SPHERICAL IMAGE DENOISING AND ITS APPLICATION TO

OMNIDIRECTIONAL IMAGING

ephanie Bigot

, Djemaa Kachi

, Sylvain Durand

and El Mustapha Mouaddib

L.A.M.F.A., UMR CNRS 6140 - U.P.J.V. 33 rue Saint Leu 80000 Amiens Cedex 1, France

C.R.E.A., EA 3299 - U.P.J.V. 7 rue du moulin neuf 80039 Amiens Cedex 1, France

Keywords:

Omnidirectional imaging, Spherical ﬁlters, Spherical harmonics, Wiener ﬁlter, Tikhonov regularization.

Abstract:

This paper addresses the problem of spherical image processing. Thanks to projective geometry, the omnidi-

rectional image can be presented as a function on sphere S

. The target application includes omnidirectional

image smoothing. We describe a new method of smoothing for spherical images. For that purpose, we in-

troduce a suitable Wiener ﬁlter and we use the Tikhonov method to these images. In order to compare their

performances, we present the most used classical spherical kernels.

We present several examples for ﬁltering real and synthetical spherical images.

This project was ﬁnanced by the D.G.A.

1 INTRODUCTION

Images deﬁned on a spherical surface rather than in

the image plane arise in various domains like en-

vironment mapping, and medical imaging (Chung,

2006). . . . Omnidirectional images appear with distor-

sion (because of the used mirror) in the image plane

and are naturally deﬁned on the sphere as shown in the

model of Geyer (Geyer and Daniilidis, 2001). Most

approaches analysing omnidirectional images applied

classical operators. However the analysis or the ﬁl-

tering of these images by traditional methods pro-

duces erroneous results. In (Daniilidis et al., 2002),

Daniilidis proposed of the very ﬁrst approaches of

adaptation of the operations of ﬁltering to the omni-

directional images. The proposal of the authors is to

project the image plane to a non-deformed space (unit

sphere) (Geyer and Daniilidis, 2000).

The spherical images are obtained by stere-

ographic projection of omnidirectional image, as

shown in Fig.1.Daniilidis then deﬁne kernels in the

space of the spherical coordinates and propose to

carry out the convolution on the surface of the sphere.

In (Strauss and Comby, 2005), the authors present

new morphological operators that use the projective

property of omnidirectional sensor. These operators

Figure 1: Omnidirectional image of Ofﬁce and the corre-

sponding spherical image.

make use of a structural element with a varying shape.

Another work (Ieng et al., 2003), advocates the prin-

ciple of kernel with varying shape to match indices

from two omnidirectional images. Except a few at-

tempts to adapt the method of processing omnidirec-

tional images, no normalization method has been es-

tablished for these images.

A simple technique for smoothing planar images is

the convolution of the image with a Gaussian kernel

g(x;t) =

4πkt

exp(−

4kt

)

Rather then just using this kernel for noise-removal, a

linear scale-space theory can be built upon the appli-

cation of this convolution kernel. This idea has been

101

Bigot S., Kachi D., Durand S. and Mustapha Mouaddib E. (2007).

SPHERICAL IMAGE DENOISING AND ITS APPLICATION TO OMNIDIRECTIONAL IMAGING.

In Proceedings of the Second International Conference on Computer Vision Theory and Applications - IFP/IA, pages 101-108

 SciTePress

introduced by Witkin (Witkin, 1984) and Lindeberg

(Lindeberg, 1994). The linear scale space of an im-

age is deﬁned as the set of smoothed images created

by convolving the image with Gaussian functions of

different scales.

The Gaussian function is the Green function of the

linear diffusion equation, i.e. the solution of equation

(1) with the δ-impulse as initial condition.

∆u(x;t) =

∂

∂t

u(x;t) (1)

In the case of omnidirectional images, the major dif-

ﬁculties of these approaches have to deal with explic-

itly include shrinkage, dependency of the smoothing

results on the mesh connectivity. In order to be able

to deal with well deﬁned globally parameterized do-

main we restrict the class of omnidirectional images

to those which can be described as functions on the

sphere, as deﬁned in the model of Geyer.

Diffusion smoothing of surfaces then corresponds

to convolution of the surface with the spherical

Gaussian.

We propose in this paper to adapt the classical Wiener

ﬁlter and the Tikhonov regularization to the spheri-

cal images. We compare the denoising performances

of these methods with the classical spherical kernels.

The remainder of this paper is structured as follows.

In the next section, we recall mathematical framework

related to spherical harmonics, which will be used for

the solution of the spherical diffusion equation, and

convolution on the sphere. In Section 3, we deﬁne

the most important classical spherical kernels, and we

describe the spherical form of Wiener and Tikhonov

ﬁltering. Section 4 shows results of the spherical ﬁl-

tering process applied to synthetical and real indoor

omnidirectional images. Finally we give some con-

clusions and future work.

2 SPHERICAL THEORY

Filtering is based on convolution operators. We

present in this section the various deﬁnitions of this

operation and the mathematical tools used for its im-

plementation.

2.1 Spherical Fourier Transform

We parametrize the unit sphere, embedded in R

, by

using the spherical coordinates η ∈ S

: η(θ,ϕ) =

(cos(ϕ)sin(θ), sin(ϕ)sin(θ),cos(θ)) with ϕ ∈ [0,2π[,

angle of longitude and θ ∈ [0,π], angle of colatitude

(latitude + π/2).

The effect of planar diffusion smoothing can be well

understood in the frequency domain as a low-pass ﬁl-

ter. Since we are going to carry out the corresponding

analysis on the sphere we need a spherical analog of

the Fourier transform. Such a tool exists in the expan-

sion of a function into a series of spherical harmonic

functions.

Notice by Y

the spherical harmonic of l(∈N) degree

and order m as follows

(θ,ϕ) =

2l + 1

4π

(l −m)!

(l + m)!

(cos(θ))e

imϕ

for m ≥ 0

where the P

(x) are the polynomials of Legendre as-

sociated with l degree and order m. We can notice that

the spherical harmonics of l degree form a subspace

of L

) of dimension 2l +1 which is invariant under

rotations of the sphere. Since the spherical harmonics

form an orthonormal basis for L

), we have

f (l,m) =

f ,Y

where the scalar product on the sphere is deﬁned as

f ,h

2π

f (θ, ϕ)h(θ,ϕ)sin(θ)dθdϕ

The set of the coefﬁcients

is called spherical

Fourier transform or spectrum of f. For the imple-

mentation, we will use the sampling theorem (Healy

et al., 1998).

Theorem: Let f ∈ L

) be a bandlimited function

of bandwith B, then:

√

2π

2B−1

∑

j=0

2B−1

∑

k=0

(B)

f (θ

,ϕ

(θ

,ϕ

)

for

≤ l < B. The sampling grid is the equiangular

or lat-lon grid with θ

π(2 j+1)

et ϕ

πk

2.2 The Convolution on the Sphere

Two deﬁnitions were proposed to carry out a prod-

uct of Convolution on the sphere : that introduced

by Driscoll-Healy (Driscoll and Healy, 1994) and that

used, by Daniilidis (Daniilidis et al., 2002) and Wan-

delt (Wandelt and G

orski, 2001).

We begin by introducing some notations.

We represent the sphere S

as the quotient

SO(3)/SO(2) where S O(3) is the group of rotations

which acts on the sphere (Vilenkin, 1969). The rota-

tion of a function f ∈L

) by an element g ∈SO(3)

is then deﬁned with the operator Λ

such as

f (η) = f (g

−1

η)

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102

Let f ∈ L

) and g ∈ L

)

The convolution product between f et g is the func-

tion on SO(3) deﬁned by

( f

∗g)(R) =

f (R

−1

η)g(η)dη

We have

∗g ∈ L

(SO(3))

with L

(SO(3)) ≡ L

(SO(3),dR) where dR is the

Haar measure on SO(3).

An important problem arises; whereas the functions f

and g are deﬁned on the sphere, the product of con-

volution is deﬁned on the group of rotations. Conse-

quently, it is clear that the product of convolution is

not associative. Moreover

∗δ

(R) = f (R

−1

)

that implies a loss of symmetry in this deﬁnition of

convolution.

Now let us see another deﬁnition of the convolution,

introduced by Driscoll and Healy. Let f , h ∈ L

we have

( f ∗h)(η) =

SO(3)

f (Rn

)h(R

−1

η)dR

where η ∈ S

, n

= (0,0, 1) represent the North Pole

of the unit sphere.

The effectiveness of this second deﬁnition becomes

obvious thanks to the theorem of convolution, shown

by Driscoll and Healy. For f , h ∈L

), we have that

( f ∗h)

= 2π

4π

2l + 1

(2)

We can observe that only the coefﬁcients of spheri-

cal transform of the ﬁlter

with m = 0 are used in

expression (2). These coefﬁcients correspond to the

zonal harmonics Y

and thus represent the invariant

part by rotation of the ﬁlter h. So the ﬁltering opera-

tor is invariant under the actions of rotations.

3 SPHERICAL FILTERS

3.1 Spherical Kernels

The question which we put is to know to deﬁne

Gaussian on the sphere. B

ulow (Bulow, 2004) pro-

posed to determine the Green function, as a solu-

tion of the spherical diffusion equation. The spherical

equation of diffusion is :

∆

u(θ,φ,t) =

∂

∂t

u(θ,φ,t)

where

∆

sin(θ)

∂

∂θ



sin(θ)

∂

∂θ



sin

(θ)

∂

Let us recall that the spherical harmonics are the

eigenfunctions of the spherical Laplace operator :

∆

l,m

= −l(l + 1)Y

l,m

Consequently, it is checked easily that the functions

l,m

(θ,φ,t) = Y

l,m

(θ,φ)exp(−l(l + 1)kt)

are solutions of the spherical equation of diffusion.

To obtain the function of Green G, we force like initial

condition :

u(θ,φ, 0) = G(θ,φ, 0) := δ

(θ,φ)

where δ is deﬁned by f (n

) =

f (η)δ

(η)dη.

By using the decomposition of spherical Dirac in the

basis of the spherical harmonics, we obtain

G(.,0) = δ

∑

l∈N

2l + 1

4π

l,0

∑

l∈N

2l + 1

4π

l,0

(.,0)

From where ﬁnally for the Green function

G(θ,φ,t) =

∑

l∈N

2l + 1

4π

(cos(θ))e

−l(l+1)kt

we carry out then the spherical Fourier transform to

ﬁnd

G(.,t)(l, m) =

(

2l+1

4π

−l(l+1)kt

if m = 0

0 if not

Let us deﬁne another functions candidates to build

Gaussian on the sphere.

3.2 Spherical Gaussian

If we start with the deﬁnition of the Gaussian on a

plane, we can obtain the spherical Gaussian using the

inverse stereographic projection

(θ,φ,t) = e

−(tan(θ/2)/t)

3.3 Poisson Kernel

This Kernel is given by the function

P(θ,φ,t) =

∑

n∈N

(2n + 1)t

(cos(θ))

1 −t

(1 −2t cos(θ) + t

)

3/2

for t ∈ [0, 1]

This equation is the solution of the equation

∆

P(θ,ϕ,t) = −t

∂

∂t

(hP(θ,ϕ,t))

SPHERICAL IMAGE DENOISING AND ITS APPLICATION TO OMNIDIRECTIONAL IMAGING

103

Let us calculate the Fourier transform of this function.

P(.,t)(l, m) =

(

(2l + 1)4π t

if m = 0

0 else

For more details, we refer to (Bulow, 2004).

3.4 Spherical Form of the Wiener Filter

We present in this section a Wiener ﬁlter adapted to

spherical functions.

We consider our original image f ∈L

), corrupted

by an additive white noise n. We denotes x = f + n,

the data, i.e. the corrupted image.

We seek to obtain the best possible estimate g of f

starting from our data x.

For that purpose we proceed to obtain the maximum

of the SNR deﬁned as

SNR = 10log

)

f −g

)

by minimizing the average quadratic error

e = E(

f −g

)

The image f and the estimator g are related to L

whose spherical harmonics form a basis. As a conse-

quence, we can use the theorem of Riesz-Fischer to

obtain

e = E(

∑

l∈N

∑

≤l



f (l,m) −

g(l, m)



)

We propose to search g as the result of a ﬁlter (of

impulse response h) applied to the data x, which is

equivalent to write g as

g =

2π

x ∗h

We apply the convolution theorem of Driscoll-Healy,

we obtain :

e = E



∑

l∈N

∑

≤l



f (l,m) −

4π

2l + 1

x(l, m)

h(l, 0)





(3)

We must thus ﬁnd the ﬁlter h which minimizes e, de-

ﬁned in (3). We wish that this ﬁlter to be invariant

by rotation, i.e. of form

h(η) =

∑

k∈N

h(k, 0)Y

k,0

(η)

That amounts determining the

h(k, 0) which mini-

mizes e, and so solve the equation

∂e

∂

h(k, 0)

= 0

By considering that noise is a white noise with

standard deviation σ, we have E|

n(k, m)|

= σ

We assume

f (k,m)|

where c is a constant to be chosen, we are led with

h(k, 0) =

2k + 1

4π

1 +

3.5 Regularization Process of Tikhonov

In 1977, Arsenin and Tikhonov (Tikhonov and Ars-

enin, 1977) have proposed a method for determining

the best possible estimator.

The idea is to ﬁnd

min

f ∈H

)

∇ f

+ λ

f −x

where as above,

represents the standard L

)

norm. We recall that for a smooth function f with

compact support, we have

∇ f

= − f ∆ f

Therefore :

∇ f

+ λ

f −x

= −

f ∆ f + λ

f −x

However the spherical harmonics, which form an or-

thonormal basis of L

), are the eigenfunctions of

Laplace-Beltrami operator. Consequently, we obtain

∇ f

+ λ

f −x

∑

l∈N

∑

≤l

(l,m)l(l + 1) + λ

f −x

Then by using the Riesz-Fischer theorem, we have

J =

∇ f

+ λ

f −x

∑

l∈N

∑

≤l

(l, m)l(l +1)+λ

∑

l∈N

∑

≤l



f (l,m) −

x(l, m)



Tikhonov methods aims to ﬁnd the function f where

the minimum is achieved. However by using the ex-

pansion of f in the basis of the spherical harmonics,

f (η) =

∑

l∈N

∑

≤l

f (l,m)Y

l,m

(η) ∀η ∈ S

We have to estimate the function J according to

f (k, p).

∂J

∂

f (k, p)

= 2k(k + 1)

f (k, p) + 2λ(

f (k, p) −

x(k, p))

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104

Then

∂J

∂

f (k, p)

= 0

⇔

f (k, p) =

λ + k(k + 1)

x(k, p)

Let g be the estimator which minimizes the function

J. We infer from the computations above that

g(k, p) =

λ + k(k + 1)

x(k, p)

As for the Wiener ﬁlter, we suppose that the estimator

is the result of a ﬁltering of the data, i.e.,

g =

2π

x ∗h

with a h that is the impulse response of the ﬁlter of

restoration. We thus ﬁnd

h(k, 0) =

2k + 1

4π

1 +

k(k + 1)

The ﬁlter h of restoration has a form similar to the

Wiener ﬁlter, that is a classical result in the framework

of the planar images denoising.

4 EXPERIMENTAL RESULTS

We have compared our denoising process on synthet-

ical, real indoor and outdoor omnidirectional images.

The used mirror is parabolic and veriﬁes the prop-

erty of single effective viewpoint. So we can ap-

(a) Ofﬁce image (b) Ofﬁce image noised

Figure 2: Noised Ofﬁce image with additive noise

N(σ, µ) = N(40,0).

ply the Geyer model and process the ﬁltering on

spherical images. For the implementation, omnidi-

rectional images are projected on S

and redeﬁned on

the lat-lon grid. The spherical transform of images

and kernels are computed by using the yawtb tool-

box (http://rhea.tele.ucl.ac.be/yawtb/). The noise is

an additive white noise with varying standard devi-

ation σ and is added directly to the spherical images

Table 1: Optimal SNR for Ofﬁce image N(σ,µ) = N(40,0).

Kernel Scale t SNR

Green 2.6650 ∗10

−4

19.7852

Gaussian spherical 0.0164 19.7842

Wiener 2903000 19.1174

Tikhonov 1900 19.1013

Poisson 0.9850 18.6118

as seen in Fig.2. The scale is an important parameter

in the process, and the result of ﬁltering depends on

this value. Fig. 3 shows an example of ﬁltering results

(Green ﬁlter) with different scale value. For each ﬁl-

ter, we search (numerically) the optimal scale, which

provides the highest SNR. This SNR is calculated on

the whole sphere, while the projected omnidirectional

image is deﬁned on the hemisphere. The optimal SNR

is approximately the same whatever the ﬁlter and the

type of images (indoor or outdoor) as proved in Fig.4,

Fig.5, and Table 1, even if the Green function and the

gaussian kernel provide often ligthly higher SNR. We

have obtained similar results for other types of noise

(like exponential noise. . . ).

However if we carry out a zoom in some images,

see Fig.6 and Fig.7 we can notice (visually) that the

Wiener ﬁlter and the method of Tikhonov give further

precision on edges. This precision is independent of

position on the sphere, (close or far from the pole of

the sphere). The SNR calculated at different regions

on the sphere are similar, see Fig. 6. We can also

show this conclusion for various images. These re-

sults have a considerable beneﬁt if we want to detect

edges, or extract important primitive from images.

5 CONCLUSION AND FUTURE

WORK

We presented a panorama on the tools and the

methods to treat the spherical images. We adapted the

Wiener ﬁlter and Tikhonov method in that case and

compared their performances to classical spherical

gaussian ﬁlters. The SNR obtained for various types

of images prove the effectiveness of these ﬁlters for

the denoising application. Moreover the treatments

can be carried out in real time. The ﬁlters are

invariant by rotation, the results are thus independent

of the position on the sphere. We also noticed that

the Wiener ﬁlter and Tikhonov regularization are

relatively better, if we want to analyse precisely some

parts of images for edge detection for example.

For the continuation of our work, we propose to

SPHERICAL IMAGE DENOISING AND ITS APPLICATION TO OMNIDIRECTIONAL IMAGING

105