Better Shading for Better Shape Recovery

Moumen T. El-MelegyElectrical Engineering Department

Assiut University, Assiut 71516, Egyptmoumen@aun.edu.eg

Aly S. Abdelrahim and Aly A. FaragCVIP Lab, University of Louisville

Louisville, KY 40292, USAaly.abdelrahim,aly.farag@louisville.edu

Abstract

The basic idea of shape from shading is to infer theshape of a surface from its shading information in a sin-gle image. Since this problem is ill-posed, a number ofsimplifying assumptions have been often used. Howeverthey rarely hold in practice. This paper presents a simpleshading-correction algorithm that transforms the image toa new image that better satisfies the assumptions typicallyneeded by existing algorithms, thus improving the accuracyof shape recovery. The algorithm takes advantage of somelocal shading measures that have been driven under theseassumptions. The method is successfully evaluated on realdata of human teeth with ground-truth 3D shapes.

1. IntroductionShape from shading (SFS) is a problem that has been

studied for about four decades in the vision literature. Itaims to recover surface orientation from local variationsin measured brightness. To make it solvable, classicalSFS methods often assumed an orthographic camera model,distant single point light source, and a Lambertian sur-face [6, 17, 4]. Unfortunately, those assumptions are notalways valid in reality. As such, the reconstruction resultsof these classical SFS approaches lack accuracy. That iswhy there have been more recent methods trying to relaxsome of these assumptions. For example, Prados et al. [9],Tankus et al. [13], and Yuen et al. [16] replaced the or-thographic camera model by a pinhole camera model per-forming a perspective projection, and they assumed that thelight source is located at the optical centre. Moreover, alight attenuation term is considered in [9]. These ideas havebeen further extended by Ahmed et al. [2], Vogel et al. [14],and Abdelrahim et el [1], where the Lambertian reflectancemodel is replaced by the more realistic model of Oren andNayar [8] or that of Wolff [10]. On the other hand, variousoptimization techniques have been employed to solve theSFS problem, see e.g. [4] for a recent survey.

The observation underpinning this paper is that although

considerable effort has gone into the development of im-proved shape-from-shading methods, there is room for fur-ther development. Here we propose a novel strategy to im-prove the surface recovery results of SFS. Instead of tryingto develop more sophisticated SFS algorithms that can takeinto account the more realistic assumptions of the SFS prob-lem, we investigate the possibility of correcting the shad-ing of the given intensity image such that its deviation fromthe typical SFS assumptions (such as orthographic projec-tion, and Lambertian reflectance model) is reduced. Thisis quite important as several of the most successful existingmethods for SFS are restricted in applicability to Lamber-tian reflectance and orthographic projection, and cannot beadapted easily to work with other assumptions. Hence, ourcorrection process can be used as a simple tool to prepro-cess the input image data for such methods, leading to moreaccurate shape recovery results. To the best of our knowl-edge, this has not been addressed before in the literature.The only exception [11] that we are aware of presented sim-plified formulas to perform radiance correction to the Lam-bertian case. The method is only applicable to the retrore-flection case when the object is illuminated in the viewingdirection, and requires the specific source reflectance modelas well as its parameters be known in advance, which is notalways possible in reality. Additionally, the conversion for-mulas in some cases are not tractable, and a lookup table isused instead.

The shading correction strategy proposed here alleviatesmany of those mentioned drawbacks. It works blindly onthe input image without the knowledge of the original re-flectance model, or the illumination direction relative to theviewing direction. It is based on a number of local shadingmeasures derived from the mathematical model of the im-age shading formation under a typical set of SFS assump-tions. These measures deviate from their theoretical aver-age values when an image does not satisfy those put forthassumptions. These derived measures are used to develop asimple algorithm to map an input image to a new, correctedimage that better satisfies the assumptions typically neededby the SFS algorithms, thus leading to more accurate shape

recovery.In a recent critique [5] of the state-of-the-art of SFS, it is

stated that the majority of SFS methods have been demon-strated on synthetic data, on which results are not gener-ally predictive of results on real data. In addition, only afew SFS methods have compared a shading based recon-struction of an image of a real object with a depth estimate.Nevertheless, a distinct characteristic of this paper is thatthe proposed SFS strategy is evaluated on a large set of realimages of 3D shapes with known ground-truth depth esti-mates. This dataset consists of real human teeth images ofvarious tooth types (maxillary deciduous, maxillary molars,mandibular molars, or mandibular third molars), which arecompiled for a research project on 3D tooth reconstructionfrom intra-oral images. The results of the new SFS strategyapplied to these images are assessed against ground-truthshapes obtained from CT scans of the real human teeth, andcompared with several existing SFS techniques in terms ofaccuracy.

The rest of this paper is organized as follows. Section 2develops our new strategy to improve SFS results based oncorrecting input images. Experimental results are presentedin Section 3, and the paper is concluded in Section 4.

2. Shading Correction for Better SFSThe SFS problem requires the recovery of shape of the

visible surface from a single intensity image I . Our strat-egy here to solve this problem poses this question: Is therea mapping F : R → R that when applied pixel-wise tothe image I , the resulting image F(I) provides better shaperecovery by classical SFS algorithms? An answer to thisquestion calls for addressing two issues. The first one is re-lated to the explicit functional form of this unknown trans-formation F , and how it can be parameterized. One canaddress this precisely if the cause(s) of the deviation of theinput image from the main assumptions typically put forthin the SFS algorithms is(are) known a priori. For example,if this deviation is attributed to an intensity non-linearity,such as γ-correction, the transformation can be explicitlydefined as [7]

F(I) = % I1/γ , (1)

which inverts the process of γ-correction. The constant %is a normalization factor which is determined by the valueof γ. Interestingly, it was shown [7] that the same simpleparametric form of γ-correction in (1) can also account fora rather complicated camera response function disturbingthe common SFS assumptions, regardless of the camera’stemporal gain and spatial non-uniformities. Similarly, anexplicit form of F can be devised if this deviation is due tovignetting [7]. However it is more often that the cause ofthis deviation is unknown, or no explicit form can be envi-sioned for it (e.g., non-Lambertian surfaces). Therefore, to

get around this, we propose to assume a more generic formbased on finite-order Taylor-polynomial

F(I(.)) =p∑k=0

ck Ik(.), (2)

where {ck ∈ R} is a sequence of p-coefficients need to bedetermined, which brings about the second issue.

The second issue is related to how to find the parame-ters ({ck} of this explicit form in (2)). A proper objectivecriterion should be formulated to measure how much an in-tensity image satisfies the typical SFS assumptions. Thesmaller this criterion, the better the image meets these as-sumptions, and accordingly, the better the recovery resultsof the classical SFS algorithms. Deriving such an objectivecriterion is rather difficult in general. However, in the nextsubsection, we formulate a proper objective criterion baseda set of local shading measures derived under typical SFSassumptions.

2.1. Local Shading Measures

In the classical SFS problem [6, 17], under a typical setof assumptions, the goal is to find a 3D Lambertian surfaceZ(X,Y ) satisfying the image irradiance equation

I(x, y) = ρ s · n+ µ, (3)

where I(x, y) is the intensity at the image position (x, y),with ρ being the surface albedo, n the surface normalat the 3D point (X,Y, Z(X,Y )), s the unit vector forthe illuminant direction, and µ an ambient lighting term.We assume no presence of interreflections among sur-face patches. Under the typical assumption of ortho-graphic projection, x = X , and y = Y . So the sur-face could be expressed as Z(x, y). It is often conve-nient to work in spherical coordinates, let’s parameterizes = (cos τ sin γ, sin τ sin γ, cos γ), where τ is the tilt angleand γ is the slant angle. Similarly, let’s express the sur-face normal as n = (cosϕ sinψ, sinϕ sinψ, cosψ), whereϕ = ϕ(X,Y, Z) and ψ = ψ(X,Y, Z) are the tilt and slantangles, respectively, of the surface normal. With these rep-resentations, (3) becomes

I(x, y) = ρ(cos τ sin γ cosϕ sinψ +

sin τ sin γ sinϕ sinψ + cos γ cosψ) + µ. (4)

Following some earlier works (e.g., [17]), assume that atany point (X,Y, Z) the surface could be approximated by aspherical patch:

X = X0 +R0 sinψ cosϕ,

Y = Y0 +R0 sinψ sinϕ,

Z = Z0 +R0 cosψ, (5)

where the sphere center (X0, Y0, Z0) and radius R0 dependon the local surface shape, and both may change from onepoint neighborhood to another. After some computation us-ing (4) and (5), the image derivative Ix at a point (x, y)could be found:

Ix =∂I

∂x=∂I

∂ϕ

∂ϕ

∂x+∂I

∂ψ

∂ψ

∂x

=ρ

R0(cos τ sin γ − cos γ tanψ cosϕ). (6)

Similarly,

Iy =∂I

∂y=

ρ

R0(sin τ sin γ − cos γ tanψ sinϕ),

Ixx =∂Ix∂x

= − ρ cos γ

R20 cos

3 ψ(sin2 ϕ cos2 ψ + cos2 ϕ),

Iyy =∂Iy∂y

= − ρ cos γ

R20 cos

3 ψ(cos2 ϕ cos2 ψ + sin2 ϕ),

Ixy =∂Ix∂y

=ρ cos γ sin 2ϕ

2R20 cos

3 ψ(cos2 ψ − 1),

∇2I = Ixx + Iyy = − ρ cos γ

R20 cos

3 ψ(cos2 ψ + 1). (7)

Then these derived quantities are normalized by the imageLaplacian ∇2I leading to the following ratios at any imagepoint (x, y):

Ixx =Ixx∇2I

=sin2 ϕ cos2 ψ + cos2 ϕ

cos2 ψ + 1,

Iyy =Iyy∇2I

=cos2 ϕ cos2 ψ + sin2 ϕ

cos2 ψ + 1,

Ixy =Ixy∇2I

=sin 2ϕ(1− cos2 ψ)

2(cos2 ψ + 1). (8)

These local measures (8) depend only on the tilt and slantangles of the surface normal at the point; They are notaffected by the illuminant direction, ambient lighting, theradius or the center of the spherical approximation in theneighborhood of the point (this minimizes any error in thislocal spherical approximation assumption). Note also thatIxx + Iyy = 1 for any point.

If the statistical distributions of ϕ and ψ are known, onecould obtain the expected estimates of these local measuresthroughout the whole surface. Such distributions were de-rived in [17] to solve for the illuminant direction. Here weassume the same distributions in order to estimate the ex-pected values of the ratios in (8). The assumed-independentdistributions fϕ and fψ of ϕ and ψ, respectively, are [17]

fϕ =1

2π, 0 ≤ ϕ < 2π; fψ = cosψ, 0 ≤ ψ ≤ π/2. (9)

Using (9), one can evaluate the expected value of Ixx:

Ixx = E{Ixx} =1

2π

∫ 2π

ϕ=0

∫ π/2

ψ=0

Ixx cosψ dϕ dψ = 0.5. (10)

Similarly, Iyy = E{Iyy} = 0.5, and Ixy = E{Ixy} = 0.In simple words, the theoretical average values of the localshading measures Ixx and Iyy are 0.5, while that of thelocal measure Ixy is 0.

The expected values of these local measures can be esti-mated from the images using the arithmetic means. In prac-tice, we employ the derivatives of a Gaussian function asfilters to compute the derivatives of a sampled image func-tion via convolution. For example, Ixx is implemented as

Ixx =1

size(I)

∑x∈I

∑y∈I

I ~GxxI ~Gxx + I ~Gyy

, (11)

where ~ denotes convolution, and Gxx is the second orderpartial derivative of a 2D zero mean Gaussian kernel. Im-age points with zero Laplacian (zero denominator in (11))are discarded from the summations. One can show, theo-retically and experimentally, that these means as computedin (11) have rather small noise sensitivity. Note also that amore robust estimate of the means of these local measurescan be obtained by replacing the arithmetic mean in (11))by the median function.

It is important to stress here that the theoretical averagesof the derived local shading measures have been verifiedfor the set of the 3D teeth surfaces obtained from the CTscans of real human teeth that are used in our experimen-tal results in Section 3. Fig. 1 shows the obtained distri-butions fϕ and fψ of the angles ϕ and ψ of the tooth sur-face’s normals. From those two distributions, the expectedvalues of the shading measures are found as Ixx = 0.507,Iyy = 0.493 and Ixy = 0.001, all of which are quite closeto the theoretical values derived in (10).

2.2. Shading Correction Algorithm

The local measures and their theoretical means are de-rived under typical SFS assumptions. They will start to de-viate from these theoretical means when one of more as-sumptions are unsatisfied. Moreover, it can be shown thatcommon imaging non-linearities, such as intensity and ge-ometric non-linearities (e.g., γ-correction and lens distor-tion), give rise to deviations in the computed local measures.We here use this to formulate an objective criterion in orderto estimate the unknown coefficients in the transformation(2). One way to do this is to define it as

εs = |Ixx(F(I))− 0.5|+ |Ixy(F(I))|, (12)

where Ixx(F(I)) denotes the mean of the shading measurecomputed for the image after transforming its intensities viathe mapping F in (2). Note that we have derived three lo-cal shading measures, but only two are independent. Theabove criterion reflects this fact. In addition, at most twocoefficients in (2) can be solved for. As the measures arealso invariant to linear transformation of image intensities,

(a)

(b)

Figure 1. Distributions fϕ (a) and fψ (b) of the angles ϕ and ψ ofthe tooth surface’s normals as computed from the set of 3D teethsurfaces used in building the shape model.

the coefficient c0 has no affect, and the remaining ones canbe determined up to a common scale. Thus for our pur-pose here, the transformation (2) is written more preciselyas F(I) = I (1 + c′1 I + c′2 I

2), with c′1 = c2/c1 andc′2 = c3/c1.

The proposed shading correction algorithm thus seeks toestimate the two parameters, namely c′1 and c′2, that mini-mize the criterion (12). In our experience, this criterion of-ten has several local minima, so traditional gradient-basedlocal optimization routines may not work properly to findthe global minimum. In order to overcome these drawbackswe use a methodology consisting of two steps: first, deter-mine good initial start values by means of a state-of-the-art global optimization technique and secondly, perform afine-tuning derivative-free simplex search using the previ-ous result as start value. To determine good initial startingvalues we use the method of Coupled Simulated Annealingwith variance control (CSA) [15]. CSA has already provento be more effective than multi-start gradient descent op-timization [12]. Another advantage of CSA is that it usesthe acceptance temperature to control the variance of theacceptance probabilities with a control scheme. This leadsto an improved optimization efficiency because it reducesthe sensitivity of the algorithm to the initialization parame-ters while guiding the optimization process to quasi-optimalruns. This initial result is then used as a starting value fora derivative-free simplex search. This extra step is a fine-

tuning procedure resulting in more optimal tuning param-eters and hence better performance. In our experiments, asearch space of [−2, 2] for either parameter was found sat-isfactory.

Having estimated the transformation parameters, the in-tensities of all pixels in the original input image are mappedinto new values that better satisfy the typical SFS assump-tions. Afterwards, the transformed image can be arbitrarilyscaled within the intensity range [0, 255], if required, as thisscaling does not affect the local shading measures (or theerror criterion).

3. Experimental Results

Several experiments are carried out to evaluate the per-formance of the proposed shading correction strategy. Inorder to quantify the gain out of this strategy, it is importantto apply it on real data whose ground-truth depth surfacesare known. Qualitative results on the popularly-used testimages in the SFS literature (mainly synthetic images) willnot be sufficient or very useful [5] for our purpose here.As such, we have constructed a set of images for real 3Dsurfaces with known depth information based on real hu-man teeth. The accuracy of the SFS outcome is assessedby comparing the resulting surface to the 3D ground-truthsurface as obtained from CT scan. This will allow us morerigorous evaluation of the proposed strategy.

We apply the proposed shading correction algorithm onthe input test images. The algorithm takes about 25s on aPC with 2.2GHz AMD processor and 4GB RAM. Then thecorrected images are fed to a standard SFS algorithm [6]assuming typical assumptions of Lambertian surface, ortho-graphic projection and distant directional light source in or-der to estimate the surfaces’ depth maps. The results arequantitatively compared to the results of the same SFS al-gorithm [6] without the correction process as well as to theresults of a more sophisticated SFS algorithm [2, 9] basedon viscosity solutions for Hamilton-Jacobi type formulationof the image irradiance equation. This algorithm was orig-inally developed in [9] for the Lambertian case assuminga perspective camera projection and near-field illumination.Later [2] it has been extended to the Oren-Nayar (O-N) re-flectance model which more properly describes the rough-ness of the real tooth surface. Table 1 summarizes the keydifferences between the methods under evaluation and com-parison.

An ICP-based rigid registration algorithm [3] is usedto align the 3D ground-truth surface as obtained from CTscan with the surface recovered from the SFS algorithm.The performance accuracy is assessed using the root meansquare (RMS) error between the corresponding 3D pointsfrom both surfaces after the 3D alignment.

Table 1. Summary of Methods Under Experimental Comparison.Camera Light source Reflectance Corrected Reference

A ortho. infinity Lamb. No [6]B pers. optical center O-N No [2, 9]C ortho. infinity Lamb. Yes Ours + [6]

3.1. Dataset

A dataset is constructed from several real invitro teethof various types (maxillary deciduous, maxillary molars,mandibular molars, or mandibular third molars) from adultsubjects with different races, gender and ages. At least30 teeth of each type have been collected. A Cone-beamCT (KODAK 9000 3D Extraoral) scanner at a resolutionof 76 µm is then used to scan the teeth. Figure 2 showssamples of the acquired CT scans. The root part of eachtooth is manually taken out from the CT scan, and the vis-ible 3D tooth surface is segmented. Furthermore, the sur-face roughness of the tooth surfaces which is needed for theO-N diffuse reflection model in Algorithm B is estimated.This is done with the help of a 3D optical surface profiler(NewView 700s from Zygo company), which is based onScanning White-Light Interferometry technology that offersfast, non-contact, accurate 3D metrology of micro surfacefeatures. A near-field imaging setup consisting of a smallcamera with a built-in bright light source is employed to ac-quire the 2D images of the occlusal surfaces for all the teeth.The camera has 1/4 Sony CCD, and an image resolution of2.0 mega pixels.

Figure 2. Sample CT scans for various human tooth types. (First row)Maxillary deciduous teeth. (Second row) Maxillary molars. (Third row)Mandibular molars. (Fourth row) Mandibular third molars.

3.2. Results and Comparisons

Sample results from the methods under evaluation aredemonstrated in Fig. 3. Fig. 3(a) shows the input test imagesfor four different teeth models (maxillary deciduous, maxil-lary molar, mandibular molar, and mandibular third molar),while Fig. 3(b) shows the corresponding ground-truth toothsurface as obtained from CT scans. Fig. 3(c)depicts the im-ages after applying the shading correction algorithm to thesame images in Fig. 3(a). The outputs from Algorithm A,shown in Fig. 3(d), are rather bumpy with too many peaks as

the algorithm fails to handle the inevitable specularity dueto the tooth surface characteristics and the near-illuminationsetup. As it assumes more realistic settings of perspec-tive projection, near light source and the O-N reflectancemodel, Algorithm B has provided rather better global shapereconstructions, see Fig. 3(e). However the algorithm isnot able to recover the geometrical details of the occlusalsurface. Although there are still some bumpy results, ourmethod C shown in Fig. 3(f) has produced better results dueto the shading correction pre-process. This can be verifiedfrom the root-mean-square (RMS) error between the recon-struction and the ground-truth after performing the 3D rigidalignment, which is given beneath each reconstruction inFig. 3(d)-(f). Consistently, the proposed strategy (methodC) of shading correction followed by a simple classical SFSalgorithm provides the smallest RMS errors. The methodhas an error as low as 50−70% of that of the more sophisti-cated SFS algorithm B but without shading correction. Thisemphasizes the role of the shading correction pre-process inshape recovery. It is important to stress here that while theimprovements are fractions of a millimeter, this is consid-ered significant for dental-related applications such as toothimplant and surface analysis.

Figure 3. Tooth reconstruction from three different methods. (a) Imageacquired by the setup camera.(b) Shading-corrected images for those in(a).(c) Ground-truth occlusal surface generated from a CT scan of thetooth. (d) Reconstruction using Algorithm A. (e) Reconstruction usingAlgorithm B. (f) Reconstruction using our proposed method (C). Beneatheach reconstruction is the root-mean-square (RMS) error when comparedto the ground-truth surface from CT.

The overall performance of the various methods on theentire data can be summarized in Table 2, which lists theaverage RMS error in mm over over all the four types ofthe dataset teeth (30 per tooth type) for the three methods.It is clear that our method C has considerably better accu-racy than the other methods. Although method C relies onthe same SFS algorithm [6] used in method A, its overall

error is just 53% of that of method A. It has also an error of63% of that of the more sophisticated SFS algorithm [2, 9]in method B. This highly emphasizes the gain out of theproposed shading correction algorithm.

Table 2. Overall surface reconstruction accuracy (RMS) in mmTooth Type A B Cmaxillary deciduous 1.34 1.11 0.74maxillary molar 0.84 0.85 0.65mandibular molar 1.60 1.20 0.60mandibular third molar 1.06 1.01 0.66

Overall 1.24 1.05 0.66

4. Conclusions

This paper has addressed the problem of correcting theshading of a given intensity image to improve the resultsof shape recovery using classical SFS algorithms. This is anovel alternative strategy to developing more sophisticatedSFS algorithms. To the best of our knowledge, this has beenseldom addressed before in the literature. We have proposeda new simple shading-correction algorithm that transformsthe image to a new image that better satisfies the assump-tions typically needed by existing SFS algorithms. It worksblindly on the input image without the knowledge of theoriginal reflectance model, or the illumination direction rel-ative to the viewing direction. It is based on a number of lo-cal shading measures derived from the mathematical modelof the image shading formation under a typical set of SFSassumptions.

The proposed method is evaluated on real teeth imagesin order to reconstruct the shape of the crowns. Our exper-imental results have shown that the shading-corrected im-ages have indeed improved the reconstruction accuracy con-siderably as compared to ground-truth CT scans (even usingthe same SFS algorithm). Moreover, a simpler SFS algo-rithm [6] applied to the corrected image has demonstrated aclearly better performance than a more sophisticated PDE-based SFS algorithm [2, 9] applied to the original images.Although in this paper, we focused on the well-known SFSalgorithm of Frankot and Chellappa [6] in order to demon-strate the gain out of the proposed shading correction algo-rithm, the correction algorithm can also be used with any ofthe classical SFS algorithms to pre-process the input imagesin order to improve their shape recovery results.Acknowledgements. This work has been supported byKentucky Imaging Technologies and the University ofLouisville.

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