were recon-

structed from gradient ﬁelds calculated using photo-

metric stereo. The OTF and SNR functions for Hans-

son, and Symmetric methods were as Hansson pro-

posed in (Hansson and Johansson, 2000). The paper

surface images for photometric stereo were acquired

using a CCD camera with a resolution of 2048 x 2048

pixels with 12 bits per pixel. In the experiments, the

image area was 15 mm x 15 mm.

Because no ground truth of the surface topology

was available, the inﬂuence of white noise in surface

reconstruction was studied using artiﬁcial noise, in

a similar fashion to the simulated data. The recon-

struction results from the two noise degraded gradient

ﬁelds were contrasted to original reconstructed sur-

faces without added noise. Similar to previous ex-

periment, Gaussian white noise was added to images,

and the gradient ﬁelds were calculated from the de-

graded images by photometric stereo. Standard de-

viations of gray-scale values of images were applied

in determination of the 5 dB and 10 dB noise levels.

In the gradient ﬁeld restoration with Wiener ﬁlter, the

white noise ﬁeld was determined using Equation 10.

The reconstruction results are presented in Fig. 3,

and Table 1. In Fig. 3, the smoothing effect of

minimization based Frankot-Chellappa and Poisson

solvers can be observed. Hansson and Symmetrical

sharpen the reconstruction result by Wiener ﬁltering,

which includes SNR and a PSF optimized for paper

surfaces. The reconstruction errors due to noise are

remarkably smaller for minimization based methods,

than for Hansson and Symmetric, see Table 1. Wiener

ﬁltering of gradient ﬁelds can be seen to enhance the

reconstruction results in 5 dB noise level. The effects

of boundary conditions are not clearly visible on this

application data.

Hansson Symmetric

Frankot-Chellappa Poisson

Poissonf

Figure 3: A fragment of reconstructed paper surfaces from

Wiener ﬁltered gradient ﬁelds.

For all the methods, the reconstruction errors in-

creased with the noise level with simulated and real

application data. In 10 dB noise level, the reconstruc-

tion results did not differ remarkably from the noise-

less surfaces. Wiener ﬁltering was able to denoise

the gradient ﬁelds at 5 dB noise level. In RMS er-

ror, the Wiener ﬁltering could not provide signiﬁcant

improvement in reconstruction result. However, the

MEASUREMENT NOISE IN PHOTOMETRIC STEREO BASED SURFACE RECONSTRUCTION

575

high-pass reconstruction error was decreased 1%–2%

with simulated data, and 5%–11% with real applica-

tion data depending on the reconstruction method. In

general, Wiener ﬁltering improved the reconstruction

result for all the methods and data sets.

4 CONCLUSIONS

In this paper, surface reconstruction techniques were

studied in the context of surfaces with high frequency

variation. The effect of imaging noise to gradient

ﬁelds was also evaluated and a denoising approach

was proposed.

The experiments demonstrated that the Wiener ﬁl-

tering based denoising of the gradient ﬁelds is useful

and applicable, if the power spectrum of the noise

is known. Minimization based surface reconstruc-

tion techniques, such as Poisson solver and Frankot-

Chellappa, are more robust against the noise with sim-

ulated and real application data compared to Hans-

son and Symmetric, which perform the integration

in Fourier domain with Wiener ﬁltering. Hansson’s

method applies only one gradient ﬁeld and was found

to be more sensitive to correct Wiener ﬁlter parame-

ters than Symmetric using two gradient ﬁelds. Pois-

son solvers with Neumann and periodic boundary

conditions provided the best results in total and high

frequency scale reconstruction, respectively. Neu-

mann boundary conditions provided correct surface

boundaries, whereas periodic boundary conditions

deteriorated the surface boundary with a non-periodic

pattern.

High frequency surfaces reconstructed using min-

imization based methods are is still not adequate for

some real applications, as they smoothen the surface

making, for example, roughness measurements in-

valid. Frankot-Chellappa and Poisson solvers provide

a robust and parameter free surface reconstruction in

many cases, but the methods have to developed fur-

ther for high frequencycontaining data. Recent devel-

opments in surface reconstruction, such as α surfaces,

M-estimators, regularization and diffusion, seem not

to provide improvement for these problems, as their

main effect is additional adaptive smoothing of the

surface, as noted in (Agrawal et al., 2006). Thus,

one problem in the reconstruction of high frequency

surfaces is in the deﬁnition of the function to be min-

imized. In future work, minimization constraints for

the surface reconstruction methodsneed to be studied.

Another approach would be to model the imaging sys-

tem more accurately, for example, using Wiener ﬁlter-

ing with SNR and PSF. In this work, only the effect of

the SNR was evaluated, and in future works also the

PSF has to be studied.

ACKNOWLEDGEMENTS

The authors gratefully appreciate the provided fund-

ing from European Regional Development Fund

(ERDF), Finnish Funding Agency for Technology

and Innovation (TEKES), Stora Enso, UPM, Metso,

and Future Printing Center (FPC).

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