K. Daniilidis, P. Maragos, N. Paragios (Eds.): ECCV 2010, Part IV, LNCS 6314, pp. 115–126, 2010. © Springer-Verlag Berlin Heidelberg 2010

Photometric Stereo from Maximum Feasible Lambertian Reflections

Chanki Yu, Yongduek Seo, and Sang Wook Lee

Department of Media Technology, Sogang University, Seoul, Korea {ckyu,yndk,slee}@sogang.ac.kr

Abstract. We present a Lambertian photometric stereo algorithm robust to specularities and shadows and it is based on a maximum feasible subsystem (Max FS) framework. A Big-M method is developed to obtain the maximum subset of images that satisfy the Lambertian constraint among the whole set of captured photometric stereo images which include non-Lambertian reflections such as specularities and shadows. Our algorithm employs purely algebraic pixel-wise optimization without relying on probabilistic/physical reasoning or initialization, and it guarantees the global optimality. It can be applied to the image sets with the number of images ranging from four to hundreds, and we show that the computation time is reasonably short for a medium number of im-ages (10~100). Experiments are carried out with various objects to demonstrate the effectiveness of the algorithm.

1 Introduction

Photometric stereo has been a subject of extensive research in computer vision since it was first introduced by Woodham [1]. Given a set of at least three images with differ-ent illumination directions, the estimation of surface orientation and albedo is a simple linear problem for Lambertian surfaces. In reality, however, complex object geometry and reflectance result in non-Lambertian behavior of reflections such as specularities and shadows. Although much research has been carried out to develop methods robust to non-Lambertian reflections using more than three images, their presence still poses serious problems to photometric stereo.

Probably the most comprehensive approach would be to account for all the reflec-tions including Lambertian using an appropriate BRDF (bidirectional reflectance distribution function) model and locally estimate albedo and non-Lambertian reflec-tance parameters as well as surface orientations [10][9][12][5]. This approach can extract the richest information from the image set when the BRDF model can physi-cally represent the image irradiance accurately. On the other hand, any limitations in the BRDF model can degrade the results and the local BRDF-based estimation cannot take cast shadows into account. Some methods use the dichromatic model for separat-ing or discounting specular reflections [7][3], but they require that dielectric object surfaces have reasonably strong chromaticities. A reference object has also been used for obtaining BRDF and surface orientations, but shadows are not considered in the recovery [8].

116 C. Yu, Y. Seo, and S.W. Lee

The earliest photometric stereo approaches use photometric reasoning with 4 im-ages to detect non-Lambertian reflections and recover albedo and shape by rejecting the images that are suspected to have non-Lambertian reflections. Coleman and Jain [2] used a set of 4 images, computed 4 albedos using combinations of 3 images and detected the presence of specular reflections if the computed albedos are substantially inconsistent. The combination that results in the lowest albedo value is taken as the set of Lambertian image irradiance. Barsky and Petrou extended this approach to shadows but with the help of color information [4]. For Lambertian photometric ste-reo with 4 light sources, Chandraker et al. employed a graph cut-based MRF (Markov Random Field) algorithm to detect light source visibility and recovered shape discarding shadows [6].

Four images hardly suffice to fully sort out non-Lambertian reflections for an object with complex shape and BRDF. Some robust photometric stereo algorithms have re-cently been developed that use a large number (>100) of dense images. Wu et al. em-ployed two MRF inference algorithms (graph cut and tensor belief propagation) for dense photometric stereo [13], but the smoothness constraints on surface orientations diminish fine details and albedos are not recovered. Wu and Tang used EM (Expectation Maximi-zation) to compute albedo while simultaneously clustering a set of initial normals ob-tained through ratio images [14]. The convergence of EM algorithms is well known, but their performance is influenced by initial conditions. They used the highest 50% intensi-ties as numerators of the ratio images to perform plane fitting. Verbiest and Van Gool adopted an ML (Maximum Likelihood) framework and modeled their inlier map as an MRF with an associated Gibbs-prior distribution [15]. They also took the 50% highest intensities as inliers to provide the prior distribution for their EM optimization.

In this paper, we present a general extension of photometric stereo with from small to large number of images based on a maximum feasible subsystem (Max FS) frame-work. The Max FS problem is to find the maximum cardinality feasible subset in an infeasible set of linear constraints such as the Lambertian constraints [19]. By solving this problem, we can obtain the maximum subset of images that satisfy the Lamber-tian constraint among the whole set of images which include specular reflections and shadows. Until recently there has been little development of algorithms for actually solving the MAX FS problem, but a small number of methods are now becoming avail-able, with more under development, inspired by several applications in fields such as radiation therapy planning, machine learning, signal processing, etc.

We use the Big-M method, a mixed-integer exact formulation, for solving the Max FS problem, and show that it is appropriate for the photometric stereo problem [20]. Unlike the dense photometric stereo methods mentioned above, the presented method is based on purely algebraic pixel-wise optimization without relying on the probabilis-tic/physical reasoning or initialization and it guarantees the global optimality. It re-quires only one parameter that can be determined once for a system depending on the image noise/distortion level. The presented algorithm does not require the redundancy that a dense image set can provide. Therefore, the number of images the presented algorithm can be as small as 4 and as large as hundreds. For a medium number of images (10~100), the computation is reasonably fast.

In multiview geometry, the SOI (sum of infeasibilities) has mainly been used for global minimization [18]. Recently, Li has developed a global optimization method for the algebraic DLT(Direct Linear Transformation) problem that has fixed bounded variables [17]. He suggeted an exact bilinear-MIP(Mixed Integer Programming)

Photometric Stereo from Maximum Feasible Lambertian Reflections 117

formulation and obtained globally optimal solutions using an LP(Linear Programming)-based BnB(Branch-and-Bound) algorithm. Although the Li’s work is a somewhat dif-ferent approach to a geometric problem than our MaxFS/Big-M approach to a photo-metric problem, the objectives are the same.

It may be noted that RANSAC [16] has been used for photometric stereo and ap-pearance-based approach [26][27]. The main difference between the RANSAC and the Max FS lies in the global optimality. The RANSAC finds some of the inliers, but not all. As RANSAC is a non-deterministic heuristic algorithm, it provides no guaran-tee to the optimality of its solution in terms of maximizing the feasible set's cardinal-ity. Li performed some RANSAC experiments and observed significant variation of results that cannot be eliminated by further nonlinear refinement [17].

The rest of this paper is organized as follows. Section 2 presents a Max FS formu-lation of Lambertian photometric stereo, and experimental results are presented in Section 3, and we conclude in Section 4.

2 Max FS Formulation for Photometric Stereo

2.1 Lambertian Photometric Stereo

In photometric stereo, surface normal vector and albedo are estimated from images captured under multiple illumination directions. We consider only calibrated photo-metric stereo where light directions and intensities are known. Images are captured at a fixed viewpoint under orthographic projection.

For the Lambertian reflection model with k illumination directions, the image ir-radiance I = [ I1 I2 … Ik ]

T∈ ℜk) at a pixel is given by:

( ) ,nLNLI ⋅=⋅= ρ (1)

where ρ is the albedo, N is the surface normal ([Nx Ny Nz]T∈ ℜ3), L is the illumina-

tion unit vectors ([L1 L2 … Lk ]∈ℜ k×3). Attached shadows are not considered in Equation 1.

We estimate the scaled normal vector n∈ℜ3 which is the normal vector N multi-plied by the scalar value ρ. When a grey-scale image is captured in m bits, the scalar value ρ ranges from 0 to 2m−1 and the scaled surface normal n has a range of value of nlb~nub (nub = 2m −1, nlb = −nub ). When the n is estimated, N and ρ can be easily ob-tained as follows:

.n,nn

N == ρ

(2)

In the Max FS formulation described below, Lambertian pixels are taken as inliers and non-Lambertian specular/shadowed pixels are as outliers. If we could select Lamber-tian pixels from input data, we could estimate the accurate scaled normal vector.

2.2 Maximum Feasible Subsystem Problem

The goal of a Max-FS algorithm is to find the largest cardinality set that constraints are feasible. In other words, it seeks a feasible subsystem containing the largest

118 C. Yu, Y. Seo, and S.W. Lee

number of inequalities for an infeasible linear system Ax ≤ b with real matrix A∈ℜk×n, real vector b∈ℜk and variable x∈ℜn. The Max-FS problem admits the following mixed integer linear programming (MILP) formulation by introducing a binary variable yi for each of the inequalities:

{ } , 10

subject to

min

1

1

1,...,kiy

iyMbxa

y

in

iii

n

jjij

k

ii

=∈ℜ∈

∀+≤∑

∑

=

=

,,,x

,

y ,x

(4)

where Mi is a large positive number that converts an infeasible inequality into a feasible one when yi = 1. The case where yi = 0 indicates that the inequality is feasible. Note that if yi = 1, then the i

th constraint is automatically deactivated. This MILP formulation is known as the Big-M method [19][20].

The performance of the Big-M method varies depending on the constant Mi. With tighter Mi, the method produces better results. If there are fixed bounds on the vari-able x (xlb ≤ x ≤ xub), we can compute a proper Mi using the following equation and obtain a tight formulation:

.,max 1,...,njxxxbxaM ubjj

lbj

n

jjiji =≤≤

⎭⎬⎫

⎩⎨⎧

−= ∑=

, i1

(5)

Generally, MILP problem are solved by the LP-based BnB(Linear Programming-based Branch and Bound ) or the LP-based BnC (LP-based Branch and Cut). LP-base BnB/BnC guarantees the global optimality of its solution [17][21].

2.3 Max FS ( Big-M MILP ) Formulation for Photometric Stereo

By taking the pixels from Lambertian reflections as inliers and taking those from non-Lambertian reflections as outliers, we can formulate the photometric stereo problem with non-Lambertian reflections as a Max-FS problem. The set of input data (D) is partitioned into the inlier-set DI (Lambertian pixels) and the outlier-set DO (specular and shadowed pixels) with DI ⊆ D, DO⊆ D, DI ∪ DO = D and DI ∩ DO = Ø.

A maximum noise magnitude (or tolerance) ε > 0 provides a bound for the equation Li�n and image intensity Ii:

.nL ε ≤−⋅ ii I (6)

The Max-FS formulation of Equation 6 using the big-M method is as follows:

{ }.32 ,

10

subject to

min

i

1

,1,jnnn

1,...,kiy

iyMI

y

ubj

lb

i

iii

k

ii

=≤≤

=∈

∀+≤−⋅

∑=

,,

,nL εy , n

(7)

Photometric Stereo from Maximum Feasible Lambertian Reflections 119

Fig. 1. Five synthetic images of a sphere

If yi = 1, the i

th data is specular or shadowed pixel. If yi =0, on the other hand, the i th

constraint satisfies the Lambertian condition and the pixel is an inlier. We can calculate the proper Mi from Equation 5. For an m-bit grey-scale image,

however, the captured image intensities are limited to nub = 2m−1 and so is | Li�

·�

n − Ii |. Therefore, we can determine the tight fixed bound considering the tolerance value

(ε) as follows:

., inM ubi ∀+= ε

(8)

The constraint equation (| Li n Ii | + Mi yi) of Equation 7 can now be replaced

by the following two inequalities:

.nL,nL iiiiiiii yMIyMI +≤+⋅−+≤−⋅ εε

(9)

If attached shadows are considered, the Lambertian reflection becomes as follows:

( ) .n,LmaxI 0 ⋅= (10)

Intensities should always be non-negative. However, Equation 9 allows that a Li ⋅ n

have the small negative value (−ε < Li ⋅ n < 0) near the boundaries of attached shadows. To solve the this problem, we add the non-negative constraints with respect to the

tolerance value ε as follows:

.nL iii yM−≥⋅ ε2

(11)

This constraint prevents shadowed pixels from being chosen as a maximum feasible set. With the constraints shown in Equations 9 and 11, the photometric stereo with the maximum feasible Lambertian reflections is given as follows:

{ } . 10

32 ,

2

subject to

min1

1,...,kiy

,1,jnnn

iyM

iIyM

iIyM

y

i

ubj

lb

iii

iiii

iiii

k

ii

=∈

=≤≤

∀−≤−⋅−∀+−≤−⋅−

∀+≤−⋅

∑=

,,

,

,nL

,nL

,nL

εε

εy , n

(12)

Note that our method does not require initial values of parameters except the tolerance value ε. Since it is a mixed-integer programming problem, we use a BnC algorithm in the GLPK (GNU Linear Programming Kit) as a solver. The GLPK package is written

120 C. Yu, Y. Seo, and S.W. Lee

in ANSI C and developed for solving large-scale linear programming (LP) and mixed integer linear programming (MILP) problems. An LP-based BnC algorithm consists of a cutting plane method and an LP-based BnB algorithm. It provides constraints by plane cutting to the BnB algorithm for faster convergence.

3 Experimental Results

3.1 Experiment with a Synthetic Sphere

For an experiment with known ground truth, we used images generated from a synthetic sphere. For simulation, a variant of the Torrance-sparrow model is adopted with the surface roughness 0.07. We sampled the illumination space for zenith angles up to 60° with approximately uniform projected solid angle and generated 196 images. We carried out experiments for several subsets of 196 images. The captured RGB images have a 24-bit color depth and we used simple grey-scale conversion I = (R+G+B)/3. The upper and lower bounds of the scaled normal vector elements are assigned to 255 and -255.

Table 1 shows the running times per pixel for 12, 24, 36, 48 and 60 random sam-ples of 196 images. Table 2 shows the running times and the average angular errors of computed surface normals in degrees for several tolerance values. The error between the estimation and the ground truth of surface normal is computed as follows:

( ) ,nncos∑=

− ⋅=pN

i

ei

gi

pNe

1

11

where Np is the total number of pixels, gin

and ein are the ground-truth and the

estimated surface normal vectors at pixel i, respectively.

Table 1. Summary of running times (tolerance ε =5)

Number of images 12 24 36 48 60

Average running time per pixel [msec]

7.834 35.474 225.568 985.625 5,624.73

Table 2. Running times and angular errors for experiment with 12 and 24 images

- Average running time per pixel. [unit: msec]

ε =1 ε =2 ε =3 ε =4 ε =5

12 images 10.073 9.319 8.762 8.305 7.834 24 images 45.657 40.577 37.863 36.051 35.232

- Average angular errors of recovered surface normals [unit: degree]

ε =1 ε =2 ε =3 ε =4 ε =5

12 images 0.138 o 0.158o 0.178o 0.199o 0.219o 24 images 0.085o 0.102o 0.118o 0.137o 0.153o

Photometric Stereo from Maximum Feasible Lambertian Reflections 121

The experiments were run on an Intel Core™2 CPU 6300, 1.87GHz with 4GB RAM and our algorithm is implemented with C language using the GLPK library which provides functions for mixed-integer programming [21]. The scaled normal vector is estimated using a least-squares method with the inlier image set estimated by the Max-FS algorithm. For these medium-sized dataset, the computation is reasonably fast and the normal errors are very small.

3.2 Experiments with Real Objects

We test the performance of our algorithm using four data sets captured by equipment. Illumination sources are made of the white LED lights and they are calibrated. The raw image data obtained with Nikon D-200 camera are used, and we convert the raw images to linear 24-bit RGB images (8 bits per channel) for processing.

Figure 2 show the results from baseball images: (a-c) show 3 images among 19 im-ages used by our algorithm, (e-g) show the 3 binary feasibility maps obtain from our

Fig. 2. (a~c) three images of a baseball, (e~g) binary feasibility maps for the images shown in (a~c), respectively, (i) estimated albedo, (j) color-encoded normal map, (k) inlier map, (d) rendered view of recovered 3D shape, (h) rendered view of 3D shape textured only with Lam-bertian albedo, and (l) environmental lighting effects shown on metallic surface.

122 C. Yu, Y. Seo, and S.W. Lee

Fig. 3. (a~c) three images of a leaf, (e~g) binary feasibility maps for the images shown in (a~c), respectively, (i) estimated albedo, (j) color-encoded normal map, (k) inlier map, (d) rendered view of recovered 3D shape, and (h,l) rendered view of 3D shape textured only with Lamber-tian albedo.

Max-FS algorithm for the 3 images shown in (a~c), respectively, where inliers in each image are denoted in white and outliers in red, (i) shows estimated albedo, and (j) shows color-coded surface normals. This color-coded map of surface normals is obtained by the RGB encoding: [R, G, B] = 255 [(nx+1)/2, (ny+1)/2, nz]. Figure 2 (k) shows the inlier map computed by the Max-FS algorithm. The regions with more than 50% of inliers are shown in blue, and those with less than 50% are in red, and those with less than 3 inliers are in green. (Only a few pixels are on green, but not clearly visible.) The amount of inliers is denoted by the brightness in each region. Figure 2 (d) shows the rendered view of the reconstructed surface, (h) shows the rendered view texture-mapped only with the Lambertian albedo shown in (i), and (l) shows the environmental lighting effects shown on metallic surface. The images (d), (h) and (l) show that the details of the complex concave regions of the baseball seams are recovered properly. We used open source code to recover the shape [22][23].

Figure 3 shows the results from a glossy leaf which has some anisotropic non-Lambertian reflectances. Figure 4 shows the results from a picture frame with com-plex shape and various materials, and the results from a highly glossy teapot is shown in Figure 5.

For the experiments with the baseball (Figure 2) and the teapot (Figure 5), we used 19 images with viewing direction (θ, φ) = (0, 0), and the illumination angles were given as follows. For θ = 30° and 60°, φ ranges from 45° to 315° at an interval of 90°. For θ = 45°, φ ranges from 30° to 330° at an interval of 30°. For the experiments with the leaf (Figure 3) and the picture frame (Figure 4), we used 15 images with viewing

Photometric Stereo from Maximum Feasible Lambertian Reflections 123

direction (θ, φ) = (0, 0), and the illumination angles were given as follows. For θ = 30°, φ ranges from 45° to 315° at an interval of 90°. For θ = 45°, φ ranges from 30° to 330° at an interval of 30°. The running times for the four real datasets are shown in Table 3.

Fig. 4. (a, b) two images of a picture frame, binary feasibility maps for the images shown in (a, b), respectively, (i) estimated albedo, (j) color-encoded normal map, (k) inlier map, (c, d) ren-dered views of recovered 3D shape, (g, h) rendered views of 3D shape textured only with Lam-bertian albedo, and (l) environmental lighting effects shown on metallic surface.

Table 3. Summary of experiments with four datasets

Running time per pixel [unit: msec]

Data set Baseball Leaf Picture frame

Teapot

# of images 19 15 15 19

tolerance ε 6 4 11 11 Average running time per pixel.

[msec] 48.205 40.673 9.882 29.692

124 C. Yu, Y. Seo, and S.W. Lee

Fig. 5. (a~c) three images of a teapot, (e~g) binary feasibility maps for the images shown in (a~c), respectively, (i) estimated albedo, (j) color-encoded normal map, (k) inlier map, (d) rendered view of recovered 3D shape, (h) rendered view of 3D shape textured only with Lam-bertian albedo, and (l) environmental lighting effects shown on metallic surface

4 Conclusion

The recent development of Max FS methods and software can be very useful for many computer vision problems, and it is shown in this paper that they are highly suitable for distinguishing non-Lambertian reflections from Lambertian in an image set that has specular reflections are shadows. For the photometric stereo problem, the irradiance and albedo values captured images are strictly bounded and thus the Big-M method is very effective. The presented algorithm is highly efficacious in rejecting non-Lambertian components of reflections due to weak specular reflections and shad-ows. It requires only one parameter that represents the system noise and distortion and does not rely on probabilistic reasoning or initial conditions. The system noise can be possibly estimated automatically from captured images [25].

We have yet to experimentally compare our algorithm with one of the recent dense photometric stereo methods. The dense photometric stereo with more than a hundred images can benefit from surface normal averaging effect and thus higher signal-to-noise results can be obtained. If this effect is of interest, we may simply capture

Photometric Stereo from Maximum Feasible Lambertian Reflections 125

several images for each light direction and average them for the similar results. This does not increase the computation time substantially.

Our future work includes the development of methods for very rough surfaces where non-Lambertian reflections from specularities are wide spread spatially. Rougher a surface is, fewer Lambertian reflections are observed in general. For the rough surface where less than three Lambertian reflections are exposed, we are inter-ested in developing methods that utilize spatial constraints from the neighborhood to solve the two-image photometric stereo [24].

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