IMAGE COMPLETION USING A DIFFUSION DRIVEN MEAN

CURVATURE FLOW IN A SUB-RIEMANNIAN SPACE

Gonzalo Sanguinetti

Instituto de Ingenier

ıa Electrica, Universidad de la Rep

ublica, Montevideo, Uruguay

Giovanna Citti

Dipartimento di Matematica, Universit

a di Bologna, Bologna, Italy

Alessandro Sarti

Dipartimento di Elettronica, Informatica e Sistemistica, Universit

a di Bologna, Bologna, Italy

Keywords:

Perceptual completion.

Abstract:

In this paper we present an implementation of a perceptual completion model performed in the three dimen-

sional space of position and orientation of level lines of an image. We show that the space is equipped with

a natural subriemannian metric. This model allows to perform disocclusion representing both the occluding

and occluded objects simultaneously in the space. The completion is accomplished by computing minimal

surfaces with respect to the non Euclidean metric of the space. The minimality is achieved via diffusion driven

mean curvature ﬂow. Results are presented in a number of cognitive relevant cases.

1 INTRODUCTION

Perceptual completion is performed by the mam-

malian visual system in a number of phenomenologi-

cal cases, deeply studied by psychology of Gestalt to

understand the underlying structure of visual process-

ing in humans. The most common examples compre-

hend modal completion, amodal completion, trans-

parency, intersection and self intersection of curves

(Kanisza, 1979). Modal completion is the process of

ﬁlling the missing part of an object and building a per-

cept that is phenomenally undistinguishable from real

stimuli. It gives rise to the well known phenomenon

of illusory boundaries (or subjective contours) and it

takes place often to complete occluding objects (in

Fig. 1(a) the completed triangle is occluding the 3

circles). Amodal completion (Fig. 1(b)) is a percep-

tual modality for integrating missing parts of partially

occluded objects. Since the occluded ﬁgure underlies

the occluding one, it is completed without any senso-

rial counterpart. In case of transparency (Fig. 1(c))

and curve intersection (Fig. 1(d)), both occluding and

occluded ﬁgures are visible in the scene and the per-

ceptual system is able to disambiguate them and rec-

ognize them as different objects. A point made clear

by the studies of phenomenology of perception is that

in all cases of completion both the occluding and the

occluded objects are perceived at the same time in the

scene and therefore there are points in the input stim-

ulus corresponding to more than one ﬁgure at the per-

ceptual level. Many computer vision techniques have

been proposed to model perceptual completion. ei-

ther heuristically based or biologically inspired. Rec-

tilinear and curvilinear subjective contours have been

modeled by D.Mumford with Euler elastica as ex-

tremality points of curvature functionals (Nitzberg

and Mumford, 1990) and by stochastic ﬁelds as so-

lution of the Fokker-Planck equation (Williams and

Jacobs, 1995). In the latter case the stochastic com-

pletion ﬁeld represents the likelihood that a comple-

tion joining two contour fragments passes through

any given position and orientation in the image. An

extension taking into account also the curvature has

been proposed in (August and Zucker, 2003). Amodal

completion has been accomplished by a number of

techniques. In (Masnou and Morel, 1998) (Ambro-

Sanguinetti G., Citti G. and Sarti A. (2008).

IMAGE COMPLETION USING A DIFFUSION DRIVEN MEAN CURVATURE FLOW IN A SUB-RIEMANNIAN SPACE.

In Proceedings of the Third International Conference on Computer Vision Theory and Applications, pages 46-53

DOI: 10.5220/0001075800460053

 SciTePress

(a) Modal completion (b) Amodal completion

Figure 1: Examples of perceptual completion.

sio and Masnou, 2005) an extension of the Mumford

functional to level lines has been used to ﬁll miss-

ing regions. Digital inpainting has been introduced

as a technique to diffuse existing information on the

boundary toward the interior region (Bertalm

ıo et al.,

2000) (Ballester et al., 2001). A total variation ap-

proach has been proposed in (Chan and Shen, 2001).

All these techniques consider the perceptual space in

which completion is performed has the same dimen-

sionality of the image. This could be a restriction

in case we are interested in the presence of recon-

structed occluding and occluded objects in the scene,

as in case of mammalian vision. To overcome this re-

striction, in (Citti and Sarti, 2006) has been proposed

a completion model based on the functional architec-

ture of the visual cortex, where completion is fully

performed in the rototranslation group R

× S

, al-

lowing the simultaneous reconstruction of occluding

and occluded objects. This model is an extension of

the one proposed in (Petitot and Tondut, 1999) where

curves are lifted in the three dimensional Heisenberg

group. Another higher dimensional model has been

introduced in (Ben-Shahar and Zucker, 2004) and in

(Medioni, 2000).

Following (Citti and Sarti, 2006), a two dimen-

sional image is lifted to a surface in the 3-dimensional

sub-Riemannian space, an occlusion is considered as

a hole in the surface, and the proposed model com-

plete the missing part of the image with a minimal

surface. Computing a minimal surface in the hole

and re-projecting it over the image domain, we ﬁnd

the same level lines as Morel and Masnou have found

in (Masnou and Morel, 1998) minimizing an elastica

based functional.

In (Hladky and Pauls, 2005) the authors proposed

a very fast method for ﬁnding the minimal surface ex-

plicitly interpolating the level lines represented in the

Sub-Riemannian space even if it is not well suited for

simultaneous representation of occluded and occlud-

ing objects.

The main objective of this paper is to propose a

computational technique for ﬁnding minimal surfaces

by diffusion driven mean curvature ﬂow. The tech-

nique is able to simultaneously construct occluded

and occluding objects. The surface is represented as a

thin concentrated mass, suitably diffused and concen-

trated with a two step algorithm adapted to the sub-

Riemannian metric. The diffusion driven method was

ﬁrst introduced in the Euclidean settings in (Merriman

et al., 1998).

The paper is organized as follows:

• In section 2 we explain the lifting of the image to

the 3D position-orientation space and describe the

subriemannian structure of the space.

• In section 3 the main model of image completion

is proposed and discussed in detail.

• In section 4 we present the numerical scheme for

the equations presented in previous sections.

• In section 5 we describe the experiments realized

and provide the results obtained.

• Finally, conclusions are presented.

2 THEORETICAL BACKGROUND

2.1 Lifting of the Image Level Lines in a

3-Dimensional Space

An image I can be represented as a bounded func-

tion deﬁned on a domain M ⊂ R

, I : M → R

. The

points of M have coordinates (x, y). At every point

of the image we detect the tangent direction to level

lines(I

,−I

), where I

and I

are the components of

the image gradient. If θ is the angle between the tan-

gent and the x-axis the tangent can be rewritten as

(cos(θ),sin(θ)).

We want to deﬁne the orientation independently of

the versus of the tangent vector. Therefore, we iden-

tify a tangent vector with its opposite one. This means

that angles which differ form π will be identiﬁed, and

θ(x,y) = −arctan(I

) , θ ∈ S

IMAGE COMPLETION USING A DIFFUSION DRIVEN MEAN CURVATURE FLOWIN A SUB-RIEMANNIAN

SPACE

where S

is the set of angles [0,π].

To every point (x,y) is associated a three dimen-

sional vector (x,y,θ), in a new space homeomorphic

to R

×S

. Since the process is repeated at each point,

each level line is lifted to a new curve in the three

dimensional space. We will call admissible curve a

curve in R

× S

if it is the lifting of a level line.

2.2 The Tangent Bundle and the

Integral Curves

Figure 2: A lifted level line.

A tangent vector to the lifted curve has the same two

ﬁrst components as the tangent vector to the level line,

i.e. a real multiple of (cos(θ),sin(θ)), and it has the

third component in the direction (0,0,1). Hence it

can be represented as a linear combination of the vec-

tors (cos(θ), sin(θ), 0) and (0,0,1) which, from now

on, will be called

and

respectively. The set of

vectors α

+α

deﬁnes a plane and every admis-

sible curve is tangent to a vector of the plane. Hence

an admissible curve satisﬁes the differential equation:

(t) = α

(t)+ α

(t)

It is well known that the ratio α

/α

is the the curva-

ture k(t) of its 2D projection, the level line of I.

2.3 Curve Length’s and Metric of the

Space

If we equip the tangent planes with an Euclidean met-

ric then the length of an admissible curve can be com-

puted as usual integrating the tangent vector.

λ(γ)(t) =

kγ

(s)kds =

kα

+ α

kds

1 + k

ds (1)

In order to deﬁne a distance in term of the length, we

need to answer the following question: Is it possible

to connect every couple of points of R

× S

using an

integral curve?

This is not a simple question taking into account

that in every point we have only directions which are

linear combinations of two vectors even if we are im-

mersed in a three dimensional space. However, the

answer is yes and it will become clear in the exam-

ple below. Otherwise, see (Citti and Sarti, 2006) for a

detailed justiﬁcation.

Consequently, it is possible to deﬁne a notion

of distance between two points p

= (x

,θ

) and

= (x

,θ

d(p

, p

) = inf{λ(γ) : γ is an admmisible curve

connecting p

and p

} (2)

In the Euclidean case this inﬁmum is realized by a

geodesic that is a segment. Here, the geodesics are lo-

cally curvilinear. The metric induced by (2) is clearly

Non-Euclidean, moreover it is not even Riemannian.

With the chosen metrics on the tangent plane, the

space co-metric is given by:

g =





cos(θ) 0

sin(θ) 0

0 1







cos(θ) sin(θ) 0

0 0 1







cos

(θ) cos(θ)sin(θ) 0

cos(θ)sin(θ) sin

(θ) 0

0 0 1





Since the matrix g is not invertible, it can not induce

a Riemannian metric on the space. Spaces equipped

with Sub-Riemannian metrics appears often when

one of the dimensions is a state variable depending

on the others. In this case the state variable is θ.

2.4 The Lifted Surface as an Implicit

Function

When every point of an entire image is lifted up, a

three dimensional surface is constructed as:

Σ =



(x,y,θ) ∈ R

× S

: θ(x, y) = − arctan(I

)



We can identify the lifting of an image with the lifting

of every level line. This point of view allows us to

understand a remarkable property of the lifted surface.

In fact, since two level lines of an image never cross,

also the lifted level lines don’t do it. Then we say

that the lifted surface is foliated by the lifted curves

(see Fig. 3). We will call rule an admissible curve

foliating a surface.

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

Figure 3: A lifted image is a surface foliated by the lifted

level lines.

Let’s now represent the surface in terms of the im-

plicit function

u(x,y,θ) = [cos (θ + arctan(I

))]

(3)

For every coordinate (x,y) this function attains its

maximum in the variable θ in correspondence to a

point (x,y,

θ) of the surface. The cosine function is

Figure 4: The lifted image can be viewed as a thick surface

and the surface obtained with eq. 4.

chosen in order to have periodicity of u in the third

coordinate since it is an angle. Note we have imposed

that the maximum value of u is 1.

The surface Σ can be represented as the zero level

set of the function u

Σ = {(x,y,θ) ∈ R

× S

: ∂

u(x,y,θ) = 0,

∂

θθ

u(x,y,θ) < 0} (4)

The condition over ∂

θθ

u is imposed in order to

avoid minima of u.

2.5 Sub-riemannian Differential

Operators

We will deﬁne differential operators acting over the

function u, in terms of the subriemannian structure

introduced before on the space R

×S

, instead of the

Euclidean one. We will need to deﬁne two differen-

tial operators X

and X

which play the role of the

Euclidean partial derivatives, and have the same coef-

ﬁcients as the vector ﬁelds

and

. Hence

= cos(θ)∂

+ sin(θ)∂

, X

= ∂

Accordingly we deﬁne the Sub-Riemannian gra-

dient as:

∇

u = (X

u,X

u).

The notation SR (Sub-Riemannian) will be used in

order to avoid confusions with the classical operators.

We deﬁne the so called sub-laplacian operator, which

is the analogous of the classical laplacian in this struc-

ture:

∆

u = X

u + X

= cos

(θ)u

+ sin

(θ)u

2cos(θ)sin(θ)u

+ u

θθ

(5)

and we deﬁne the subriemannian diffusion equation

as:

= ∆

Despite of the fact the sublaplacian operator is built

just with two directional derivatives in a 3 dimen-

sional space, the diffusion process reaches every

point due to the connectivity property of the sub-

riemannian geometry.

2.6 Differential Geometry of the

Surface

Since the surface Σ, is the zero level set of the function

= X

u, it is possible to deﬁne geometrical prop-

erties of Σ, in terms of the function u

and its sub-

riemannian derivatives. The subriemannian gradient

∇

is orthogonal to the surface (w.r. of the subrie-

mannian metric), and an admissible tangent vector is

(−X

). Correspondingly the rules on the sur-

face have the expression

= −X

+ X

. (6)

Analogously the diffusion on the surface, which is

the diffusion along the rules, is expressed in terms of

∇

The foliation feature suggests a natural notion

of area in the sub-riemannian structure R

× S

Indeed the area of a lifted surface can be deﬁned as

IMAGE COMPLETION USING A DIFFUSION DRIVEN MEAN CURVATURE FLOWIN A SUB-RIEMANNIAN

SPACE

the integral of the lengths of every rule. With this

deﬁnition, a minimal surface with assigned boundary

conditions is obtained requiring every rule to have

minimal length.

3 THE COMPLETION MODEL

3.1 Basic Model

In this section we present our completion model in the

rototraslation group, (see also (Citti and Sarti, 2006))

Let’s consider an image with an occlusion and let

us call D the missing part in the two dimensional do-

main. In order to complete it, we lift the image to a

surface in the Sub-Riemannian space. This lifted sur-

face will have a hole, which will be completed with a

minimal surface. Indeed, using relation (1), in (Citti

and Sarti, 2006) it has been proved that the subrie-

mannian minimization of the surface area gives rise to

the minimization on the rules on the surfaces, whose

projection are the elastica curves. Hence the mini-

mization of the ﬁrst order area functional on R

× S

correspond to the minimisation of a second order cur-

vature functional on the image plane (Ambrosio and

Masnou, 2005) (Masnou and Morel, 1998).

The method we will use is the following: ﬁrst we

lift the non occluded part of the image with eq. (3) to

a function u deﬁned on (R

\D)× S

. In the occluded

region D × S

we assign value zero to the function u.

Later we built an initial surface in the missing region.

Finally we evolve this surface with an approximated

diffusion driven mean curvature ﬂow until it becomes

minimal. This is a two step algorithm of diffusion and

concentration, as shown in (Citti and Sarti, 2006):

• Diffusion of existing information in the subrie-

mannian space with the sub-laplacian.

• Concentration of diffused information on the ﬁber

over every point (x, y).

3.2 Algorithmic Implementation

The image I is lifted to a surface, represented by the

maxima over the ﬁber S

of a function u, by using

equation (3) The ﬁrst step is to propagate existing in-

formation from the boundary of the missing region

D × S

with sub-riemannian diffusion:







∂

u =

∆

u if (x,y, θ) ∈ D × S

∂

θθ

u if (x,y,θ) ∈ (R

\ D) × S

,t ∈ [0,h]

u(0) = u

(7)

This ﬁrst step is necessary to initialize the func-

tion u to be a rough solution, which will be reﬁned by

diffusion driven mean curvature ﬂow.

In fact after the initial propagation, a mean cur-

vature evolution of the function u is implemented by

using a two step iterative algorithm consisting in al-

ternative diffusion and concentration:

• Diffuse with the Sub-Laplacian operator (5) for a

short time with ﬁxed boundary conditions in the

boundary of D × S

In the occluded region we diffuse using the sub-

Laplacian operator. This operator propagates data

in the direction of the vectors X

and X

. The dif-

fusion in the direction of X

alone would expand

into the occlusion the information taken from the

boundary just in a straight line parallel to the (x,y)

plane. By adding the diffusion in the X

direc-

tion, we allow propagation on curvilinear paths

on R

× S

, even if we make thicker the surface

represented by u as a side effect. Outside D × S

we use the equation u

= u

θθ

just to keep the same

thickness of the surface as in the interior of D×S

Note that if we just use this equation for a short

time the maximum of u is not moved and there-

fore the surface Σ does not change. For the dis-

occlution problem it is only necessary to consider

values of u near the boundary of D × S

. Only

this values will be propagated inside D ×S

. Nev-

ertheless, for improving the visualization we will

consider a larger domain outside D × S

• Concentrate the function u over the surface, i.e.

make thinner the thick version of the surface.

After diffusing u for a period of time h, we per-

form a concentration over its maximum and denote ¯u

the new function which implicitly deﬁne the concen-

trated surface:

¯u(x,y,θ) =



u(x,y,θ)

max

(x,y)



, γ > 1 (8)

where:

max

(x,y) = max

θ∈S

{u(x,y,θ)} (9)

This procedure renormalize the function u in such a

way that the maximum over each ﬁber is 1. The con-

centration, obtained elevating the function u to a suit-

able power greater than one, preserves the value of the

maximum and reduces all the other values of u. Thus

this mechanism concentrates the function around its

maximum.

3.3 Multiple Concentration

The three dimensionality of the space allows the co-

existence of occluded and occluding objects at the

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

Figure 5: Concentration.

same time. In terms of the function u it means that

we expect to have more than one maximum in each

ﬁber. However, the equation described before (9), al-

lows only one maximum per ﬁber. The method de-

scribed above could be slightly modiﬁed in order to

avoid this limitation. In particular we propose the fol-

lowing renormalization criterion.

We ﬁrst detect the maxima on a ﬁber over the point

(x,y) as the set {θ ∈ S

,∂

u(x,y,θ) = 0, ∂

θθ

u(x,y,θ) <

0}. We call them θ

,...,θ

with θ

< θ

i+1

. Then we

construct a piecewise linear function u

norm

(Fig. 6 )

connecting every local maximum detected and peri-

odic in the variable θ:

norm

(x,y,θ) = u(x, y,θ

) + (10)

(θ − θ

)

u(x,y,θ

j+1

) − u(x, y,θ

)

j+1

− θ

with θ ∈ [θ

,θ

j+1

We use eq (10) to re-normalize every single col-

umn of u as follows:

¯u(x,y,θ) =



u(x,y,θ)

norm

(x,y,θ)



, γ > 1

After renormalization the function ¯u keep the same

points of maximum as the function u and attains value

1 at each of these points.

As we mentioned before, this modiﬁcation allows

more than one maximum on each ﬁber. Hence

applying iteratively this improved concentration

technique and the sub-riemannian diffusion, we

compute minimal surfaces, in R

× S

which are

union of graphs of the variable (x, y), which can

partially overlap. It corresponds to the completion of

both occluding and occluded object.

Figure 6: Improved re-normalization.

4 NUMERICAL SCHEME

For the diffusion we use a ﬁnite difference scheme.

Let us consider a rectangular grid in space-time

(x,y,θ,t). The grid consist of a set of points

,θ

) = (l∆x,m∆y, q∆θ, n∆t).

Following the standard notation, we denote by

lmq

the value of the function u at a grid point. We

use forward differences in order to approximate the

time derivative:

u =

n+1

lmq

− u

lmq

∆t

and center differences for the spatial ones:

lmq

(l+1)mq

− u

(l−1)mq

2∆x

lmq

(l+1)mq

− 2u

lmq

+ u

(l−1)mq

(∆x)

The second directional derivatives are approximated

with:

lmq

= cos(θ

)

lmq

+ sin(θ

)

lmq

+2cos(θ

)sin(θ

lmq

= D

θθ

lmq

We impose Neumann boundary conditions on x and

y and periodic boundary conditions on the third di-

rection θ. The time step ∆t is upper bounded by the

usual Courant-Friedrich-Levy condition that ensures

the stability of the evolution [11].

5 EXPERIMENTS AND RESULTS

5.1 Macula Cieca

In this experiment we consider the completion of a

ﬁgure that has been partially occluded. This example

IMAGE COMPLETION USING A DIFFUSION DRIVEN MEAN CURVATURE FLOWIN A SUB-RIEMANNIAN

SPACE

(a) Original Image.

(b) Initially lifted surface and the minimal surface com-

puted.

Figure 7: Macula cieca example.

mimics the missing information due to the presence

of the macula cieca (blind spot) that is modally com-

pleted by the human visual system. As described in

the previous section the occluded image is lifted to a

surface with a hole in the three dimensional space and

an initial surface is deﬁned in the missing part with a

classical Euclidean diffusion equation. Then the sur-

face is evolved applying iteratively equations (7) and

(9) until a steady state is achieved.

The image dimensions are 100 × 100 pixels, and

we use 100 values to discretize the variable θ. For

the preprocessing step 100 iterations of the Euclidean

heat equation were made using a time step of ∆t = 0.1.

The steady state was reached after 20 iterations with a

concentration power in (8) of γ = 2 and 20 steps with

∆t = 0.1 of the subriemannian heat equation (7).

At this point we have completed the missing in-

formation of the lifted surface with a minimal surface

in the Sub-Riemannian space. The lifting and com-

pletion processes take into account just the direction

of the level lines of the image, as a geometric infor-

mation. Then the intensity information of the image

is completely missed.

Let’s deﬁne a function v extending the values of

the image I on the 3 dimensional space, and constant

in the variable θ:

v(x,y,θ) =



I(x,y) (x,y,θ) ∈ (R

\D) × S

0 (x,y,θ) ∈ D × S

We will use a Laplace Beltrami diffusion algorithm in

the sub-riemannian setting to propagate the function v

along the rules of the minimal surface. Since the rules

of the surface, deﬁned in (6) only depend on ∇

the Laplace Beltrami operator is a linear operator in

the variable v whose coefﬁcients depend on ∇

v + |X

+ X

−

v − X

+ X

5.2 Occlusion

In Figure 8 an occlusion problem is considered. The

initial image (top) shows an underlying object par-

tially occluded by a vertical stripe. The human vi-

sual system simultaneously segments the occluding

object and amodally completes the occluded one, tak-

ing both at the same time as perceived units. In the

numerical experiment ﬁrst the image is lifted in the

sub-riemannian space and the missing information is

completed. The result shows that the partially oc-

cluded object has been completed and the occluding

one has been segmented. Both objects are present at

the same time in the three dimensional space.

For this example the dimensions were again 100×

100 × 100 pixels. Non preprocessing step is needed.

The steady state was reached after 10 iterations with

a concentration power of γ = 2 in equation 8 and 10

steps with ∆t = 0.1 of the subriemannian diffusion

step.

6 CONCLUSIONS

In this paper we utilized a model of perceptual com-

pletion inspired from the visual cortex to perform

completion of occluding and occluded objects in im-

ages. In particular we achieved the task by computing

minimal surfaces in sub-riemannian space via diffu-

sion driven mean curvature ﬂow. The implementation

has been performed with a two steps iterative algo-

rithm of diffusion and concentration. A new concen-

tration technique allowing more than one maximum

over the ﬁbers has been proposed. This allows to com-

pute a set of graphs partially overlapped representing

the occluding and the occluded objects. Computa-

tional results on cognitive images have been achieved.

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

(a) Original Image.

(b) Some steps of the evolution.

Figure 8: Occlusion example: Mean Curvature Evolution

with 2 simultaneous surfaces.

ACKNOWLEDGEMENTS

This work was partially supported by ALFA

project II-0366-FA and NEST project GALA (Sub-

Riemannian geometric analysis in Lie groups) num-

ber 028766.

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