# Recovering Orientation of a Textured Planar Surface Using Wavelet Transform

**ABSTRACT** Shape from texture has received a great deal of attention in the past few decades. This paper analyzes the spectral variations of texture spatial frequencies as a function of orientation and depth of a 3-D planar surface. Based on this relationship we attempt to derive an expression for the extraction of 3-D surface orientation using texture features alone. Using experimentation on simulated texture images, we illustrate the advantage of using 1-D wavelets for this purpose.

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**ABSTRACT:**In this correspondence, we propose a direct method for estimating the orientation of a plane from a single view under perspective projection. Assuming that the underlying planar texture has random phase, we show that the nonlinearities introduced by perspective projection lead to higher order correlations in the frequency domain. We also empirically show that these correlations are proportional to the orientation of the plane. Minimization of these correlations, using tools from polyspectral analysis, yields the orientation of the plane. We show the efficacy of this technique on synthetic and natural images.IEEE Transactions on Image Processing 09/2007; 16(8):2154-60. · 3.20 Impact Factor

Page 1

Recovering Orientation of a textured planar surface

using wavelet transform

Thomas Greiner

School of Engineering, University of Applied Sciences

Tiefenbronner Str. 65, D-75175 Pforzheim, Germany.

Email: Tgreiner@fh-pforzheim.de

Sukhendu Das

School of Engineering, University of Applied Sciences

Tiefenbronner Str. 65, D-75175 Pforzheim, Germany.

Email: sdas@fh-pforzheim.de

Abstract: Shape from texture has received a great deal of

attention in the past few decades. This paper analyzes the

spectral variations of texture spatial frequencies as a

function of orientation and depth of a 3-D planar surface.

Based on this relationship we attempt to derive an

expression for the extraction of 3-D surface orientation

using texture features alone. Using experimentation on

simulated texture images, we illustrate the advantage of

using 1-D wavelets for this purpose.

Key Words: Texture, orientation, wavelet, spectral energy,

separable analysis, variance

1. Introduction

In the field of computer vision and pattern recognition, the

problem of extracting 3-D surface orientation from a

monocular texture image has received a great deal of

attention [1-3, 6, 7, 10, 11, 14-16, 18, 19]. If the surface

exhibits a textured pattern, it is often easy for a human

being to extract the structure information from the image.

But the problem is ill-posed for a machine to solve. This

paper presents analytical expressions using separable 1-D

analysis of 2-D images for extraction of surface orientation

of a textured planar surface.

In this paper we use a model of the viewing geometry

which is similar to that used by Nayar et. al. [4, 12, 13] in

Curet database {www.cs.columbia.edu/CAVE/curet}, but

different from the one used by Super and Bovik [18-19],

Ribeiro and Hancock [14] and Leung and Malik [11]. Dana

and Nayar [4, 12] however proposed this model of the

viewing geometry for measurements of surface reflectance

of textured surfaces, based of different viewing angles and

illumination. A few others [8, 17] have proposed

visualization of surfaces using 3-D textures. The work by

Malik [11] used density, height and occlusion to derive the

shape from textures. Bovik [18], and Hancock [14] in their

work have attempted to extract surface orientation using

spectral gradient, peaks and distortion. Most of the earlier

work [1, 2, 6, 7, 18, 19] involved an exhaustive numerical

search. Ribeiro and Hancock [14] uses the eigenstructure of

an affine distortion matrix to extract orientation. The most

recent work by Clerc and Mallat [3] uses Warplets (affine

transformation of the mother wavelets) to recover shape

using statistical estimates. Texture is realized as a stochastic

process. Deformation gradient is estimated using the

‘texture gradient equation’ which models the ‘Warpogram’

(variance of the wavelet coefficients) of the image.

We present a simpler algorithm and analysis based on 1-D

wavelets to study the spectral variations in a textured planar

surface as a function of its depth and orientation. The

advantage with our method will be the separable analysis in

both dimensions of the image to extract the individual

components of the surface orientation parameters. This will

be evident in the next section, when we present the viewing

geometry and derive the analytical expressions. Due to

separable analysis, errors in computation of one of the

orientation angles will not effect the other, which is a

drawback in the method suggested in [14]. We then present

the need and use of a multi-resolution filter (wavelet) for

spectral analysis of texture surfaces to extract the surface

orientation.

2. Basic texture projective equations

Figure 1. shows the viewing geometry and coordinate

system used. Consider a surface element S, containing a

simple sinusoidal texture. This simplicity is assumed to

derive the relationship between spectral features of the

texture pattern in the image space and 3-D surface

parameters (depth and surface orientation) of S.

Figure 1. Viewing geometry and coordinate system.

Let N be the surface normal on S. This vector is defined

using polar and azimuth angles of N w.r.t the world

coordinate system. The azimuth angle φ, is the angle

between the vector N and its projection, V, on the

horizontal ZW-XW plane. The polar angle, θ (in the ZW-XW

plane), is the angle between the vector V and ZW axis,

clock-wise looking along the Y-axis from the origin. This

model is similar to that used in the CURET database [4,

12]. The view axis is along the –ZW axis and F be the focal

length of the viewing system, assuming a pin-hole camera

configuration and perspective geometry of the viewing

setup. The image plane (2-D) coordinate system is aligned

with the XW-YW axis of the world-coordinate system.

Assume a simple sinusoidal texture pattern on a planar

surface with frequency fr. Let the surface S be at a distance

Z0, from the origin. Let the surface inclination be such that

either θ or φ is zero and the other be a non-zero value

Page 2

α. The advantage of the viewing geometry can be seen

here. Local spectral gradient or variations along the x-axis

(horizontal direction) of the image plane vanish when

θ = 0. Similarly, when φ = 0, the spectral variations of the

texture along the y-axis (vertical direction) of the image

plane is zero.

When only one of the components of the angles of the

surface orientation is non-zero (say, φ), and the other zero,

the local spectral variations will be only along one of the

principal axis (in this case, y-axis) of the image plane. Let

the frequency content of the sinusoidal texture on a planar

surface, at a distance Z0 from the origin and oriented

orthogonal to the viewing direction, as observed in the

image space be a local spectral peak at fr. Let fr be assumed

to be known initially (this constraints will be relaxed later

on). Assume orthogonal projection initially, without any

loss of generality. Perspective projection and effect of depth

will be considered and incorporated next. If the surface is

inclined such that, θ = 0 and φ = α, then the observed

frequency peak will be, fr’ = fr sec(α). (1)

This relation is illustrated in figure 2. Eqn. (1) gives the

relation between the observed frequency of the texture

surface and inclination of the surface w.r.t. viewing

direction. Thus if θ = β and φ = α, then the observed

frequency will be, fr” = fr sec(β). sec(α) (2)

???

α

??

S

S

(A)(B)

Figure 2. (A) A simple sinusoid texture pattern on a planar

surface S, oriented orthogonal to the viewing direction (θ =

φ = 0). (B) The surface is oriented at φ = α and θ = 0. The

projection of one period of the sinusoid is, T’ = T

cos(α) . Orthogonal projection is assumed

sinusoidal texture pattern is shown as a dotted curve on the

planar surface S.

???????????

Let us now observe the effect of depth of the surface from

the viewer on the frequency peak. From figure 3(A), we can

write using perspective projection models, the following

equation:

H/Z0 = h/F

where H is the time period of the sinusoid on the object

surface, and h (= 1/ fr ) is the observed time period on the

image plane. If the surface is moved away from the viewer

by a distance ∆Z, then

H/Z’ = h’/F

where Z’ = Z0 + ∆Z, and h’ is the observed time period of

the sinusoid under perspective projection. Combining

equations (3) and (4), we have:

(3)

(4)

00

0

0

1)1 (

)('

'

Z

Z

h

Z

Z

Z

FH

ZZ

FH

+

Z

FH

h

∆

+

=

∆

+

=

∆

==

(5)

The observed frequency, fz, now is:

Z

∆

+

==

)1 (

r

1

'

1

h

0

0

Z

Z

f

h

Z

f

z

∆

+=

(6)

?

??

?????

?

∆

?????????

?! "$#?%

&('*),+.-(/?0?1

2

354

6

6?7

89;:

<>=.?

S

S

@ A?B$C?D

E!FB$GHD

-ZW

-ZW

YW

YW

Figure 3. A planar object surface, S, with a simple

sinusoidal texture pattern and oriented orthogonal to the

viewing direction is projected on the image plane. (A)

Surface is at a distance Z0 from the camera and length of

the projected segment is h. (B) Surface is at a distance (Z0 +

∆Z) from the camera and length of the projected segment is

h’. F is the focal length of the camera.

From equations (2) and (6), we get the locally observed

frequency foi, of a planar surface at depth (Z0 + ∆Zi), and

orientation θ = β, φ = α, as:

Z

ff

roi

+=

))(sec

α

)(sec(

))(sec

α

)(sec1 (

0

0

β

β

Z

Z

f

Z

i

r

i

=

∆

(7)

where, Zi = Z0 + ∆Zi.

Equation 7 is the basic equation for the observed texture

frequency in the image plane, depending on the surface

parameters (depth (Z0 + ∆Zi) and orientation α, β). The

plot in Figure 4, illustrates the nature of the frequency

variation, given in equation (7), as a function of relative

depth (∆Zi/Z0) and one of the orientation angles (α or

β), where the normalized observed frequency is (foi/fr).

If any texture pattern can be considered as a superposition

of several sinusoids (band-limited), then all the individual

components of the signal will also be effected in a similar

manner as in equation (7). We will now derive equations

which estimate these parameters from the observed

frequency foi, on the image plane. Henceforth, the term

Page 3

‘frequency’ will mean the observed local spectral peak of

the texture around a neighborhood of a point in the image

plane.

3. Method to estimate orientation parameters

Observe Figure 5, which shows the perspective geometry of

a planar surface S with orientation φ = α and θ = 0. Select

two points I and J, on the vertical axis on the image plane,

with coordinates (xi, yi) and (xi, yj) respectively. These may

be considered to be the projections on the image plane of

points Pi and Pj on S, with depth Zi and Zj respectively.

Figure 4. Normalized observed frequency (foi/fr), in the

image texture pattern, as a function of relative depth ∆Zi

and orientation angle α.

In equation(7), substituting φ = α and θ = 0 at a point (xi, yi)

in the image plane, we have:

Z

ff

roi

+=

) )(sec1 (

0

α

Z

i

∆

(8)

and at a point (xi, yj) in the image plane:

Z

ff

roj

+=

) )(sec1 (

0

α

Z

j

∆

(9)

Since the polar angle θ = 0, the spectral gradient along the

horizontal (x) axis of the image is zero. Hence we look for

the variations in spectral values along the vertical (y) axis

of the image plane. This role is reversed if φ = 0 and θ = β.

From equations (8) and (9), we can obtain the difference in

the observed frequencies as:

Z

ffff

rpoiojo

=−=∆

0

,

Z

ij

ji

∆

(10)

where, ∆Zij = Zj - Zi (= ∆Zj -∆Zi ) and frp = fr (sec α).

Setting ∆Zi = -∆Zj = -∆Zij /2

We have, ∆Zij = Zj – Zi = ∆Zj -∆Zi = 2∆Zj = -2∆Zi

(11)

Let the factor (∆Zij/Z0) be represented by K. K denotes the

relative difference in depth of two points (Pi and Pj) on S,

which are equidistant from a reference point P0 with depth

Z0. A method to identify such a pair of points from the

image plane, will be discussed later.

From the constraint given in equation (11) , we can write

from equations (8) and (9):

rpji

fff

2

00

=+

(12)

Using equations (10) and (12) we can write:

f

f

f

]

1

+

][ 2

00

0000

ij

ij

rp

1

ij

ff

fff

K

+

−

=

−

=

i

j

d

d

d

f

f

fwhere

f

0

0

,[ 2

=

−

=

(13)

?

?

∆

∆

?

∆

?

?

?

∆

∆

?

∆

?

?

?

?

??

-ZW

YW

α

??

?

?

?

?

?

?

??

I

J

? ???????

????????

Figure 5. Surface S is oriented at an angle φ=α. Two points

I and J, selected on the image plane correspond to points Pi

and Pj on the surface. The viewing axis intersects the

surface at P0, with depth Z0. The image plane is orthogonal

to the view direction, and is viewed in the figure as a line.

It is necessary to identify a pair of points on the image

plane which are projections of a pair of points, say P1 and

P2, on S. Once this is done, K may be computed from the

corresponding spectral values, fo1 and fo2, at the projections

of P1 and P2 on the image plane.

Next we discuss, how K is used to obtain the surface

orientation. From Figure 5 select two appropriate points on

the vertical axis on the image plane, with coordinates (xi, yi)

and (xi, yj) (algorithm for the selection of this pair of

appropriate points, is discussed later on in this section).

These may be considered to be the projections of points Pi

and Pj on the surface S.

From figure 5, using similarity of triangles we can write:

)(tan

ijijij

Z

Z

−

2

0

ij

ZY

F

y

∆

∆−∆

=

∆γ

(14)

where, tan(γ) = yj/F, ∆yij = yj – yi and ∆Yij = Yj – Yi.

From equation (14) we obtain (see Appendix):

)2 (

cot

+=

KF

) 2 (

,

+=

KF

Then the expression of the azimuthal angle is:

C

y

=

)arctan(1/-

π

Similarly the polar angle θ is:

γα

tan

2

−

∆−

yK

ij

γ

tan

2

∆

yK

Clet

ij

y

veis

C

C

C

yy

y

−

+==

if

ve is if )/ 1arctan(

αφ

(15)

Page 4

Fx

KF

xK

Cwhere

veis

C

C

−

C

C

j

ij

x

xx

xx

/

2

)2 (

,

if )arctan(1/-

π

ve is if )/ 1 arctan(

+

∆

=

−=

+=θ

(16)

For determining the value of K, we need to choose a

suitable pair of points on any one of the orthogonal axis of

the image plane (vertical axis for obtaining α, horizontal

axis for obtaining β). Let us consider the case of the

vertical axis (obtain the value α). We need to identify a

pair of points (x1, y1) and (x1, y2), with spectral frequencies

fo1 and fo2, which satisfy equation (12). The frequency at

coordinate (x1,0) is frp. The steps are as follows:

a) First, select a point (x1, y1) (y1 <> 0) and observe its

frequency fo1.

b) Observe the frequency frp at coordinates (x1,0).

c) Search for yk ( yk > 0 if y1 is –ve,

else < 0 if y1 is +ve)

and select y2 = yk, where fo2 = fok = (2 frp – fo1)

K is computed using equation (13), F is known from

camera calibration, or may be considered to be unity, in

which case the coordinates of the image plane must be

scaled w.r.t. the focal length of the camera (focal length

normalized image coordinates).

The advantage of the proposed method is the separable

analysis in x and y directions which gives the polar and

azimuth angles of the surface orientation respectively. The

depth information (upto a scale factor [9, 15], in the

absence of any additional information) can also retrieved

using the analytical expressions derived (see equations (8) -

(11) ). In the next section, we illustrate the advantage of

using wavelet transform in detecting spectral differences in

texture images with experimentation on simulated data.

4. Wavelet based texture analysis and results

Let us consider two 1-D signals obtained by scanning along

the horizontal lines of two texture images, of the same

texture surface with difference in one of the angles of

orientation. The images (see figure 6) are simulated as a

superimposition of two simple sinusoidal patterns. Typical

plots (signal I and signal II) of the horizontal scan lines of

the pair of images in figure 6, are shown in figure 7. The

corresponding spectral plots are shown in figure 8. Most

spectral based methods [14, 18, 19] involves locating and

finding the difference in the local spectral peaks of the

signals. This process if often erroneous and difficult even

for simple signals as shown in figure 8 (note that there are

two separate and almost identical peaks in each plot).

Hence it is necessary to use a multi-rate and multi-

resolution filter bank to discriminate these features, rather

than the use of a simple Fourier based analysis.

We suggest the use of wavelet transform for this purpose.

The plot of the wavelet coefficients for the pair of signals in

figure 7, are shown in figure 9 (detail coefficients at level 1,

i.e. D1, are negligible and are hence not illustrated).

Daubechies 6-tap dyadic filters [5] with 3 levels of

decomposition are used for this purpose. The wavelet

features exhibit a distinct difference in the response

noticeably at detail levels 2 and 3 (compare the responses

among the pairs of plots in figures 8 and 9). The process of

Figure 6. Two simulated texture images of the same surface

with a difference in the orientation angle (φ=0 in both, θ

42 deg. for Image I and 40 deg. for Image II).

?

???

Figure 7. Two horizontal intensity profiles of the pair of

images shown in figure 6 respectively.

Figure 8. Plots of spectral power for the pair of 1-D

intensity profiles shown in figure 7 respectively.

Page 5

Figure 9. Plots of the wavelet coefficients (level 3 decomposition) of the signals shown in figure 7 respectively.

Figure 10. Features of the signals at the corresponding levels derived from the wavelet coefficients (in figure 9).

feature extraction from the wavelet coefficients consists of

two steps. The first step of processing involves mean

subtraction, squaring and Gaussian smoothing. The second

step involves computing the variances of the post-processed

signals, for each level of decomposition (namely, A3, D3

and D2) separately. Plots of the variance of the post-

processed signals are illustrated in figure 10, which are

derived from the wavelet coefficients shown in figure 9.

A weighted (obtained empirically) sum of the differences of

the variances of the post-processed coefficients in the

corresponding bands of the wavelet decomposition is

related to the orientation angle and depth of the surface, as

illustrated in figure 4. A ratio of these weighted sums

computed at the pair of points P1 and P2, gives the value of

fd which is used to compute K as in equation (13).

Normalized observed frequencies, (foi/fr), are computed

using the proposed method for seven different orientations

of the texture surface, with the simulated texture pattern as

in figure 6. The actual and estimated values of the

normalized observed frequencies are shown in figure 11.

Discussions on accuracy and experimentation with real

world data are beyond the scope of this current paper.

5. Conclusion

This paper illustrates the advantage of using wavelet

transform to extract the orientation (in 3-D) of a textured

planar surface. It promises to be powerful than the spectral

based methods used in [1, 2, 6, 10, 11, 14, 16, 18, 19]. The

weighted sum of the differences of the variances of the

post-processed wavelet coefficients in the respective bands,

is used to obtain the orientation of the texture surface.

Expressions relating the orientation and depth of a texture

surface with the spectral contents of the image texture have

been derived. The method will be helpful in cases where the

spectral characteristics of a texture (fr) is known.

The proposed method has some drawbacks. It is assumed

that surface S intersects the X-Z plane of the world

coordinate system. This is necessary for detecting a pair of

points in the image plane which satisfy equations (11-12).

Errors in estimation are high when any of the angles of

orientation of the surface and image resolution are small.

Results are shown using superimposed sinusoidal signals to

illustrate the effectiveness and utility of the proposed

method.

Appendix

From equation (14)

y

∆

=

2

)(tan

Z

∆

0

ij

ijijij

Z

ZY

F

−

∆−∆γ

(A.1)

Page 6

Figure 11. Actual and estimated values of the normalised observed frequncy for a set of seven images of the simulated

texture pattern, with orientation angles (φ = 0 in all cases) θ = -49, -35, -21, 0, 19, 33 and 47 degrees respectively.

Estimation is based on the feature derived from detail level 2 (i.e. D2) coefficients only.

Since

α

tan

=

∆

∆

∆

ij

ij

Y

y

Z

, from (A.1), we have:

ij

ij

2

ij

ZZ

Z

F

∆−

−∆

=

0

) tan )(cot( 2

γα

Thus,

γα

tan

)(

∆

2

K

) 2 (

cot

0

+

∆

∆∆−

F

=

ij

ijij

Z

y

yZZ

γ

tan

2

)2 (

12+

−

=

KF

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Acknowledgement: This work was sponsored by the

"Bundesministerium für Bildung und Forschung",

Germany, (Program "aFuE").

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