SELF-CONSISTENT 3D SURFACE RECONSTRUCTION AND

REFLECTANCE MODEL ESTIMATION OF METALLIC SURFACES

Steffen Herbort and Christian W

ohler

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

Keywords:

3D Surface Reconstruction, Active Range Scanning, Image-based 3D Reconstruction, Data Fusion, Re-

ﬂectance Model Estimation.

Abstract:

3D surface reconstruction data measured with active range scanners typically suffer from high-frequency noise

on small scales. This poses a problem for highly demanding surface inspection tasks and all other applications

that require a high accuracy of the depth data. One way to achieve increased 3D reconstruction accuracy is

the fusion of active range scanning data and photometric image information. Typically, this requires modeling

of the surface reﬂectance behavior, which, in turn, implies the surface to be known with high accuracy to

determine valid reﬂectance parameters as long as no calibration object is available. In this study, we propose

an approach that provides a detailed 3D surface reconstruction along with simultaneously estimated param-

eters of the reﬂectance model. For 3D surface reconstruction, we employ an algorithm that combines active

range scanning data for large-scale accuracy with image-based information for small-scale accuracy. For in-

ferring the reﬂectance function, we incorporate the estimation of the reﬂectance model into a self-consistent

computational scheme that successively increases the resolution and thus determines the reﬂectance parame-

ters based on reﬁned depth information. We present results for a homogeneous dark rough metallic surface,

which is reconstructed based on a single coarse 3D scan and 12 images acquired under different illumination

conditions.

1 INTRODUCTION

Active range scanning approaches typically suffer

from high-frequency noise and thus lack the capa-

bility to perceive and resolve ﬁne surface details es-

pecially for non-diffusely reﬂecting surfaces. While

ﬁltering may reduce high-frequency noise and re-

cover some of the the underlying details, the over-

all quality and accuracy is not improved signiﬁcantly

while the effective lateral resolution decreases. If

one wants to truly enhance the amount of surface de-

tail, it is desirable to supplement the absolute depth

data obtained by active range scanners with gradient

information obtained using image-based approaches

like shape from shading (Horn, 1970) or photometric

stereo (Woodham, 1980). The crucial problem lies in

ﬁnding an approach that fuses both data sources and at

the same time exploits the mutual advantages: Range

scanning approaches provide robust large-scale data

with high-frequency noise, while image-based data

provide accurate small-scale details but tend to devi-

ate systematically from the true large-scale shape. A

well-known example for that fusion process has been

proposed by Nehab et al. (Nehab et al., 2005) and

their results clearly demonstrate the improved small-

scale accuracy. However, their approach only deals

with pre-existing data and omits the gradient deter-

mination stage, which is challenging e.g. for metal-

lic surfaces. Later approaches for fusing depth data

and photometric image information (cf. e.g. (W

ohler

and d’Angelo, 2009) and references therein) include

an estimation of the surface gradients but still assume

the reﬂectance function to be known in advance.

It is thus interesting from a theoretical and rele-

vant from a practical point of view to develop a self-

consistent approach that incorporates all steps nec-

essary for 3D surface reconstruction, including an

estimation of the reﬂectance function. Apart from

surface inspection, the demand for highly accurate

surfaces comes from other ﬁelds as well. In com-

puter graphics, the problem of determining the re-

ﬂectance function from arbitrarily shaped surfaces

commonly lacks accuracy due to the fact that the ex-

amined surface shape is not known to the required

level of detail (Weyrich et al., 2008). While sev-

eral image-based methods solve that problem by as-

suming known shapes (Matusik et al., 2003b), this

is only possible for surfaces which provide all illu-

114

Herbort S. and Wöhler C..

SELF-CONSISTENT 3D SURFACE RECONSTRUCTION AND REFLECTANCE MODEL ESTIMATION OF METALLIC SURFACES.

DOI: 10.5220/0003819701140121

In Proceedings of the International Conference on Computer Vision Theory and Applications (VISAPP-2012), pages 114-121

ISBN: 978-989-8565-04-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

mination and viewing geometries required for a re-

liable reﬂectance parameter estimation. Once this

precondition is fulﬁlled and the surface is known

to the required level of detail, the determination of

the reﬂectance function becomes a problem of non-

linear model adaptation using methods such as the

Levenberg-Marquardt algorithm (Mor

e, 1978). As a

result of previous and current research, a large vari-

ety of reﬂectance models have been developed, which

are typically selected based on the material at hand.

The most popular ones include the classical Lamber-

tian model (Lambert, 1760; Horn, 1989) for strictly

diffuse surfaces, the Phong model (Phong, 1975) for

empirically modeled specularities, its more physically

motivated version (Lewis, 1994), its generalized ver-

sion (Lafortune et al., 1997), and more specialized

models for metals and rough surfaces (Cook and Tor-

rance, 1981; Beckmann and Spizzichino, 1987) and

anisotropic surfaces (Ward, 1992).

Image-based algorithms analyze the object ap-

pearance and thus commonly depend on a reﬂectance

model that is known a priori as accurately as possi-

ble. Unfortunately, this usually requires the surface

shape to be known in advance with very high accu-

racy, as discussed above. To overcome that drawback,

we present a self-consistent approach for the simulta-

neous determination of surface shape and reﬂectance

function parameters for strongly non-Lambertian sur-

faces. In contrast to methods from the ﬁeld of com-

puter graphics, e.g. (Lensch et al., 2003), we do not

only change local surface normals to model the ap-

pearance of the object in a rendered image, but actu-

ally incorporate that information into the 3D recon-

struction of the surface.

The critical aspect lies in the determination of re-

ﬂectance parameters without ﬁne surface shape data

being available. Our approach thus uses strongly

downsampled absolute depth and image data to de-

termine initial reﬂectance function parameters. These

are reliable since the absolute depth and image data

are reasonably accurate on that scale. The surface re-

construction algorithm then uses that information and

computes a reﬁned surface. The process then itera-

tively continues on a higher resolution scale (cf. Sec-

tion 2). The alternating scheme of reﬂectance func-

tion estimation and surface reconstruction thus suc-

cessively reﬁnes the 3D surface reconstruction and

the estimated reﬂectance parameters (cf. Section 3).

For data acquisition, we use a calibrated range scan-

ning system with 12 attached LED light sources with

known (calibrated) illumination directions and a sin-

gle camera position. Our experimental results are de-

scribed in Section 4.

downsample 1:2

images

images depth

BRDF estimation

surface reconstruction

, q

, z

upsampling (factor 2)

BRDF

K=K - 1

optimized

K <

0 ?

full resolu-

tion result

initial BRDF estimation

median-

filter

Figure 1: Overview of the self-consistent algorithm for

3D surface reconstruction and reﬂectance parameter esti-

mation.

2 SELF-CONSISTENT

RECONSTRUCTION

Fig. 1 illustrates the proposed self-consistent 3D re-

construction algorithm. The main elements are the

estimation of the reﬂectance parameters (green) and

the surface reconstruction (blue). The algorithm starts

with the given image and depth data, which are ini-

tially subsampled by a factor of 2

with appropriate

spatial low-pass ﬁltering to avoid aliasing. The sub-

sampling stage ensures the removal of spurious high

spatial frequency components from the range scan-

ner data and thus provides the basis for robust re-

ﬂectance estimation on that scale. Once the initial

reﬂectance parameters are known, the reconstruction

exploits the reﬂectance information to incorporate the

image-based depth data. The successive steps “sur-

face reconstruction” and “BRDF estimation” can be

iterated in an inner loop (cf. dashed line in Fig. 1)

without increasing the current resolution scale. Af-

terwards, the result of the 3D reconstruction stage is

upsampled by a factor of 2 using bicubic interpola-

tion and then serves as the initialization for the next

iteration. The algorithm terminates when the full res-

olution scale is reached.

SELF-CONSISTENT 3D SURFACE RECONSTRUCTION AND REFLECTANCE MODEL ESTIMATION OF

METALLIC SURFACES

115

3 SURFACE RECONSTRUCTION

AND BRDF ESTIMATION

In this paper, we will use u ∈ [1...N] and v ∈ [1...M]

to denote the integer pixel coordinates of an image

I ∈ R

N×M

. The image data I contain intensity mea-

surements for each pixel, and the available range

scanner data z

(u,v) provide pixel-synchronous ab-

solute depth measurements for most but generally not

all image pixels, i.e. there may be gaps in the absolute

depth data.

Local illumination directions, viewing directions,

and surface normals are denoted by the vectors

s(u,v),

~v(u,v), and ~n(u,v), respectively, with k

= k~vk

k~nk

= 1. The vector ~r denotes the incident light di-

rection mirrored at the respective surface normal. The

reﬂectance model parameters P are introduced later

when the applied reﬂectance model M is discussed.

The reﬂectance function itself is termed BRF (Bidi-

rectional Reﬂectance Function) or BRDF (Bidirec-

tional Reﬂectance Distribution Function).

The algorithm computes the optimized surface

gradient ﬁelds

p(u,v) =

∂z(u,v)

∂x

= ∂

z(u,v) = z

(u,v) (1)

q(u,v) =

∂z(u,v)

∂y

= ∂

z(u,v) = z

(u,v) (2)

and the optimized surface z

∗

(u,v). A rendered image

of the surface obtained using the reﬂectance function

of the surface, a set of surface gradients, illumination

and viewing directions is denoted “reﬂectance map”

R according to

R = R (p(u,v),q(u,v),

s(u,v),~v(u,v),~n(u,v),P,M).

(3)

An approach for recovering and fusing absolute depth

and gradient data (p,q) has been proposed by us

previously (Herbort et al., 2011). The algorithm is

an extension of Horn’s method for the simultaneous

recovery of height and gradients (Horn, 1989). While

Horn’s approach operates solely on image data, the

extension regards the fusion with absolute depth data.

To give a complete background for the approach

presented in this study and to provide better expla-

nations, we summarize the main ideas and give an

overview of its capabilities: The algorithm as such

minimizes the overall error

E = E

+ γ E

int

+ δ E

(4)

according to

∗

= argmin

p,q,z

+ γ E

int

+ δ E

) (5)

by ﬁnding an optimal surface z

∗

(u,v), which is com-

posed of the gradient ﬁeld (p(u,v),q(u,v)). The

weight parameters γ and δ have to be determined em-

pirically, i.e. by manually choosing a set of parame-

ters that lets the optimization scheme iterate and con-

verge. Each component of the error function E con-

tributes to different aspects that enforce certain re-

strictions upon the optimal surface. The intensity er-

ror

∑

u,v

(I − R)

(6)

determines the difference between the observed origi-

nal image I and the reﬂectance map R. The extension

towards several images is straightforward by evalua-

tion of the mean error over all images and their re-

spective reﬂectance maps. The error term E

causes

the optimized surface to alter its gradients until the

image and the reﬂectance map match as closely as

possible. The integrability error

int

∑

u,v

− p)

+ (z

− q)

(7)

denotes the deviations of the estimated gradient ﬁeld

from the gradients of the determined surface, i.e. from

an integrable gradient ﬁeld, and thus prevents the oc-

currence of local gradient spikes. The range scanner

depth gradient error

∑

u,v



(∂

(G ∗ z

) − G ∗ p)

+(∂

(G ∗ z

) − G ∗ q)



(8)

measures the deviation of the estimated surface gra-

dients from those derived from the range scanner data

on large spatial scales. This is achieved by removing

small surface details by convolution with a (Gaussian)

low-pass ﬁlter G, which then allows an adaptation of

the low spatial frequency components of the recon-

structed surface gradients to those of the range scan-

ner data (Herbort et al., 2011).

The generation of the reﬂectance map R requires

the reﬂectance properties of the surface to be known.

This is typically achieved in a data driven or model

driven way (Matusik et al., 2003a). In our algorithm,

we apply a reﬂectance model, since there is only a

limited range of viewing directions available if, as in

our case, a ﬁxed camera is used. The estimation of the

model from sparse data is usually possible and robust,

while the inevitable interpolations of data driven ap-

proaches possibly produce unexpected results. In the

following, the chosen model (cf. Section 1 for other

examples) is discussed.

The three-component Lambert/Phong model (Na-

yar et al., 1990) has been applied to isotropic surfaces

with non off-specular reﬂectance behavior (W

ohler

and d’Angelo, 2009). The observed intensity is de-

scribed by

I = I

ρ [~n ·

s + σ

(~v ·~r)

+ σ

(~v ·~r)

]. (9)

VISAPP 2012 - International Conference on Computer Vision Theory and Applications

116

Figure 2: Overview of the experimental setup. The object

has a height of about 50 mm, the distance between the ob-

ject and the scanner (1), camera (2), and illumination (3)

amounts to approximately 250 mm.

Its parameters are the intensity I

of the incident light,

the surface albedo ρ, the specular lobe strength σ

, the

specular lobe width m

, the specular spike strength

, and the specular spike width m

ohler and

d’Angelo, 2009). A more generalized form allows a

directional diffuse behavior according to

I = I

ρ [(~n ·

s) + σ

(~n ·

+ σ

(~v ·~r)

+ σ

(~v ·~r)

] (10)

with the directional diffuse width m

and strength

. In our experiments (cf. Section 4), this model

has been shown to be ﬂexible enough to represent the

reﬂectance behavior, while having a feasible number

of 7 parameters (the term I

ρ can be set to the “ef-

fective albedo” ρ

eff

). The directional diffuse term has

empirically proven to have a favorable effect on the

3D reconstruction accuracy when few light sources

(i.e. images) are available. Note that the system de-

scribed in this study poses almost no restrictions re-

garding the applied reﬂectance model. The only re-

quirements are the capability of the reﬂectance func-

tion to model the reﬂectance behavior and the solv-

ability of the ﬁtting problem, i.e. it must be possible

to obtain the reﬂectance model parameters from the

given/obtained data I, p, q, and z.

4 EXPERIMENTAL RESULTS

This section initially provides an overview of the ex-

perimental setup. The obtained results are then pre-

sented and discussed.

4.1 Experimental Setup

Since our range scanning system

already contains

a camera

and records pixel-synchronous depth and

image data, there is no need for data registration

prior to the reconstruction. All 12 attached LED

light sources

have been calibrated using a white

diffuse sphere

and solving Lambert’s law I(u,v) =

ρ (~n(u,v) ·

s) for the global light direction

s and

the intensity I

. In contrast to the popular method

to use a specularly reﬂecting sphere, using the Lam-

bertian sphere yields more robust results due to the

larger number of measurements being involved in the

optimization. The obtained phase angles, i.e. the an-

gles between the illumination directions and the view-

ing direction, range from 18

◦

to 67

◦

. Note that the

bright regions in the intensity images exhibit very

strong specular spikes, as they are typical for metal-

lic surfaces. To account for these large dynamic vari-

ations within the image due to the dark surface and

the strong specular spikes of the metallic material, we

recorded high dynamic range (HDR) images.

4.2 Results

The results for the 3D reconstruction of an emboss-

ing on the surface of a metallic dumbbell consisting

of dark cast iron are presented in Fig. 3. The object

and the area of interest are depicted in Fig. 3(a), the

input data for the algorithm are shown in Fig. 3(b)

and 3(d). Four of the 12 images acquired under dif-

ferent illumination directions (cf. Fig. 3(b)) and the

raw scanner data (cf. Fig. 3(d)) are shown as well.

Figs. 3(j)–3(m) illustrate how the surface evolves over

4 iterations (K = 3) from a coarse surface at 1/8

of the full scale (cf. Fig. 3(j)) over the intermediate

scales 1/4 (cf. Fig. 3(k)) and 1/2 (cf. Fig. 3(l)) to the

full scale where the resolution of the 3D surface re-

construction reaches the full resolution of the images

(300 × 700 pixels at a scale of 42 µm per pixel) as

shown in Fig. 3(m). Note that with each iteration,

an increasing amount of surface detail becomes vis-

ible and is thus incorporated into the reconstructed

surface. The comparison of the images and the sur-

face shows correspondences between surface bumps

and their bright or dark counterparts in the images.

ViALUX zSnapper Vario, structured/modulated light

range scanner

AVT pike 421B, 14 Bit monochrome CCD camera,

2048 × 2048 pixels

Seoul P4 LED, λ ≈ 525nm (green), luminous ﬂux ap-

proximately 120 lm

30 mm diameter, manufactured by Optopolymer, Mu-

nich

SELF-CONSISTENT 3D SURFACE RECONSTRUCTION AND REFLECTANCE MODEL ESTIMATION OF

METALLIC SURFACES

117

≈3cm

(a) Metallic dumbbell, the ROI for the reconstruction is indi-

cated in red.

(b) 4 out of 12 HDR images, 400×700 pixels, lateral

resolution 0.042mm per pixel.

tion plots.

(d) Raw range scanner data.

−50 0 50 100 150 200 250

100

120

140

160

180

observed intensity (x component)

observed intensity (y component)

(e) Development of the full reﬂectance function.

−0.4 −0.2 0 0.2 0.4 0.6

0.5

1.5

observed intensity (x component)

observed intensity (y component)

(f) Diffuse component 1.

−0.4 −0.2 0 0.2 0.4 0.6

0.5

1.5

observed intensity (x component)

observed intensity (y component)

(g) Diffuse component 2.

−4 −2 0 2 4 6 8 10 12 14 16 18

observed intensity (x component)

observed intensity (y component)

(h) Specular lobe.

−4 −2 0 2 4 6 8 10 12 14 16 18

observed intensity (x component)

observed intensity (y component)

(i) Specular spike.

(j) Reconstruction result at scale 1/8. (k) reconstruction result at scale 1/4. (l) reconstruction result at scale 1/2.

depth profile,

see Fig. (n)

(m) reconstruction result at full scale (400 ×700 pixels.)

0 5 10 15 20 25 30

−1

−0.5

0.5

1.5

x [mm]

z [mm]

0.69mm

(n) Depth difference measurement on a cross-sectional surface proﬁle

(cf. arrow in (m)).

Figure 3: Experimental results for an embossing in a metallic dumbbell consisting of dark cast iron. In (f)–(i), the diffuse

components are plotted with the direction

s of incident light varying over the upper hemisphere, while the specular components

are plotted with the viewing direction~v being varied.

VISAPP 2012 - International Conference on Computer Vision Theory and Applications

118

(a) Scale 1/8SSIM = 0.85962 (b) Scale 1/4SSIM = 0.84261 (c) Scale 1/2SSIM = 0.88546 (d) Scale 1/1SSIM = 0.92896

Figure 4: Details regarding the ﬁnal reﬂectance maps for different scales under illumination from one selected LED. For each

image, the upper half shows the image data and the lower half the reﬂectance map.

Table 1: Determined reﬂectance function parameters for each reconstruction scale and the ﬁnal result.

scale ρ

eff

initial 17.24 0.12 0.0001 1.46 1.85 7.10 9.21

1/8 10.52 0.61 0.45 2.62 1.79 11.75 9.88

1/4 11.59 0.92 0.91 4.83 3.65 11.83 15.03

1/2 12.42 0.98 1.24 5.01 4.05 13.40 18.17

1/1 14.27 1.07 1.45 8.73 7.39 18.22 49.80

The same is true for the correspondences between im-

ages and reﬂectance maps in Fig. 4. An analysis of a

cross-sectional proﬁle of the surface with an indicated

depth difference measurement is shown in Fig. 3(n),

where the reconstructed surface has a depth difference

of 0.69 mm, whereas a tactile reference measurement

with a caliper gauge yields 0.67 ± 0.02 mm.

Figs. 3(e)–3(i) show the estimated reﬂectance

functions for each resolution level. The full re-

ﬂectance function (cf. Fig. 3(e)) is decomposed into

its four components as shown in Figs. 3(f)–3(i), where

a normalization with respect to the effective albedo

eff

has been performed in order to demonstrate the

development of the respective component without the

inﬂuence of the albedo. Note that the diffuse compo-

nents are plotted with the direction

s of incident light

varying over the upper hemisphere, while the specu-

lar components are plotted with the viewing direction

~v being varied. The numerical results for each com-

ponent are listed in Table 1.

The plots show an increasing strength of the spec-

ular reﬂectance components while their widths de-

crease, which causes the characteristic sharp and in-

tense specular reﬂections on the surface apparent at

full resolution. This behavior can also be observed in

the resulting reﬂectance maps shown in Fig. 4. Both

diffuse components are signiﬁcantly lower in their in-

tensities compared to the specular components, which

is the typical behavior of metallic surfaces. In Fig. 4,

the structural similarity (SSIM) measure known from

the domain of video coding (Wang et al., 2004) is

used to illustrate the similarity between the acquired

images and the corresponding reﬂectance maps. The

SSIM is a real number from the interval [0,1] and in-

creases with increasing similarity.

4.3 Validation on Synthetic Data

Since we lack the availability of a ground truth with

the required accuracy, we can only evaluate the ac-

curacy of our approach using synthetically gener-

ated data. For this purpose, we use the result of

the described surface reconstruction algorithm as the

ground truth surface and the obtained reﬂectance

maps as the corresponding synthetic images. Our

algorithm is then initialized with these data, where

Gaussian noise with a standard deviation of 60 µm is

added to imitate the inaccuracies of the range scan-

ner. The obtained results are shown in Fig. 5. Note

that the RMSE of the reconstructed surface with re-

spect to the synthetic ground truth only amounts to

10.2µm, which corresponds to approximately 1/4 of

the lateral pixel extent of 42µm, where the highest

deviations occur near the margin of the reconstructed

surface section.

The reﬂectance function estimated based on the

synthetic data set is shown in Fig. 6. The estimated re-

ﬂectance function resembles the ground truth closely

to within a few percent. Fig. 6 shows the results for

different numbers of subiterations (cf. Fig. 1), which

have a very small effect on the inferred shape of the

reﬂectance function. The surface is reconstructed at

a high accuracy (cf. Fig. 5), and the rendered re-

ﬂectance maps closely resemble the synthetic images

SELF-CONSISTENT 3D SURFACE RECONSTRUCTION AND REFLECTANCE MODEL ESTIMATION OF

METALLIC SURFACES

119

(a) Reconstruction result using synthetic data for validation. (b) Deviations of the reconstructed surface from the synthetic ground truth.

The RMSE amounts to 10.2µm

Figure 5: Validation of the proposed algorithm based on synthetic ground truth data.

used for 3D reconstruction as the SSIM corresponds

to a very high value of 0.985 on the full resolution

scale (cf. Fig. 7).

5 SUMMARY AND

CONCLUSIONS

In this study, we have described an approach that

provides a detailed 3D surface reconstruction along

with simultaneously estimated parameters of the re-

ﬂectance model based on a combination of active

range scanning data for large-scale accuracy with

image-based photometric information for small-scale

accuracy. The simultaneous estimation of the 3D sur-

face proﬁle and the reﬂectance model is incorporated

into a self-consistent computational scheme that suc-

cessively increases in resolution. We have presented

results for a dark rough metallic surface, which has

been reconstructed based on a single coarse 3D scan

and 12 images acquired under different illumination

conditions. The experimental evaluation has shown

that the obtained surface exhibits a high level of visi-

ble detail. A comparison of a depth difference on the

reconstructed 3D surface proﬁle with a simple tac-

tile measurement has shown deviations of the order

of some 10 µm, while a validation based on synthetic

image data has revealed a RMSE of 10.2 µm or about

1/4 of the lateral extent of a pixel. However, a better

validation of the absolute accuracy is still required,

e.g. using data from a highly precise tactile measure-

ment device.

Additionally, it has been shown that the applica-

tion of a parametric reﬂectance model allows to de-

termine the reﬂectance parameters along with the re-

constructed surface. Since the estimated strengths of

−50 0 50 100 150 200 250

100

120

140

160

180

observed intensity (x component)

observed intensity (y component)

Figure 6: Full reﬂectance function determined using syn-

thetic data. Black: ground truth; blue: 4 sub-iterations.

There are no differences visible within the thickness of the

blue line for 1 and 2 sub-iterations.

Figure 7: Full-scale synthetic image (top) and correspond-

ing reﬂectance map (bottom) of the reconstructed surface.

The SSIM amounts to 0.985, thus indicating a very high

similarity.

the specular lobe and the specular spike already in-

crease with increasing resolution level, it might be fa-

VISAPP 2012 - International Conference on Computer Vision Theory and Applications

120

vorable to choose a more suitable reﬂectance model

or to use a data-driven, non-parametric approach to

model the observed complex behaviors and/or to ac-

quire images from several viewpoints. Nevertheless,

the accurate 3D reconstruction results show that the

applied reﬂectance function is suitable for integrating

the image-based photometric information with the ab-

solute depth data.

A somewhat critical aspect lies in the generaliza-

tion of the presented approach with regard to inter-

reﬂections. Currently, the algorithm assumes only

ﬁrst-order reﬂections, which induces errors if inter-

reﬂections occur. Hence, future work will address the

development of a mechanism for the compensation

or the exploitation of the effects of interreﬂections at

specular surfaces.

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