Simultaneous Surface Segmentation and BRDF Estimation

via Bayesian Methods

Malte Lenoch, Thorsten Wilhelm and Christian W

¨

ohler

Image Analysis Group, TU Dortmund University, Otto-Hahn-Strasse 4, Dortmund, Germany

Keywords:

Image Segmentation, BRDF Estimation, Reversible Jump Markov Chain Monte Carlo.

Abstract:

We present a novel procedure that achieves segmentation of an arbitrary surface relying on the maximum

a-posteriori estimation of its reﬂectance parameters. The number of surfaces segments is computed by the

algorithm without user intervention. We employ Markov Chain Monte Carlo algorithms to compute the prob-

ability distributions associated with the model parameters of a Blinn reﬂectance model based on the input

images. The fact that parameters are treated as probability distributions enables us to directly draw addi-

tional information about the certainty of the estimation from the results of both parameters and segmentation

borders. Reversible jump MCMC allows us to include an unspeciﬁed number of change points in the computa-

tion, such that the algorithm explores model and parameter space at the same time and derives a segmentation

of the surface from the input data. To accomplish this, we extend the existing concept of change points to

two dimensions introducing a number of necessary new regulations and properties. The performance of the

segmentation and reﬂectance estimation is evaluated on a synthetic and a real-world dataset.

1 INTRODUCTION

Many man-made objects or natural materials exhibit

a reﬂectance that is non-uniform across the surface.

The estimation of a spatially varying bi-directional

reﬂectance distribution function (SVBRDF) is a well-

approached research task in computer vision as well

as in photogrammetric backgrounds. Additionally, it

is of interest in certain tasks, to determine the surface

areas that consist of the same material and are there-

fore associated with identical reﬂectance properties.

Our goal is to segment the surface of an arbitrary

object into meaningful regions based on the reﬂection

properties of those particular regions. We apply meth-

ods from Bayesian statistics to compute the posterior

of the acquired BRDF parameters and the patch bor-

ders based on the given input data and the speciﬁed

prior distributions, reﬂecting knowledge which is at

hand prior to the experiment. Since we handle every

parameter as a stochastic distribution, we know the

uncertainty of the estimate as well, instead of only

handling a point estimate as do common gradient de-

scent procedures. Furthermore, conﬁdence analysis

in classical statistics is inferred from asymptotic the-

ory assuming large datasets, which simply does not

always hold in practice, e.g. (King et al., 2010). The

combination of Bayesian statistics and reﬂectance es-

timation has had little attention in recent years.

In our approach, we focus on patchwise BRDF ac-

quisition that ensures sufﬁcient data even when not

many input images are available. The estimation of

pixelwise BRDF parameters requires a large amount

of data to be available at every surface point to reli-

ably ﬁt the desired BRDF model, e.g. (Lensch et al.,

2003; Goldman et al., 2005). In fact, a dense angular

representation is necessary to capture narrow specular

lobes and prevent aliasing. Due to shadowing effects,

a complex surface structure can reduce the available

intensity information per pixel to be smaller than the

number of captured images. We assume isotropic re-

ﬂectance and neglect the inﬂuence of subsurface scat-

tering or translucency.

2 RELATED WORK

The scope of BRDF reproduction in previous work

ranges from the acquisition of pure albedo informa-

tion to the reconstruction of per-pixel sets of param-

eters of complex BRDF models, e.g. Weyrich et al.

(2008); Aittala et al. (2013).

A common approach to estimate the SVBRDF of

an unknown material is to compute mixture maps as-

Lenoch, M., Wilhelm, T. and Wöhler, C.

Simultaneous Surface Segmentation and BRDF Estimation via Bayesian Methods.

DOI: 10.5220/0005722600390048

In Proceedings of the 11th Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2016) - Volume 4: VISAPP, pages 39-48

ISBN: 978-989-758-175-5

Copyright

c

2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

39

suming weighted sums of base materials to construct

an approximation of the measured intensity Gold-

man et al. (2005); Alldrin et al. (2008); Lenoch et al.

(2014). These base materials are either taken from

densely acquired datasets, e.g. the MERL database

Matusik et al. (2003), or composed of few analyti-

cal BRDF models. In the latter case, the set θ

θ

θ of pa-

rameters of the model function has to be estimated.

Goldman et al. (2005) assume two base materials per

surface and represent those with a mixture of two

Ward-BRDFs. Similarly, Lensch et al. (2003) cap-

ture reﬂectance samples that yield salient information

and apply mixtures of up to ﬁve Lafortune BRDFs to

clustered lumitexels. This yields more possibilities in

the model space, but requires the estimation of multi-

ple sets of parameters and a mixture map at the same

time.

The combination of BRDF estimation and prob-

abilistic methods is very rare in literature. Aittala

et al. (2013) represent reﬂectance that may vary along

the surface by a Gaussian mixture model in the fre-

quency domain. The initial solution is improved

with a maximum a-posteriori estimation. Louw and

Nicolls (2010) use a Markov random ﬁeld to imitate

the reﬂectance behavior of a surface. The MRF is es-

timated with a Population Markov Chain Monte Carlo

algorithm Laskey and Myers (2003).

3 PROPOSED PROCEDURE

The BRDF ﬁtting procedure requires the illumina-

tion geometry to be known. This can be ensured

easily, if 3D information about the surface is avail-

able. However, the approach works independent of

the source of the 3D data and could be integrated in

a self-consistent algorithm similar e.g. to Herbort and

W

¨

ohler (2012), that has no prior knowledge about the

surface shape.

3.1 Bayesian BRDF Modeling

We choose the BRDF model according to Blinn

(1977) in the physically plausible modiﬁcation by

Giesen (2009) combined with a Lambertian term

(Lambert, 1760) to account for diffuse reﬂection.

We assume that this relatively simple model sufﬁces,

since the unknown surface will be segmented during

the procedure and complex reﬂectance can be mod-

eled by small segments of different parameters. Note

that we do not require physical materials to coincide

with segmented patches, since even within the same

material subtle changes can require changing BRDF

parameters to achieve accurate modeling.

In terms of Bayesian statistics, the BRDF model

is the mean of a normal distributed likelihood. We

assume the residuals of the BRDF model to follow a

Gaussian distribution with an unknown variance σ

2

.

Since σ

2

is unknown, we estimate it as a part of the

procedure and are thus able to quantify the deviation

of the model from the observed data which based on

the BRDF model formulation by Giesen (2009) leads

to

y

y

y ∼ N (µ

µ

µ, σ

2

) (1)

µ

µ

µ =

I

0

r

2

k

d

π

+ k

s

(γ + 2)(γ + 4)

8π(2

−

γ

2

+ γ)

·

h

h · n

i

γ

h

l · n

i

(2)

1 = k

d

+ k

s

. (3)

Here

h

l · n

i

= cos∠(l

l

l, n

n

n) denotes the vector prod-

uct between the normalized vectors l

l

l and n

n

n, with sur-

face normal n

n

n, direction l

l

l of incident light, viewing

direction v

v

v and half-way vector h

h

h =

l

l

l+v

v

v

||l

l

l+v

v

v||

2

.

The chosen model thus depends on four parame-

ters. Light source intensity I

0

, diffuse and specular

weight k

d

(since k

s

= 1 − k

d

), cosine lobe exponent γ

and variance σ

2

. Each of these parameters requires

a prior to deﬁne the distribution of the parameter be-

fore any data was seen. We assume the priors to be

normal distributions; exemplarily for the light source

intensity

p(I

0

) ∼ N (µ

I

0

, σ

2

I

0

) (4)

The prior yields the possibility to involve certain

knowledge about the possible range of the parameters

into the estimation process. This fact can be exploited

if, for example, a coarse estimate of the light source

intensity exists. Yet, this estimate might be prone to

simpliﬁcation errors. If there is no prior knowledge

available or only at a high level of uncertainty, a large

variance is used to reduce the inﬂuence of the prior,

since its probability is equal in almost the entire pa-

rameter space. In a Bayesian framework this is termed

an uninformative prior. A uniform distribution would

cancel out the effect of the prior completely.

Concluding from Bayes’ law we can now compute

the probability distribution of the set θ

θ

θ of model pa-

rameters for the input intensity data Y

Y

Y and the a-priori

probability p(θ

θ

θ).

p(θ

θ

θ|Y

Y

Y ) =

p(Y

Y

Y |θ

θ

θ) · p(θ

θ

θ)

p(Y

Y

Y )

(5)

The normalization factor p(Y

Y

Y ) is difﬁcult to esti-

mate and can be omitted, since it is constant as long

as the input data does not change. Therefore the

posterior-density can be expressed as

p(θ

θ

θ|Y

Y

Y ) ∝ p(Y

Y

Y |θ

θ

θ) · p(θ

θ

θ) (6)

All information required to determine the proba-

bility distribution of the set of model parameters for

VISAPP 2016 - International Conference on Computer Vision Theory and Applications

40

the given input data is the likelihood and the prior,

which is both at hand.

3.2 Markov Chain Monte Carlo

Actually, the chosen priors can be based on additional

available information or be completely uninformative,

in any case the real distribution of θ

θ

θ remains unknown

and subject to the estimation. Markov chain Monte

Carlo (MCMC) methods as e.g. introduced by Nt-

zoufras (2011) can be employed to sample the model

parameters’ unknown distribution. To achieve this, a

Markov chain is constructed whose stationary distri-

bution corresponds to the desired unknown distribu-

tion. Thus, after a converging time, the state of the

chain can be used sample the unknown posterior.

More speciﬁcally, we use the Metropolis-Hastings

(MH) algorithm introduced by Metropolis et al.

(1953) and Hastings (1970). Accordingly, we con-

struct a Markov chain which proposes in each step

new values of the chain for the respective BRDF pa-

rameters. The likelihood of the values is assessed with

the newly generated values and if the likelihood of the

parameters increases they are probably accepted as a

new value of the chain, if not they are probably dis-

carded. The term probably refers to the acceptance

probability α that is compared to a random number,

such that there is a possibility to accept a new value

even if the likelihood decreased and vice-versa. Sim-

ilar to the priors, starting values θ

θ

θ

(0)

have to be sup-

plied by the user and can bring in additional informa-

tion.

The number T of iterations of the algorithm

is taken to be ﬁxed. A proposal distribution

p(θ

θ

θ

(t)

|θ

θ

θ

(t−1)

) and the acceptance probability α are

necessary for the generation process; given for I

0

I

(0)

0

initial value,

p(I

(t)

0

|I

(t−1)

0

) ∼ N (I

(t−1)

0

, (σ

p

I

0

)

2

) (7)

α

I

0

= min(1, A)

A =

p(y

y

y|I

(t)

0

, k

d

, γ, σ)p(I

(t)

0

)p(k

d

)p(γ)p(σ)

p(y

y

y|I

(t−1)

0

, k

d

, γ, σ)p(I

(t−1)

0

)p(k

d

)p(γ)p(σ)

!

= min

1,

p(y

y

y|I

(t)

0

, k

d

, γ, σ)p(I

(t)

0

)

p(y

y

y|I

(t−1)

0

, k

d

, γ, σ)p(I

(t−1)

0

)

!

(8)

It is assumed here that there is no interdependence

between the parameters, such that we can apply a

componentwise MH and each parameter is proposed

and accepted individually (Ntzoufras, 2011). This

yields the general advantage, that simple proposal dis-

tributions can be applied, whereas the construction of

a combined proposal distribution can require careful

design. Still, a combined proposal distribution can

yield a faster converging chain.

After a user-deﬁned burn-in phase the chain con-

verges to the actual unknown posterior distribution if

according to Gilks et al. (1996, p.46) the following

properties are met:

• Irreducibility

• Positive Recurrency

• Aperiodicity

At the state of convergence, statistics can be com-

puted from the drawn samples. Thusly, the adap-

tion of the BRDF model to unknown data immedi-

ately yields a conﬁdence region for each parameter

and with σ

σ

σ

2

a measure of the accuracy of the model.

3.3 Reversible Jump Markov Chain

Monte Carlo

So far we can adapt a BRDF model to unknown in-

tensity data, but the segmentation of the surface has

not yet been regarded. Image segmentation is usually

performed in a generative way (e.g. using Gaussian

Mixture Models). We use instead a discriminative ap-

proach and directly infer the boarders of the regions

from the data. Effectively, a change in surface struc-

ture or material enforces a change of the model pa-

rameters to reproduce the received impression accu-

rately.

So-called Reversible Jump Markov chain Monte

Carlo (RJMCMC) methods as described by Green

(1995) are able to explore the parameter and model

space simultaneously. That means in terms of statis-

tics, that there is a certain probability in each itera-

tion to create, delete or modify a change point τ that

separates two sets of model parameters. Using these

change points it is possible to divide unknown data

into meaningful segments.

We have extended the concept of change points to

the two-dimensional application of image segmenta-

tion, with the following new properties:

• Change points are possible both in u and v direc-

tion (referring to image coordinates).

• A change point becomes a vector of change

points, since it will affect the entire row/column

of image pixels.

• Each pixel coordinate of the change point vector

can be moved individually.

• Surface segments are constructed from the regions

where the same change points in u and v direction

coincide.

Simultaneous Surface Segmentation and BRDF Estimation via Bayesian Methods

41

We allow change points to lie outside of the image

coordinate frame, since this is a helpful property when

segment borders are not covering a complete row or

column.

A possible maximum of change points in each di-

rection has to be deﬁned by the user. Based on these

maxima, we construct an indicator matrix that assigns

the actual regions based on the combinations of the

change points. For example, τ

max

(u) = τ

max

(v) = 3

yield the following assignment:

(1, 1) (1, 2) (1, 3)

(2, 1) (2, 2) (2, 3)

(3, 1) (3, 2) (3, 3)

ˆ=

1 4 7

2 5 8

3 6 9

(9)

Each change point itself has a statistical distribu-

tion, such that conﬁdence intervals can be computed

directly from the data of the Markov chain. Com-

pared to other segmentation procedures this is a note-

worthy advantage, since the algorithm gives a mea-

sure for the quality of the segmentation. The change

points are assumed to follow a uniform distribution,

s.th. τ ∼ U(−1, imagesize+1), since they are allowed

to lie outside the image plane

3.3.1 A Change in Dimension

The possibility to create or delete change points cre-

ates needs of a procedure to split and merge parame-

ter sets θ

θ

θ

i

. A new region could be initialized with the

parameters of a randomly chosen neighboring region,

yet this makes it impossible to revert the step without

storing a large amount of data. A better solution is to

have a bijective transformation M

M

M which is indepen-

dent of the actual number of change points.

A random number r is drawn from the distribu-

tion q(r) = N (0, 1) and symmetrically added and

subtracted from the existing change points. In-

spired by Pascal’s triangle the coefﬁcients P

n

(k) =

(−1)

k

n

k

, k = 0, .., n of the last column of M

M

M create a

unique transformation when jumping from dimension

n−1 to n. For the step to n = 3 and former parameters

θ

θ

θ

(t−1)

1

and θ

θ

θ

(t−1)

2

this means

θ

θ

θ

(t)

1

θ

θ

θ

(t)

2

θ

θ

θ

(t)

3

=

1 0 −1

0.5 0.5 2

0 1 −1

θ

θ

θ

(t−1)

1

θ

θ

θ

(t−1)

2

r

. (10)

Note that the transformation has to be applied

componentwise to θ

θ

θ

i

, thus the random number r can

be scaled to the dimension of the parameter. The in-

verse of M

M

M is deﬁned and the Markov chain can be

reset to its state in the lower dimension. This can be

helpful to reduce computational time, since the chain

can continue from its former state, possibly near con-

vergence.

The proposed change in dimension is accepted

with an acceptance probability α similar to the propo-

sition of new parameters. The dimension is repre-

sented by a model number m that is determined from

the indicator matrix. For the sake of readability the

iteration index t will be omitted in what follows and

the proposed value is indicating by an apostrophe.

α(θ

θ

θ, θ

θ

θ

0

) = min(1, B),

B =

p(θ

θ

θ

0

, m

0

|X

X

X)p(m|m

0

)q(r

0

)

p(θ

θ

θ, m|X

X

X)p(m

0

|m)q(r)

∂θ

θ

θ

0

∂(θ

θ

θ, r

r

r)

(11)

p(m

0

|m) = p(m|m

0

) is the probability to change from

one model to the other, e.g. add or delete a change

point respectively and in our approach assumed to be

equal.

∂θ

θ

θ

0

∂(θ

θ

θ,r

r

r)

is the Jacobian of the partial deriva-

tives, necessary since what we do is basically a co-

ordinate transformation. Yet, the Jacobian is only a

scaling factor, which may be omitted and thus B sim-

pliﬁes to

B =

p(θ

θ

θ

0

, m

0

|X

X

X)q(r

0

)

p(θ

θ

θ, m|X

X

X)q(r)

. (12)

The acceptance of the inverse move is simply cal-

culated as

α = min

1,

1

B

. (13)

After the predeﬁned number of T iterations, it has

to be decided which combination of change points is

the most probable. The integration in the model space

is turned into a summation in the discrete case and the

model with the most iterations is accepted as the best

solution, e.g. (King et al., 2010). However, compar-

ing the ”time spent“ in different models yields infor-

mation about the certainty of the solution.

3.4 Synthetic Data

The algorithm is evaluated ﬁrst on a synthetic dataset.

Figure 1(a) shows one of the input images and Fig-

ure 1(b) the resulting reﬂectance map. The similar-

ity to the result expected by the human viewer under-

lines the success of the algorithm in determining the

reﬂectance parameters of the surface and segmenting

its regions. The parameters of the algorithm, (σ

p

i

)

2

of

the proposal distribution, the prior parameters and the

starting values of the Markov chain θ

(0)

i

are stated in

Table 1.

The histogram of visited models is depicted in

Figure 2. The histogram counts indicate strongly that

model m = 8 is the best choice. That equals one

VISAPP 2016 - International Conference on Computer Vision Theory and Applications

42

(a) (b) (c) (d)

Figure 1: (a) Exemplary input image of synthetic dataset and (b) reﬂectance map computed from the algorithm. Same grey

value scaling in both images. (c) Assignment of regions is almost perfect. (d) Misaligned pixel in lower right region.

Table 1: Initial values θ

(0)

i

of Markov chain, (σ

p

i

)

2

of pro-

posal distribution and prior parameters µ

prior

and (σ

prior

)

2

.

θ

(0)

i

(σ

p

i

)

2

µ

prior

(σ

prior

)

2

I

0

80 0.5 0 100

k

d

0.5 0.05 0 100

γ 1 0.07 0 100

σ

2

3 0.09 0 100

1

7

8 9

0

200

400

600

800

4

24

14,405

567

model number m

# iterations

Figure 2: Histogram of visited models in model space of

artiﬁcal dataset. Counting indicated that model 8 is the most

likely one.

change point in u and v direction and is highly consis-

tent with the dataset, as becomes apparent from Fig-

ure 1. The certainty of the algorithm is about 96%.

The sampled a-posteriori distribution of the set θ

θ

θ

of model parameters is shown along with the itera-

tions of the Markov chain in Figure 3. Convergence

of k

d

and γ is slow which may be caused by the small

(σ

p

i

)

2

of the proposal distribution (see Table 1). To

ensure that the ﬁnal sampling is based on the correct

a-posteriori distribution, the burn-in phase is set to

50%, which is a conservative estimation.

To underline the efﬁciency of our procedure, the

inferred reﬂectance parameters and their uncertainties

are listed in Table 2 together with the true values that

have actually been used to generate the synthetic test

data (green values in each cell). The averages of the

parameter estimates are very similar to the actual pa-

rameter values, which is also indicated by Figure 1(a)

and (b). Moreover, the uncertainties of the estimate

0

0.5

1

1.5

·10

4

0

1

2

3

iteration t

parameter value

I

0

/100 k

d

γ σ

Figure 3: Change of parameters of upper region for m = 8

(most likely model) throughout iterations. The light source

intensity has been scaled to improve visibility. I

0

and σ

converge very fast to a stationary distribution, whereas k

d

and γ converge slowly. This can be caused by the small

values of (σ

p

i

)

2

of the proposal distribution. The red dashed

line indicates the end of the assumed burn-in phase after

50% of the total iterations.

are very low, such that the conﬁdence band is a narrow

tube that shows the high accuracy of the procedure on

the data set.

The uncertainties of the values of the lower right

region are notably higher. This might be caused by

the data set itself, since the light sources are not dis-

tributed symmetrically around the sphere to create re-

alistic conditions. Thus, the lower right region pos-

sibly yields less salient information than the remain-

ing surface, e.g. intensity data that deﬁnes the cosine

lobe. However, the synthetic data should be regarded

as a proof of concept, and applying the method to real-

world data will be a more valuable assessment.

4 APPLICATION

The proposed method is applied both to artiﬁcial and

real world data. The artiﬁcial data consists of 10 im-

ages of a sphere that are rendered with the Lambert-

Blinn BRDF model lighted from different positions,

the surface is divided into four segments. The shading

BRDF parameters of the individual patches are given

Simultaneous Surface Segmentation and BRDF Estimation via Bayesian Methods

43

Table 2: Maximum a-posteriori estimate of parameters and 1σ deviation for upper and lower region after burn-in phase of

50%. The green values have been used render the synthetic images and are therefore the target values of the optimization.

θ

i

upper left lower left upper right lower right

I

0

100.0 ± 0.001 100 100.0 ± 0.007 100 100.0 ± 0.002 100 99.87 ± 0.048 100

k

d

0.499 ± 4.2 · 10

−5

0.5 0.899 ± 1.3 · 10

−4

0.9 0.599 ± 6.9 · 10

−5

0.6 0.363 ± 0.012 0.4

γ 4.99 ± 9.6 · 10

−4

5 9.99 ± 0.015 10 4.99 ± 1.6 · 10

−3

5 0.91 ± 0.026 1

σ

2

0.10 ± 2.2 · 10

−4

0.1 0.10 ± 2.3 · 10

−4

0.1 0.10 ± 1.6 · 10

−4

0.1 0.11 ± 0.004 0.1

Table 3: Initial values θ

(0)

i

of Markov chain, (σ

p

i

)

2

of pro-

posal distribution and prior parameters µ

prior

and (σ

prior

)

2

.

θ

(0)

i

(σ

p

i

)

2

µ

prior

(σ

prior

)

2

I

0

1 0.1 1 1

k

d

0.5 0.01 0.5 1

γ 25 10 25 100

σ

2

1 0.1 0.1 100

in Table 2 (green values). Noise is added to the image

data to create more realistic conditions.

The real-world dataset contains 18 HDR images

of the test object. We used the measurement equip-

ment in our laboratory

1

to acquire both image and

depth data of a painted plaster object. The illumi-

nation environment is calibrated according to Lenoch

et al. (2012).

4.1 Real-world Data

The test object consists of shaped plaster that has been

divided into three patches of which two are painted

with green and blue acrylic paint, respectively. The

object was designed to be of a geometrically com-

plex shape and yield multiple reﬂection characteris-

tics at different surface areas. Due to the acquisition

with a monochrome camera both painted regions ap-

pear very similar even to the human viewer. Thus,

two regions can be segmented. One is bright white

and exhibits diffuse reﬂection while the other is dark

grey and shows increased specular reﬂection from the

shiny paint. Table 3 displays the initial values of the

Markov chain θ

(0)

i

, the (σ

p

i

)

2

of the proposal distribu-

tion and the prior parameters.

The object is depicted in Figure 4(a). Figure 4(b)

and (c) display the corresponding reﬂectance map

based on the Lambert-Blinn BRDF model (Blinn,

1977) and the estimated parameters and segmentation

as well as a pixelwise error map. For the sake of sim-

plicity only one image of the object from the entire

real-world data set is given. The largest deviations

can be spotted at the dents in the upper region and at

1

3D Scanner: zSnapper Vario with AVT Pike 421-F, CCD

sensor, resolution 2048x2048 pixel. Light sources: Seoul

P4 Power LED, centered at a wavelength of 525 nm.

the shiny component of the lower region which is not

bright enough. These ﬁndings are supported by Fig-

ure 4(c). The high difference in the dents is caused by

masking and interreﬂection effects on the real surface

that are not modeled by the BRDF. The errors in the

segmentation will be addressed later.

First of all, it has to be stated that the segmenta-

tion of the surface is not intensity based, which one

might consider suitable for this object, but based on

the likelihood that a certain set of parameters ﬁts the

designated surface patch best. The shadowed parts of

the dents might be assigned to the lower patch if it

was purely intensity based. Furthermore, the glossy

specular component of the lower region yields an in-

tensity which is close to the bright areas of the upper

region. This demonstrates one great advantage of the

segmentation based on reﬂectance functions: A shiny

surface is not separated based on the fact that the im-

age intensity varies signiﬁcantly from diffuse to spec-

ular component. This property is elaborated further in

Section 5.

Secondly, the procedure is able to detect that a

segmentation of the surface into two regions yields

the best results. Figure 5 indicates that the model

m = 2, one change point vector in u direction, was

visited T

2

= 13178 times and thus has a probability of

≈ 87, 85% to be the best choice.

The estimated parameters of the two BRDFs and

their 1σ uncertainties are listed in Table 4. The in-

tensity value I

0

accounts for the changing brightness

of the surface, whereas the diffuse component is al-

most equal. The estimated noise σ

2

of the likelihood

is very small, indicating that the estimated BRDF val-

ues cover the measurements well. The cosine-lobe

parameter γ shows a high uncertainty that is caused

by the large diffuse component. Since k

s

= 1 − k

d

⇒

k

s

≈ 0.04 . . . 0.06 the cosine lobe has very little effect

on the ﬁnal reﬂectance map. However, the larger ex-

ponent of the lower region is consistent with the im-

pression of the shinier surface, although the specular

component in the reﬂectance map is too small.

The progression of the Markov chain of θ

θ

θ (upper

region) is depicted in Figure 6. The dotted red line

marks the end of the burn-in phase, after which the

chain is supposed to have converged to their true dis-

VISAPP 2016 - International Conference on Computer Vision Theory and Applications

44

(a) (b) (c)

−0.2

−0.1

0

0.1

0.2

Figure 4: (a) One of the captured images, (b) reﬂectance map of segmented surface with computed parameters. Same scaling

of grey values in both images. (c) Absolute difference of intensity values.

1 2 3 4

5 6

7 8

0

500

1,000

682

13,178

1,037

79

7

3

12

2

model number m

# iterations

Figure 5: Histogram of visited models in model space.

Counting indicated that model 2 is the most likely.

Table 4: Maximum a-posteriori estimate of parameters and

1σ deviation for upper and lower region after burn-in phase

of 20%.

θ

i

upper Region lower Region

I

0

0.899 ± 0.029 0.241 ± 0.004

k

d

0.939 ± 0.098 0.968 ± 0.012

γ 0.338 ± 0.747 32.65 ± 4.068

σ

2

0.056 ± 0.007 0.016 ± 0.002

tribution. It is visible here as well that all parameters

except γ have converged fast to an almost stationary

value, whereas γ suffers from the small specular com-

ponent which makes its estimation difﬁcult.

As already seen in Figure 4(b) the estimation of

the change points yields errors at the outer parts of the

object. The mode of the distribution of each change

point (blue line) and the 95% conﬁdence level (ma-

genta dashed line) is shown in Figure 7(a). The yellow

box frames the region of high certainty that coincides

with the correct segmentation. The erroneous change

points yield a drastically increased uncertainty and as

such the algorithm itself is able to determine when to

trust a change point and when not.

A closer look at the estimated change points is

possible with Figure 7. The histogram of each change

point is given in Figure 7(b) and it is evident that the

correct segmentation has been part of the proposed

0 0.2 0.4

0.6

0.8 1 1.2

·10

4

0

2

4

6

iteration t

parameter value

I

0

k

d

γ

σ

2

Figure 6: Change of parameters of upper region for m = 2

(most likely model) throughout iterations. Most parame-

ters converge fast to an almost stationary value, γ shows a

higher uncertainty, the red dashed line indicates the end of

the assumed burn-in phase after 20% of the total iterations.

change point values and more often than other val-

ues as indicated by the color. Yet, many other change

points have been proposed and accepted such that the

resulting conﬁdence level is low and the mode of the

distribution does not always coincide with the seg-

mentation ground truth.

A possible cause of the reduced quality of the seg-

mentation is the lack of data at the outer regions of

the object. Figure 8 shows the number of valid im-

ages per surface pixel. It is apparent that the outer ob-

ject surface yields less available information and thus

increases the difﬁculties of ﬁnding the correct change

points and BRDF parameters.

5 EVALUATION

To assess the prospects of the presented method,

we compare the results of the segmentation to com-

mon image segmentation procedures. We apply k-

means, e.g. (Seber, 1984; Spath, 1985), Gaussian

mixture model

2

(GMM) (McLachlan and Peel, 2000),

2

Matlab 2015a implementation of k-means and GMM.

Simultaneous Surface Segmentation and BRDF Estimation via Bayesian Methods

45

(a) (b)

0

20

40

60

Figure 7: (a) Position of change points (blue line) and 95%-conﬁdence tube (magenta dashed line) showing the uncertainty.

The part inside the yellow box is segmented at high accuracy. The outer parts of the object show erroneous segmentation in

combination with an increased uncertainty. (b) 2D Histogram of change points after burn-in phase. Each column represents

the histogram of the respective change point in u direction, the color indicates the histogram counts per pixel. The image has

been zoomed to enlarge the area of interest and the square root has been applied to all counts to improve visibility.

0

6

12

18

Figure 8: Valid images per pixel available to the BRDF es-

timation. The outer region lacks data, which is caused by

shadowing or outlying illumination geometry and makes the

accurate BRDF estimation more difﬁcult.

multichannel k-means (Pichler and Hosticka, 1995)

and normalized graph cuts

3

(NCuts) (Shi and Ma-

lik, 2000) segmentation procedures. The cluster al-

gorithms are applied to the synthetic data, since the

varying intensity and specular highlights are the more

challenging dataset to be accurately segmented.

Note that the comparison of these algorithms to

ours is questionable since the similarity criterion used

to dissect the surface is not the same in every pro-

cedure. Our methods operates on the average simi-

larity of measured grey values and a rendered image.

The segmentation procedures compared against our

method use grey values to dissect the image. In the

case of GMM we extended the input data with spatial

information about the pixel location. Furthermore,

NCuts can not handle multiple input images without

modiﬁcation. However, to our knowledge there is no

3

We used the existing implementation provided at

http://www.cis.upenn.edu/ jshi/software/, last updated Jan-

uary 22, 2010.

method that yields a similar approach to image seg-

mentation as the one presented in this paper.

k-means. Multichannel image data can be ex-

ploited in a k-means clustering when treating each

of the P images as a separate feature of every pixel,

such that we get N · M samples (assuming I

NxM

) each

yielding P features. Since k-means requires that the

desired number of clusters is provided by the user,

we tried k ∈ [48]. The k-means segmentation of the

synthetic dataset is displayed in Figure 9. The seg-

mentation is erroneous for every considered number

of clusters. Since k-means operates on grey values,

the specular highlights of the surface determine the

dominant center of the clusters.

The multichannel k-means algorithm as described

by Pichler and Hosticka (1995) did not converge to a

solution on the given data.

Gaussian Mixture Model. Similar to the k-

means algorithm we supplied the multiple intensi-

ties per image pixel as features to the ﬁtting of the

GMM. Moreover, we provided the spatial position of

the pixel as additional feature. The GMM is not able

to dissect the surface into the correct areas of different

reﬂectance properties as can be seen in Figure 10.

Normalized Cuts. Since the normalized cuts al-

gorithm is not designed to handle multispectral im-

age data only one image from the dataset (Fig 11(a))

is used to test the segmentation. To account for a

background-cluster the number of clusters k is in-

creased by one. Based on k ∈ {5, 6} the segmentation

appears reasonable, yet the borders are not perfectly

met. Increasing the number of clusters leads to false

segments instead of subdividing the correct segments

into smaller patches.

Note that the above mentioned segmentation pro-

cedures are superior to our method in computational

VISAPP 2016 - International Conference on Computer Vision Theory and Applications

46

(a) k = 4 (b) k = 5 (c) k = 6 (d) k = 7 (e) k = 8

Figure 9: k-means segmentation of synthetic data, the colors code different surface segments. The segmentation fails for

every considered setting.

(a) k = 4 (b) k = 5 (c) k = 6 (d) k = 7 (e) k = 8

Figure 10: Gaussian mixture model segmentation of synthetic data, the colors code different surface segments. k is the number

of Gaussian distributions ﬁt the data. The segmentation fails for every considered setting.

(a) (b) k = 5 (c) k = 6 (d) k = 7 (e) k = 8

Figure 11: a) Input image taken from the synthetic dataset. b)-e) Normalized cuts segmentation of synthetic data, the colors

code different surface segments. The number of clusters k is increased by one with regards to the actual number of surface

segments to account for a background cluster.

time. However, the general beneﬁt of the Bayesian

approach is the inherent knowledge of the uncertainty

of the computed results. Furthermore, the reﬂectance

parameters of the segmented patches need to be com-

puted separately if e.g. used in surface reconstruction

applications such as shape from shading (Horn, 1986)

or photometric stereo (Woodham, 1980). Since the

surface segmentation in the presented algorithm is di-

rectly based on the reﬂectance estimation, the param-

eters of the surface segments including their uncer-

tainty are already known. Additionally, the methods

used for comparison require the number of desired

clusters as user input, which is not necessary in our

approach since it achieves unsupervised surface seg-

mentation.

6 CONCLUSIONS

In this paper we have shown that the segmentation

of an arbitrary surface is possible based on the maxi-

mum a-posteriori estimation of its reﬂectance param-

eters. The novelty of our procedure is the direct infer-

ence of patch borders and BRDF parameters from the

data and the computation based on Bayesian statis-

tics, which inherently allows us to deduce the accu-

racy of the computed results. The experimental eval-

uation has demonstrated that the amount of available

data has to be high enough to draw the correct con-

clusions from the input data.

In comparison to existing surface segmentation

procedures our method based on the surface re-

ﬂectance properties achieves the best segmentation on

the synthetic dataset.

The proposal of new change points in the existing

Simultaneous Surface Segmentation and BRDF Estimation via Bayesian Methods

47

implementation is randomized and therefore favors

lines, which makes it difﬁcult to segment circle-like

surface patches, since those have to be based on lines

that converge from opposing sides of the circle. Con-

sequently, a different concept of change point propo-

sition adapted from prior knowledge of the type of

surface segments would enhance the performance of

the procedure.

Furthermore, we would like to extend our method

to other BRDF (likelihood) functions which consider

Fresnel effects and shadowing/masking, e.g. (Cook

and Torrance, 1981), or introduce a more general co-

sine lobe model to cope for anisotropic reﬂection, e.g.

(Lafortune et al., 1997). Moreover, it is of great inter-

est to include the estimation of surface normals in the

likelihood, which would introduce two more param-

eters per pixel. This will probably increase the un-

certainty of the overall likelihood considerably, but at

the gain of freeing the procedure from the necessity

of any prior knowledge about the surface.

REFERENCES

Aittala, M., Weyrich, T., and Lehtinen, J. (2013). Practical

svbrdf capture in the frequency domain. ACM Trans-

actions on Graphics, 32.

Alldrin, N., Zickler, T., and Kriegman, D. (2008). Pho-

tometric stereo with non-parametric and spatially-

varying reﬂectance. Conference on Computer Vision

and Pattern Recognition.

Blinn, J. F. (1977). Models of light reﬂection for computer

synthesized pictures. Association for Computing Ma-

chinery’s Special Interest Group on Computer Graph-

ics and Interactive Techniques, 11(2):192–198.

Cook, R. L. and Torrance, K. E. (1981). A reﬂectance

model for computer graphics. Association for Com-

puting Machinery’s Special Interest Group on Com-

puter Graphics and Interactive Techniques, 15(3):307

– 316.

Giesen, F. (2009). Phong and blinn-

phong normalization factors. online

http://www.farbrausch.de/ fg/stuff/phong.pdf, 1:1–2.

Gilks, W., Richardson, S., and Spiegelhalter, D. (1996).

Markov Chain Monte Carlo in Practice. Chapman &

Hall.

Goldman, D. B., Curless, B., Hertzmann, A., and Seitz, S.

(2005). Shape and spatially-varying brdfs from photo-

metric stereo. International Conference on Computer

Vision, pages 341–348.

Green, P. J. (1995). Reversible jump markov chain monte

carlo computation and bayesian model determination.

Biometrika, 82:711–732.

Hastings, W. K. (1970). Monte carlo sampling methods us-

ing markov chains and their applications. Biometrika,

57:97–109.

Herbort, S. and W

¨

ohler, C. (2012). Self-consistent 3D sur-

face reconstruction and reﬂectance model estimation

of metallic surfaces. International Joint Conference

on Computer Vision, Imaging and Computer Graph-

ics Theory and Applications, pages 1–8.

Horn, B. K. P. (1986). Robot Vision. MIT Electrical Engi-

neering and Computer Science.

King, R., Morgan, B. J., Gimenez, O., and Brooks, S. P.

(2010). Bayesian Analysis for Population Ecology.

CRC Press.

Lafortune, E. P. F., Foo, S.-C., Torrance, K. E., and Green-

berg, D. P. (1997). Non-linear approximation of re-

ﬂectance functions. Association for Computing Ma-

chinery’s Special Interest Group on Computer Graph-

ics and Interactive Techniques, pages 117–126.

Lambert, J.-H. (1760). Photometria, sive de mensura et

gradibus luminis, colorum et umbrae. Vidae Eber-

hardi Klett.

Laskey, K. and Myers, J. (2003). Population markov chain

monte carlo. Machine Learning, 50:175–196.

Lenoch, M., Herbort, S., Grumpe, A., and W

¨

ohler, C.

(2014). Linear unmixing in brdf reproduction and 3d

shape recovery. International Conference on Pattern

Recognition.

Lenoch, M., Herbort, S., and W

¨

ohler, C. (2012). Robust and

accurate light source calibration using a diffuse spher-

ical calibration object. Oldenburger 3D Tage, 11:212–

219.

Lensch, H. P. A., Kautz, J., Goesele, M., Heidrich, W., and

Seidel, H.-P. (2003). Image-based reconstruction of

spatial appearance and geometric detail. ACM Trans-

actions on Graphics, 22(2):234–257.

Louw, M. and Nicolls, F. (2010). Surface classiﬁcation via

brdf parameters, using population monte carlo for mrf

parameter estimation. Proc. IASTED Int. Conf. Com-

puter Graphics and Imaging, 11:145–154.

Matusik, W., Pﬁster, H., Brand, M., and McMillan, L.

(2003). A data-driven reﬂectance model. ACM Trans-

actions on Graphics, 22(3):759–769.

McLachlan, G. and Peel, D. (2000). Finite Mixture Models.

John Wiley & Sons, Inc.

Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A.,

and Teller, E. (1953). Equation of state calculations

by fast computing machines. Journal of Chemical

Physics, 21:1087–1092.

Ntzoufras, I. (2011). Bayesian modeling using WinBUGS.

John Wiley & Sons.

Pichler, O. and Hosticka, A. T. B. J. (1995). A multichannel

algorithm for image segmentation with iterative feed-

back. In Image Processing and Its Application.

Seber, G. A. F. (1984). Multivariate Observations. John

Wiley & Sons, Inc.

Shi, J. and Malik, J. (2000). Normalized cuts and image

segmentation. Trans. Pattern Analysis and Machine

Intelligence, 8:888–905.

Spath, H. (1985). Cluster Dissection and Analysis: Theory,

FORTRAN Programs, Examples. Halsted Press.

Weyrich, T., Lawrence, J., Lensch, H., Rusinkiewicz, S.,

and Zickler, T. (2008). Principles of appearance ac-

quisition and representation. Association for Comput-

ing Machinery’s Special Interest Group on Computer

Graphics and Interactive Techniques, pages 1–70.

Woodham, R. J. (1980). Photometric method for determin-

ing surface orientation from multiple images. Optical

Engineering, 19(1):139–144.

VISAPP 2016 - International Conference on Computer Vision Theory and Applications

48