3D Reconstruction of Interreﬂection-affected Surface Concavities using

Photometric Stereo

Steffen Herbort, Daniel Schugk and Christian Wöhler

Image Analysis Group, TU Dortmund, Otto-Hahn-Straße 4, 44227 Dortmund, Germany

Keywords:

3D surface Reconstruction, Photometric Stereo, Interreﬂections, Surface Concavities.

Abstract:

Image-based reconstruction of 3D shapes is inherently biased under the occurrence of interreﬂections, since the

observed intensity at surface concavities consists of direct and global illumination components. This issue is

commonly not considered in a Photometric Stereo (PS) framework. Under the usual assumption of only direct

reﬂections, this corrupts the normal estimation process in concave regions and thus leads to inaccurate results.

For this reason, global illumination effects need to be considered for the correct reconstruction of surfaces

affected by interreﬂections. While there is ongoing research in the ﬁeld of inverse lighting (i.e. separation

of global and direct illumination components), the interreﬂection aspect remains oftentimes neglected in the

ﬁeld of 3D shape reconstruction. In this study, we present a computationally driven approach for iteratively

solving that problem. Initially, we introduce a photometric stereo approach that roughly reconstructs a surface

with at ﬁrst unknown reﬂectance properties. Then, we show that the initial surface reconstruction result can

be reﬁned iteratively regarding non-distant light sources and, especially, interreﬂections. The beneﬁt for the

reconstruction accuracy is evaluated on real Lambertian surfaces using laser range scanner data as ground

truth.

1 INTRODUCTION

The problem of image-based shape reconstruction has

been examined by Horn (Horn, 1970) using a single

image and regularization constraints for “Shape from

Shading (SfS)". The Photometric Stereo (PS) princi-

ple evolved later on (Woodham, 1980), favorably re-

moving the constraints, but needing at least three in-

tensity measurements per pixel obtained under vary-

ing illumination directions. Both, the SfS and the PS

setup, assume distant light sources with known illu-

mination direction and intensity, which is nowadays

termed “calibrated Photometric Stereo". Addition-

ally, the observed surfaces are assumed to be strictly

Lambertian (Lambert, 1760).

The case of uncalibrated PS has been examined

extensively, starting with Hayakawa’s seminal work

(Hayakawa, 1994), where the problem of an ambigu-

ity in the determinable light source position from PS

image data becomes exposed. That “Generalized Bas-

Relief (GBR) ambiguity" (Belhumeur et al., 1999)

was subject to extensive research and has been solved

using e.g. the generic viewpoint constraint (Freeman,

1994; Yuille et al., 2000), minimization of the surface

albedo distribution entropy (Alldrin et al., 2007), by

modeling the illumination conditions using spherical

harmonics (Basri et al., 2007), or with special relative

light source arrangements (Zhou and Tan, 2010).

In parallel, non-Lambertian surfaces have been

approached by different authors. (Ikeuchi, 1981)

assume purely specular surfaces, (Nayar et al.,

1988; Nayar et al., 1990a) assume hybrid sur-

faces with Lambertian and superimposed specu-

lar components. For these methods, the im-

age formation and object reﬂection properties be-

come relevant. Well-known reﬂectance function

models are e.g. Lambert (Lambert, 1760), Oren-

Nayar (Oren and Nayar, 1994), Blinn (Blinn,

1977), Phong (Phong, 1975), Lafortune (Lafortune

et al., 1997), Beckmann-Spizzichino (Beckmann and

Spizzichino, 1987), Torrance-Sparrow (Torrance and

Sparrow, 1967), Ward (Ward, 1992), Ward-Dür-

Geisler-Moroder (Geisler-Moroder and Dür, 2010).

Since PS works robustly on Lambertian surfaces,

there has been research regarding the separation of

diffuse and specular components, which then al-

lows performing PS on the diffuse component only.

These use e.g. polarization (Wolff, 1989; Wolff and

Boult, 1991), color (Tan and Ikeuchi, 2005; Thomas

and Sugimoto, 2010), or both (Nayar et al., 1997).

208

Herbort S., Schugk D. and Wöhler C..

3D Reconstruction of Interreﬂection-affected Surface Concavities using Photometric Stereo.

DOI: 10.5220/0004213702080214

In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2013), pages 208-214

ISBN: 978-989-8565-48-8

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

The model-based reﬂectance description has been re-

laxed by BRDF

-independent approaches like (Zick-

ler et al., 2002; Alldrin et al., 2008).

A new formulation of the problem using a projec-

tive framework has been published by (Tan and Zick-

ler, 2009; Tankus et al., 2005). Further variations of

the PS framework regard e.g. exploiting the effects of

non-distant light sources (Iwahori et al., 1990; Clark,

1992).

PS itself determines only pixelwise surface nor-

mals or gradients, while absolute depth determination

requires a postprocessing step, which was elaborately

examined by (Agrawal et al., 2006).

Due to the common assumption of single sur-

face reﬂections (direct illumination) and disregarding

higher orders (interreﬂections, a subset of global il-

lumination), it is not possible to infer the underly-

ing surface normal from the observed intensity cor-

rectly. Fig. 1 depicts an image separated into direct

and global illumination components and illustrates

the intensity aggregation in concavities.

100

120

140

160

180

(a) camera image

100

120

140

160

180

(b) direct component

100

120

140

160

180

Figure 1: Measured illumination components of a camera

image using the method by (Seitz et al., 2005) for (b) and

a variant of that method with an inverse illumination pat-

tern for (c). Note that all images have the same grayvalue

scaling.

Systems for synthetically computing global illu-

mination exist in a large variety. One of the open

source ray tracers that additionally employs physi-

cally plausible BRDFs, which is crucial for correct

Bidirectional Reﬂectance Distribution Function

real world object representation, is PBRT

(Pharr and

Humphreys, 2010).

While extensive research has been carried out re-

garding (non-) Lambertian surfaces and uncalibrated

PS, there is hardly any research regarding the image-

based reconstruction of 3D shapes in the presence of

interreﬂections. Basic thoughts have been published

by (Nayar et al., 1990b) who iteratively reﬁne concave

v-shapes. The case of inversely computing global

illumination with (at most) second order reﬂections

along its path has been examined by (Yu et al., 1999),

who use their data for realistic scene modeling and

relighting. The effect of interreﬂections and the re-

sulting inaccuracies on structured light laser scanning

proﬁles have been discussed by (Gupta et al., 2011)

and (Couture et al., 2011), but photometric aspects

remain untreated therein.

The work by (Gu et al., 2011) stresses the inﬂu-

ence of interreﬂections for PS. Their method is based

on multiplexed illumination (Nayar et al., 2006) for

the separation of global and direct illumination in the

image acquisition stage. They apply high frequency

patterns under different incident directions for esti-

mating the two scene illumination components. The

direct component is then used to estimate shapes with

strongly increased accuracy using PS.

To the authors’ best knowledge, there is no publi-

cation that actually deals with a method for the recon-

struction of shapes with interreﬂection-affected con-

cavities under illumination with simple point light

sources (LEDs).

2 CONSIDERATION OF

INTERREFLECTIONS

We propose a novel method that accounts for inter-

reﬂections in a fully calibrated photometric stereo en-

vironment. The approach as such imposes no restric-

tions upon their reﬂective behavior and thus allows

arbitrary and possibly even measured BRDFs. Fur-

thermore, the light sources are not assumed to be dis-

tant, which is not a prerequisite for our algorithm

to work but it improves the reconstruction accuracy.

Calibration of the (point) light source positions and

intensities is achieved using the method described by

(Lenoch et al., 2012). Absolute surface depth data

is computed using Agrawal’s M-estimator approach

(Agrawal et al., 2006). In the following, the con-

catenation of PS and Agrawal’s method is denoted

“PS+Z".

Our key to the consideration of interreﬂections lies

Physically Based Ray-Tracer v2, pbrt.org

3DReconstructionofInterreflection-affectedSurfaceConcavitiesusingPhotometricStereo

209

PS+Z w/ Lambert and

global light directions

images

scene

configuration

PS+Z w/ Lambert and

local light directions

remove

interreflections

PS+Z w/

local light directions

final surface

PS+Z w/ BRDF and

local light directions

estimate BRDF

Figure 2: Algorithm overview. “PS+Z" denotes photomet-

ric normal estimation and followed by depth integration.

Note that each group of components (green, orange, blue)

can possibly be iterated.

in an iterative scheme: (1) Our method initially recon-

structs the surface based on distant light sources and

Lambertian reﬂectance. (2) That result becomes re-

ﬁned assuming non-distant light sources. The result-

ing surface is afterwards used (3) to obtain the param-

eters of a parametric BRDF, which additionally re-

ﬁnes the surface in another reconstruction step. Lastly

(4), the consideration of interreﬂections is incorpo-

rated by determination of the amount of global illu-

mination with a physically plausible rendering sys-

tem. This allows us to remove a (signiﬁcant) amount

of interreﬂection-based radiance from the captured

scene image data and thus to obtain a reﬁned recon-

struction result. Note that it may be beneﬁcial to re-

peat some of the stages described above in order to

ensure their convergence. Fig. 2 gives an overview of

these elements.

We demonstrate the potential of our method on

real data for a Lambertian and a non-Lambertian sur-

face. The validation of the correctness of these results

is achieved on the one hand by comparison with the

surface data measured by an industry-standard laser

scanner and on the other hand based on synthetically

rendered images with a known underlying surface.

In contrast to related publications, we model point

light sources and are thus not required to illuminate

the scene with specialized projectors like (Gu et al.,

2011); furthermore, there is no shape restriction like

the v-shapes examined by (Nayar et al., 1990b).

3 ALGORITHM DETAILS

In stage 1 (yellow), the surface normals are computed

for each pixel (u, v) based on the standard PS equation

for Lambertian surfaces,

obs

(u,v) = I

ρ(u,v) L ~n(u,v), (1)

where u and v denote the pixel indices,

obs

(u,v) ∈

k×1

denotes the vector of k observed intensity sam-

ples under i = 1...k varying (distant) normalized il-

lumination directions

with |

= 1. The light

source relative radiance is referred to as I

and the

light source directions are aligned row-wise in L =

;...]. The surface normals are

~n = [n

]

with |~n|

= 1 and ρ denotes the sur-

face albedo. The surface normal is thus determined in

the least-mean-square (LMS) sense using

~n(u,v) =

ρ(u,v)

−1

obs

(u,v) (2)

with a pseudo-inverse L

−1

. Note, that locally varying

albedos can be obtained when solving for~n and ρ with

known irradiance I

∗

= L

−1

obs

(u,v), I

ρ = |~n

∗

, ~n =

∗

|~n

∗

(3)

The obtained surface normal ﬁeld is then integrated to

absolute depth values using the M-estimator approach

described by (Agrawal et al., 2006). For the trans-

formation into a metric depth space, the algorithm is

given the approximate object distance from the cam-

era, which sufﬁces for the determination of pixel ex-

tents and thus metric depth scaling.

In stage 2 (green), the light sources are regarded as

non-distant. This alters the image intensity equation

such that

obs

(u,v) =

(u,v) r

(u,v)

L(u,v) ~n. (4)

Note that we assume locally varying illumination di-

rections

(u,v) for each pixel (u, v) and each light

source i = 1...k, which is due to the scene being il-

luminated by non-distant point light sources. Addi-

tionally, each pixel is assigned the light travel dis-

tance from the light source to the object (r

) and

from the object to the camera sensor (r

). Even on

this small-scaled framework (object height ≈ 50mm,

mean(r

) ≈ 300mm and mean(r

) ≈ 310mm), this

shows a signiﬁcant effect. Up to this point, the sur-

face has been assumed to be perfectly Lambertian and

VISAPP2013-InternationalConferenceonComputerVisionTheoryandApplications

210

free of global illumination.

In stage 3 (orange), the BRDF of the underlying ma-

terial is estimated globally based on the extracted sur-

face from stage 2 and the obtained images. This

is achieved by minimizing the RMSE of the inten-

sity data I = I(u,v) and the rendered intensities R =

P,u, v) of the surface with respect to the parameter

P of the chosen BRDF:

opt

= argmin

U ·V

∑

u=1

∑

v=1

[I − R]

. (5)

For Lambertian surfaces, the only parameter is the

surface albedo ρ, which can be computed by deter-

mination of I

using (Lenoch et al., 2012) and thus

having white balance data available.

In stage 4 (blue), the interreﬂection component is (it-

eratively) compensated before starting the PS+Z algo-

rithm: Two separate images of the last stage’s PS+Z

surface result are rendered under (1) direct R

d,i

and (2)

global R

g,i

illumination for each light source i = 1...k.

Afterwards, the global component is determined as

the difference between R

g,i

and R

d,i

, which is used to

compensate these in the real camera images I

new,i

= I

− (R

g,i

− R

d,i

)

| {z }

interreﬂections

i = 1...k (6)

As a last step for this stage, the PS+Z algorithm is

started on the new image data basis I

new,i

4 EXPERIMENTS AND RESULTS

Figure 3 gives an overview of the setup used for data

acquisition. Due to using the same camera for im-

age acquisition and ground truth depth measurement

(structured light laser scanning system), it is possible

to capture pixel synchronous depth and image data,

which facilitates the evaluation process signiﬁcantly.

The description and discussion of our experiments is

structured as follows: Initially, we analyze the pro-

gressing change in PS+Z depth with increasing al-

gorithm iterations (Sec. 4.1). Afterwards (Sec. 4.2),

we show the beneﬁcial effect of our approach quali-

tatively based on an analysis of the reﬂectance maps

with global illumination.

Sec. 4.3 then shows the comparison of the PS+Z

depth result with independently measured scanner

depth data z

for a region of the surface that is es-

pecially affected by interreﬂections. To conclude the

evaluation, the aspect of algorithm convergence is dis-

cussed in Sec. 4.4 along with suitable criteria that ter-

minate the execution at a reasonable state.

Figure 3: Overview of the experimental setup. The object

has a height of about 50 mm, the distance between object

and camera (2) / object and illumination (3) amounts to ap-

proximately 250 mm. Note that the laser range scanner (1)

is used for validation only.

4.1 Progressing Change in

Reconstructed Depth

Fig. 4 shows the differences between the naive PS+Z

depth z

PS,1

(no interreﬂection compensation) and its

iterations z

PS,2...5

using image data with compensated

interreﬂections. It becomes clear that the compen-

sation of the global (interreﬂection) component pro-

duces deeper concavities compared to the uncompen-

sated case. The results show that less deep concav-

(a) z

PS,2

− z

PS,1

(b) z

PS,3

− z

PS,1

PS,4

− z

PS,1

(d) z

PS,5

− z

PS,1

Figure 4: Difference between the naive PS depth z

PS,1

(no

interreﬂection compensation) and the iterations using im-

age data with compensated interreﬂections z

PS,2...5

. Scale:

−0.5mm (dark blue) to 2.5 mm (red).

ities on the one hand (e.g. neck and ear region) ex-

hibit only a change in depth over the early iterations

3DReconstructionofInterreflection-affectedSurfaceConcavitiesusingPhotometricStereo

211

of the algorithm, but remain static on the late itera-

tions. Deep concavities on the other hand progress in

their depth over several iterations. This already gives

a hint that the algorithm converges suitably, since re-

gions with shallow and deeper concavities converge

independently.

4.2 Qualitative Evaluation

Fig. 5 shows reﬂectance maps from different stages

of the algorithm. Note that at iteration 5 (Fig. 5(c)),

the object surface appears more similar to the physi-

cal image (Fig. 5(a)), especially in the interreﬂection-

affected regions. For a human observer, it is obvious

(a) image

(b) stage 4, iteration 1 rendering re-

sult (global illumination)

Figure 5: Rendered PS+Z results. Note the appearance dif-

ference at surface concavities.

that the underlying surface resembles the one of the

physical image more closely due to the perceived in-

tensities and due to the plausibility of where and how

shadows are cast. In summary, the qualitative assess-

ment result of the surface is very favorable, due to ﬁne

surface details being visible and surface concavities

being shaped as expected from the real world object

image.

4.3 Quantitative Evaluation

Fig. 6 shows a comparison of the reconstructed depth

PS,n

and quantitatively measured depth data obtained

using a laser range scanner z

for the iterations

n = 1...8. The registration of both (already pixelsyn-

chronous) depth proﬁles has been achieved by align-

ing the absolute depth of a convex surface part in

the “hand"-region. This ensures that the PS+Z depth

PS,n

is unaffected by interreﬂections in that area, and

thus allows unbiased registration. With naive photo-

metric reconstruction (no compensation of interreﬂec-

tions, z

PS,1

, Fig. 6(a)), there are very strong differ-

ences visible between the reconstructed surface z

PS,1

and the ground truth scanner depth z

, which are in

the range of some mm. The differences initially reach

(a) z

− z

PS,1

(b) z

− z

PS,2

− z

PS,3

(d) z

− z

PS,4

(e) z

− z

PS,5

(f) z

− z

PS,6

(g) z

− z

PS,7

(h) z

− z

PS,8

(i)

scale

Figure 6: Ground truth validation. Scale: −0.5 mm (dark

blue) to 2.5mm (red).

more than 2 mm (Fig. 6(a)), which is unacceptable in

terms of measurement accuracy for e.g. industrial ap-

plications. Tn the course of the iterations, the differ-

ences decrease quickly and already after the second

iteration with interreﬂection-compensated image data

PS,3

, Fig. 6(c)), the error has decreased over a wide

range of the concavity.

4.4 Convergence Criterion

Although it seems to be of less importance when to

stop iterating due to e.g. shallow concavities remain-

ing unchanged while deeper concavities still deepen,

it makes sense to include another criterion for robust-

ness and for saving computational time. We found

it particularly useful to examine the mean absolute

VISAPP2013-InternationalConferenceonComputerVisionTheoryandApplications

212

1 2 3 4 5 6 7

0.1

0.2

0.3

0.4

0.5

0.6

iteration

mean absolute depth difference

Figure 7: Average change in depth ∆z

= mean(|z

PS,n

−

PS,1

|) per iteration in the “bag"-section.

change in depth of the photometrically reconstructed

depth data of iteration n (i.e. z

PS,n

) compared to the

initial PS+Z result without compensation of inter-

reﬂections (i.e. z

PS,1

) such that ∆z

= mean(|z

PS,n

−

PS,1

|). Once that change is lower than a certain

threshold (here: ∆z

< 0.02mm), the computation is

regarded as complete. In our case, convergence is

then reached after the 4

iteration.

5 SUMMARY

AND CONCLUSIONS

We have presented a novel, iterative approach for

dealing with interreﬂections in the context of PS

based shape reconstruction. The algorithm initially

reconstructs the depth proﬁle without consideration of

interreﬂections using a PS approach and subsequent

absolute depth estimation based on the PS gradient

ﬁelds. Since that proﬁle mainly bares errors in re-

gions affected by excessive light emission due to in-

terreﬂections, we have introduced an iterative scheme

that computes the intensity of the global illumination

at surface concavities and then removed these from

the physical image data before using that improved

data for PS and absolute depth reconstruction.

The evaluation showed a qualitatively and quan-

titatively beneﬁcial effect for the extracted surface.

The results show furthermore, that convergence is

reached already after ≈ 4 iterations (for the given sur-

face). Furthermore, a criterion (mean absolute depth

change) has been presented that allows to determine

when the computation has ﬁnished.

In future work, there is mainly the need to perform

experiments with non Lambertian surfaces, which are

signiﬁcantly more common in the real world than the

applied Lambertian surfaces. Conceptually, this re-

quires the algorithm to handle the BRDF estimation

and absolute depth estimation step with greater ro-

bustness, but the basic functionality and algorithm

principle will remain unchanged.

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