Illuminant-tilt estimation from images of isotropic texture

M.J. Chantler G.B.Delguste

Indexing terms: Three-dimensional textures, Illuminant-tilt estimation

Abstract: The authors present a new illuminant- tilt-angle estimator that works with isotropic textures. It has been developed from Kube and Pentland’s frequency-domain model of images of three-dimensional texture, and is compared with Knill’s spatial-domain estimator. The frequency and spatial-domain theory behind each of the estimators is related via an alternative proof of the basic phenomena that both estimators exploit: that is that the variance of the partial derivative of the image is at a maximum when the partial derivative is taken in the direction of the illuminant’s tilt. Results obtained using both simulated and real textures suggest that the frequency-domain estimator is more accurate.

1 Introduction

The orientation of the illumination incident upon three-dimensional textures has considerable impact on their appearance. The direction of the ground-plane component of the illuminant vector (defined as the illuminant-tilt angle) is particularly important. It is one of the main factors affecting the directional characteristics of images of sidelit or partially sidelit textures. Many images of three-dimensional textures have been captured using at least partial sidelighting in order to enhance the texture, see Brodatz’s album [l] for instance. It is not surprising, therefore, that variation of the illuminant-tilt angle can have disastrous consequences for texture classification schemes that use directional features [2]. The tilt angle is therefore an important factor in texture classification and segmentation [3]. It is also important for shape from shading and other vision tasks.

This paper presents a new algorithm for estimating the illuminant-tilt angle. It exploits Kube and Pentland’s frequency domain model of images of three- dimensional texture [4].

Other tilt estimators have been developed previously [5-81. However, these area-based estimators either assume that the surface is convex or made up of hemispheres. The consequence being that the means of 0 IEE, 1997 IEE Proceedings online no. 19971302 Paper first received 19th August 1996 and in revised form 26th March 1997 The authors are with the Department of Computing and Electrical Engi- neering, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK

the partial derivatives of the image may be assumed to be biased by the illuminant direction, and it is this systematic phenomena that these estimators exploit. Knill [9] points out that such estimators will not work with surfaces that are not predominantly convex. He develops an estimator based upon the variance of the partial derivatives that removes this restriction, and presents results obtained using images of synthetically generated isotropic textures.

In contrast to Knill’s spatial-domain treatment, this paper develops a tilt estimator in the frequency domain. Both algorithms assume that the surface textures are isotropic, They are tested using synthetic and real images of three-dimensional textures.

2 A frequency-domain model of images of three-dimensional texture

This Section presents a frequency-domain model of the image of an illuminated three-dimensional texture. A more detailed derivation of this model may be found in [2 and 31. The theory is summarised here because it is used in the alternative proof of Knill’s estimator pro- vided in Section 3, and because it leads directly to the frequency-domain tilt estimator presented in Section 4.

The model uses theory developed by Kube and Pent- land [4] and further developed by Chantler [2, 31. It provides an expression for the spectrum of an image in terms of the corresponding surface texture’s spectrum and the illuminant vector. The Lambertian reflectance model is expressed in terms of partial derivatives, line- arised and applied to a frequency-domain representa- tion of the surface texture. The normalised image intensity Z(x, y ) of the surface V d x , y ) is

-pcos.rsina - qsin-rsino +cos0 @-T-FTT

l ( z , g ) = n.L =

(1) where n = the unit vector normal to the surface at the point ( X ? Y ) p = dvH/dx q = dvH/dy V d x , y ) is the height-map of the surface L = (cos z. sin 0, sin z. sin a, cos 0) is the unit vector towards the light source cr and z are the illuminant slant and tilt angles, respec- tively [Note 11. Note 1: T h e slant angle (cr) is the angle that the i l l m a n t vector makes with the z-axis, which itself is perpendicular to the ground plane of the three-dimensional texture. Its complement, the angle of tilt (T) is the angle that the illuminant vector makes with the x-axis when projected onto the ground, i.e. x-y-plane

IEE Proc.-Vis. Image Signal Process., Vol. 144, No. 4, August 1997 213

Now, without loss of generality, choose a new axis (x', y ' , z) which is rotated z about the z-axis such that the projection of L onto the x-y-plane will be parallel to the .x' axis. In this new axis system the expression for intensity simplifies to

(2) -7- sin 0 + cos a I ( z , y) = n . L = d7-2 + t 2 + 1

where

Taking the MacLaurin expansion yields I ( z , y ) = ( - rs ina + cosa)

(7-2 + t 2 ) 9(r4 + t 4 ) +...I x [I- 2! + 4!

L

( 3 1 Now, if the surface slope angle is less than 15", then r2, t' << 1, the quadratic and higher order terms may be neglected, and eqn. 3 becomes

Now, the partial derivative r is a linear operator and may be represented in the frequency domain by

I ( r , y ) = (-rsina + cosa) (4)

where w is the angular frequency of the Fourier com- ponent and 8 is its direction w.r.t. the x-axis.

If the mean is ignored then, from eqns. 4 and 5 , the Fourier transform of the image intensity is

FI(W,Q) = F [ -~ 21 sina

= [--id cos(8 - .)FH(w, e)] sin 0 (6) where F d w , 8) is the Fourier transform of the height map of the surface and i represents a 90" phase change.

F I ( w , ~ ) = - - Z ~ F ~ ( ~ , e ) . s i n a . c 0 ~ ( 8 - T ) (7) From eqn. 7 it can be seen that directional characteris- tics of an image of an isotropic surface texture are described by the last term, cos(8 - z). This predicts that the image is directional, that the directionality may be described by a cosine function, and that the highest texture energy lies in the direction of the illuminant tilt,

Hence

z.

Fig. 1 Rendered illtensity image o a synthetically generated isotropic sicrjbee illustrating the directional f'&ing effect predicted bv the linear. model (z= 04 o= 50")

214

This effect can be more clearly understood by examining Figs. 1 and 2 which show polar plots of a synthesised isotropic surface, its rendered image, and the best-fit cosine. These plots were derived from discrete Fourier transforms of the image and surface; each point on a plot being the sum of the magnitudes of the Fourier components in that direction, 8. The plots represent how texture energy is distributed over a 180" range of directions at 1" increments.

2 X 1 O 6

IO6

0 -90 -60 -30 0 30 60 90

angle of frequency components, deg Fig.2 Polar lots and best-@ eosine o j a synthetically generated iso- tropic surjace illstrating the divectioiicilplteriig efect predicted bv the lin- ear model (z = 04 o = 50")

Lambert's cosine rule was used to render height- maps of the isotropic fractal surfaces simulating lumi- nation at z = 0", o = 50". The surface was generated using Fourier filtering [lo] with a power roll-off factor /3 = 3.5, i.e. the random-phase spectrum was generated with the form IFH(w, 8)l = (0/2~)-/3/2.

From Fig. 2 it can be seen that, as predicted by the linear model, eqn. 7, the directional characteristics of the image approximate a cosine function peaking at the illuminant-tilt direction, z = 0". If the direction of tilt is changed then the position of the peak changes accord- ingly [2]. All that would seem to be required to produce a tilt estimator, therefore, would be the FFT, polar plot and cosine-fitting routines used to generate Figs. 1 and 2. The tilt estimate would then be taken from the angular off-set of the best fit cosine.

However, before this estimator is described in detail, Knill's alternative will be presented with reference to the above frequency-domain theory.

3 Knill's illuminant-tilt estimator

In contrast to the theory presented above, Knill develops a tilt estimator entirely in the spatial domain. He first shows that the angle of tilt is that value q~ which maximises the variance of d m r , (the partial derivative of the image in the direction q). He derives an expression for this angle in terms of the x and y partial derivatives and uses it as the basis for his estimator.

Here an alternative proof that the variance of aIldr, reaches a maximum in the direction of the illuminant tilt is developed. It uses the frequency-domain model of image texture developed in the preceding Section. Note, however, that the spatial-domain estimator used in the experiments presented here is that proposed by Knill, i.e. the alternative proof provides the same estimator.

As the proof is based on Kube's linear model, eqn. 7, it has the disadvantage of assuming linearity. However, it is simpler and provides a link between Kube's and Knill's theories.

IEE Pror.-Vis. Image Signal Process.. Vol. 144, No. 4, August 1997

First, an expression for the Fourier transform of partial derivative W a r , must be derived. From eqn. 7,

IWI/%J I =I(-Wcos(8 - cp))F1(w,@)I

=I(-wcos(8 - cp))(-i~F~(~,8).sina.cos(8 -.))I = cos(9 - cp).cos(8 -TI. (w2)FH(W,8)l.sina)

(8) If this process is assumed to be wide-sense stationary and zero mean, the above magnitude spectrum may be related to the expectation of the square of the partial derivative via the power spectral density function S(w, 8).

00 2n

- - J’ J’ S(w,B)d8dw 4n-2

W=O e=o M 2 x

W=O

x 7 [cos(8 - c p ) . cos(8 - .).I2 dedw

(9) e=o

[Note 21 but

[cos(8 - c p ) . cos(8 - 7)]”8

2n

e=o I =I J 1 + cos2(8 - $7) + cos2(8 - T )

= 1 / 1 + - [cos2(28 - p - 7 ) + cos2(r - p)] de

+ COS 2(8 - q) COS 2(Q - T)dO 4

e=o 2x

1 4 2

e=o 27r

(10) 1

4 2 = 1 J’ 1 + - cos 2 (7 - cp)d8

e=o This integral is a maximum when q = z. Thus the expectation of the square of the partial derivative

x j- [ 1 + ~ c o s 2 ( $ 7 - 7 ) I /4d8dw e=o

(11) is also a maximum for cp = z. Therefore an illuminant- tilt estimator may be developed by establishing an Note 2 The Fourier transform of the surface height fmction is repre- sented solely as a function of o to emphasise the fact that it is assumed to be isotropic.

expression which maximises this expectation w.r.t. q. This is exactly what Knill does in [9] and it is repeated here for completeness:

E [

Taking the derivative of eqn. 12 with respect to q, set- ting it equal to zero, and substituting sin(2cp) = 2 sin cp cos q, Knill obtains

(13) 1 E [E) (E)] - tan(2cp) = 2

{ E [(E,’] - E

and the tilt is given by

This provides the basis for a relatively simple and effi- cient method for estimating the illuminant tilt from images of isotropic textures. For digital images it of course requires discrete estimates of the partial deriva- tives, which are known to be problematic at higher sig- nal frequencies and in the presence of noise. Knill uses a seven-point derivative estimator that attenuates the higher frequencies, and therefore goes some way towards alleviating these problems [l 11. Results obtained using Knill’s estimator on synthesised and real images are presented in Section 6. First, however, the details of an alternative estimator are provided which are based upon Kube’s frequency-domain model of texture.

4

A tilt estimator may be derived directly from the frequency-domain model presented in Section 2. Eqn. 7 predicts that the directional characteristics of the Fourier transform of an illuminated isotropic texture are a cosine function of the illuminant tilt angle (cos(8 - z)). Chantler showed that this relationship held for a variety of real and synthesised isotropic textures [3]. Fig. 2 shows this relationship graphically in the form of a polar plot. It was obtained by summing the magnitudes of the coefficients of an FFT at each of a range of angles. Thus a tilt estimator could be produced by implementing an algorithm which locates the angle at which the maximum magnitude occurs in a polar plot. This estimator would be limited to use with images of isotropic texture as is Knill’s.

Such an estimator does not work well in practice because of the noisy nature of FFTs of images of real surfaces. The variance of the coefficients and the spectral leakage introduced by a standard FFT were reduced by using a Welch periodogram [12] coupled with a circular Hann window [13]. The Welch

A frequency-domain illuminant tilt estimator

215 IEE Proc.-Vis. Image Signul Process., Vol. 144, No. 4, August 1997

periodogram used 49 square segments cut with 50% overlap from the original 512 x 512 images. Each of the resulting 128 x 128 segments was windowed using a circularly symmetric Hann window (centred on the centre of the segment) and processed with an FFT. The 49 two-dimensional FFT files were averaged, coefficient by coefficient, to produce a single FFT for input to the polar-plot routine.

This process considerably reduced the spectral leak- age and variance of the coefficients. However, they were still not suitable for direct application of a simple maximum locator. Hence three separate tilt-estimation methods that all use the polar-plot information were investigated [ll].

Method I : The first method convolves the polar plot with its shape as predicted by the linear model, that of an absolute cosine: Icos(8 - z)l. The tilt estimate is taken to be that which occurs at the maximum output of the convolution.

Method 2: The second method was developed because the non-zero elements at 8 = z ~ 9 0 " made the polar plots of real textures resemble a full sinusoid at twice the frequency of the absolute cosine used in method 1. Hence this method convolves the polar plots, minus their mean, with a double frequency sinusoid. As before, the tilt estimate is taken as that which occurs at the maximum output of the convolution.

Method 3: The third method is very similar to method 2. However, Fourier analysis is applied to the polar plots. The Fourier analysis is only performed at a frequency of twice the absolute cosine frequency. The tilt estimate is obtained from the phase calculated for this frequency. That is the tilt estimate is given by

.i = tan-' (:) where

b = J"tP(0). sin (2 e) dB and P(8) is the polar plot. Methods 2 and 3 gave very similar results that were superior to method 1. Method 3 was straightforward to implement and fast so it was chosen over method 2. It was used to produce all the results presented in Section 6. Results for the other two methods are available in

However, before the results are presented the likely effects of terms neglected in the development of the fre- quency-domain model will be investigated.

= J;P(O). COS(^ e) de

[ I l l .

on I i nea r effects

The linear model, eqn. 7, was derived assuming that surface height variance is low and that slant angles do not approach 0". This was necessary to allow Lambert's law to be linearised and applied in the frequency domain. Of particular concern is the fact that linearisation neglects second and higher order terms in eqn. 3 which are at 90" to the angle of tilt. Such terms, if significant, could reduce the directional filtering effect which both the tilt estimation schemes presented above exploit. An insight into the behaviour of these terms may be gained by considering two uni- directional surfaces: surfice $1: containing only components in the direction of z; that is t = 0 over the surface, and

surface s2: containing only components in the direction z - 90"; that is r = 0 over the surface. Now, if the linear and second-order terms of eqn. 3 are considered then

(16)

(17)

r2 2 !

1,1 (x, y) = -r sin CJ - -r cos CJ

t2 2!

Is2(x,y) = ---coscr

Clearly for surface sl, when the slant angle (0) is high, the first term will dominate and the image will be rea- sonably approximated by the linear model. However, for the second surface there is no linear term, only a square term which represents energy at right angles to the illuminant's tilt. This suggests that the texture energy in images of isotropic surfaces will be non-zero at 8 = z *9O0, contrary to that predicted by the linear model. In addition, for slant angles, 0, near 0", i.e. where the source of the illumination is vertically above the texture, the effect of the linear term is reduced com- pared with that of the square terms, and again the directional filtering effect will be attenuated. Further- more, for surfaces with higher height variances, we would expect the energy at 8 = z 290" to be non-zero because of the increased contributions of the second and higher-order terms.

11.106 slant = 70'

12x10

(U U

U 5 & 8 x 1 0

z 5 6 x 1 0 z 0

2 ~ 0 ~

I

-90 -60 -30 0 30 60 90 angle of frequency component,deg

Fig.3 Polar plots of images obtainedfrom an isotropic surface (p = 3.5, height variance = 7) rendered over a range of Illuminant-slant angles, z = 0"

2.10~

-90 -60 -30 0 30 60 90

Fig.4 Polar plots of images obtained by illuminating isotro ic surfaces (p = 3 5 ) with a range of surface height variances (illumimnt-&nt angle U = 5: tilt angle z = 04,

angle of frequency component,deg

These effects can be confirmed through the use of simulation. Fig. 3 shows the effect of decreasing the slant, clearly the directional filtering effect reduces as 0

IEE Proc -Vis Image Signal Process, Vol 144, No 4, August 1997 216

160

8

6

XI % e a E TI

9 2l 120 L

Y

LO

0 20 LO 60 80 Do 120 1LO 160 180 0

tilt ongle,deg Fig.5

~ simulated tilt; slant = 50": 0 slant = 30": + slant = 10": 0 slant = 5"; A slant = 3" Remlts obtained using the .xpectral-tilt e,Ttimator.from a synthetic surface (p = 3.5) illuminated over a range of slant and tilt ang1e.s

approaches 0". Fig. 4 shows the effect of increasing the surface height variance. As predicted, the energy at 8 = z +90° increases as the variance of the surface is increased (as does the energy in the direction of tilt). Furthermore, it can be seen that the variation of the polar plot increases with surface variance, which will tend to obscure the directional filtering effect. Both these effects could adversely affect the tilt-estimation schemes presented above, and hence should be borne in mind when considering their performance.

6 Results

The preceding Sections have presented a new illuminant tilt estimator and Knill's alternative. The performance of both of these estimators on images of synthetic and real textures is now discussed. Of particular concern are the issues raised in the above Section on nonlineararities. Low slant angles (i.e. where the illumination vector approaches the vertical) and large surface height variances were both shown to reduce the directional filtering effect which these estimators rely upon. Hence the experiments presented here cover both effects.

Images generated from simulations were used because of their controllability, in particular, they allowed the systematic investigation of the effects of increased surface height variances. A library of real images collected under controlled lighting conditions was also used. The surface characteristics of these textures could not be easily specified; rather the experiments were limited by the availability of suitable isotropic textures of the right scale that could be painted matte white. However, experiments with real textures provide tests that do not rely on the assumptions made, either implicitly or explicitly, for the simulations.

6.1 Simulation results (spectral estimator) The images used to generate the results presented in this Section were produced via Fourier filtering. Iso- tropic spectra of the desired roll-off (p = 3.5) and ran- dom phase were inverse Fourier transformed to provide height maps of the desired characteristlcs. The height maps were rendered using Lambert's law to pro- vide images of textures with the specified illuminant vector. Shadowing was ignored.

Fig. 5 presents results obtained from applying our estimator to a series of images generated over a range of illuminaat slant and tilt angles. It shows that the estimator is surprisingly accurate but that, as predicted earlier, its performance deteriorates as the slant angle decreases and the directional filtering effect is reduced. These results were obtained using surfaces with modest height variances. Hence, a set of surfaces with a range of height variances was synthesised and used to gener- ate the results displayed in Fig. 6. This bargraph shows that, as expected, the accuracy of the estimator reduces as the surface height variance rises. A height variance of 1.75 corresponds to an average slope angle of approximately 4" while a variance of 700 corresponds to an average slope angle of 48" [2].

r---l n L

g 5

0 variance=175 variance-700 variance=28 0 vanance=700

surface height variance

Fig.6 Standard-error devultions of the spectral estimator m a f i t i o n of Furface height variance and illuminunt-slant angle, synthetic surfaces with p = 3 5

6.2 Simulation results (Knill's estimator) The above set of experiments was repeated using Knill's estimator. Fig. 7 shows that for high slant angles the results are similar to those obtained using the frequency-domain estimator. However, compari- son with Fig. 5 shows that at low slant angles Knill's estimator is outperformed by its frequency-domain counterpart. Fig. 8 presents the results of varying the surface height variance. As before, the estimator's accuracy tends to deteriorate with rougher surfaces and lower slant angles. Comparison with Fig. 6 again shows that the frequency-domain estimator is more accurate for low slant angles.

IEE Proc.-Vk. Image Signal Process., Vol. 144, No. 4, August 1997 217

Fig.7

I

i 160+

I

I

180t P

0 20 LO 60 80 100 120 1LO 160 180 tilt angle, deg

Results obtained using h i l l ' s tilt estimator fiom a synthetic surface (/3 = 3.5) illuminated over a range of slant and tilt angles simulated tilt; slant = SO"; 0 slant = 30"; + slant = IO"; 0 slant = 5"; A slant = 3"

6.3 A comparison using four real textures A set of experiments using real textures was performed to further investigate the performances of the estima- tors. Four textures were chosen that were isotropic in

35 - slant =50"

30.- ms~ant.30' m

c'

P g 25.- oslant=5'

5 20.- U

variance ~1.75 variance :7.00 variance=28.0 variance ~ 7 0 0 surface height variance

Fig.8 Standard-error deviations of h i l l ' s estimator as a function of surface height variance and illuminant-slant angle ( synthetic surfaces with p = 3.5)

appearance. The textures were sprayed matte white to eliminate albedo texture and to provide an approxi- mately Lambertian reflectance characteristic. The tex- ture samples were mounted perpendicularly to the camera's line of sight at a distance of 3.3m, and illumi- nation was provided by a 500W lamp 1.6m from the subject. The tilt angle of the illumination was varied in 10" steps while all other parameters were kept constant. Sample images of the textures are shown in Fig. 9. Fig. 10 summarises the results of this experiment; for each of the textures the frequency domain estimator gives more accurate results.

Figs. 11 and 12 investigate the effect of reducing the illuminant slant. These results confirm those achieved using synthetic images, in that the accuracy of both estimators tends to reduce as the slant approaches the vertical. Again, the results achieved with the frequency- domain tilt estimator are superior to those obtained using Knill's algorithm.

a b C d

Fig.9 source above images) a Tray of butter beans, 'beans 1' b Tray of gravel chips, 'chips 1' c Small piece of conglomerate, 'rock 1' d Tray of beach pebbles, 'stones 1' 218

Sample images of the test textures: top row captured at z = 0" (i.e. the light source to the right of the images); bottom row captured at z = 90" (light

IEE Proc.-Vis. Image Signal Process., Vol. 144, No. 4. August 1997

I beans 1 11 chios 1 rack 1 1 stones 1

Fig. 10 faces ((5 = 50 ”)

spectral estimator 0 Knill’s estimator

Standard-error deviitions of the two estimators using real sur-

60 701 Y

.- F !i LO

0 10 20 30 LO 50 60 70 80 illuminant slant angle, deg

Fig. 11 (real texture images) -.-beans 1 -& chips 1 -+- rock 1 -0- stones 1

Spectral estimator error as a function of illuminant-slant angle

\

0 10 20 30 50 60 70 80 illuminant slant angle,deg

Fig.12 (real texture images) -.- beans 1 -& chips 1 -+- rock 1 -0- stones 1

frill's estimator error us a function of illuminant-slant angle

7 Conclusions

This paper has presented a novel illuminant-tilt estima- tor. Its major restriction is that it must be used on images of isotropic texture. Experiments using synthetic and real textures suggest that its accuracy is superior to that of Knill’s estimator which also requires isotropic textures. In contrast with earlier approaches [5, 6, 81, both the estimator presented here and Knill’s estimator exploit the directional filtering effect of imaging proc- esses that employ illumination which is not perpendicu- lar to the texture’s reference plane. This allows the hemispherical surface assumption to be avoided. How- ever, these directional phenomena are reduced by the use of low illuminant slant angles and surfaces of high height variance. Experiments showed that the accuracy of both estimators was adversely affected by such con- ditions, although it is surprising that reasonable results were still produced at the lower slant angles.

8 Acknowledgments

The authors would like to thank Mr. G. McGunnigle for his help and ideas during this work.

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10 SAUPE, D.: ‘Algorithms for random fractals’ in PEITGEN, H.- O., and SAUPE, D. (Eds.): ‘The science of fractal images’ (Springer Verlag, 1988), pp. 71-1 13

11 DELGUSTE, G.: ‘A comparison of spatial and frequency domain illuminant vector estimators’. Research memorandum RM/96/8, Department of Computing & Electrical Engineering, Heriot-Watt University, Edinburgh, June 1996

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