Page 1

SNAKES WITH ELLIPSE-REPRODUCING PROPERTY1

Snakes with Ellipse-Reproducing Property

Ricard Delgado-Gonzalo, Philippe Th´ evenaz, Chandra Sekhar Seelamantula, and Michael Unser

Abstract—We present a new class of continuously defined

parametric snakes using a special kind of exponential splines as

basis functions. We have enforced our bases to have the shortest-

possible support subject to some design constraints to maximize

efficiency. While the resulting snakes are versatile enough to

provide a good approximation of any closed curve in the plane,

their most important feature is the fact that they admit ellipses

within their span. Thus, they can perfectly generate circular

and elliptical shapes. These features are appropriate to delineate

cross sections of cylindrical-like conduits and to outline blob-like

objects. We address the implementation details and illustrate the

capabilities of our snake with synthetic and real data.

Index Terms—Exponential B-spline, parametric snake, active

contour, parameterization, segmentation.

I. INTRODUCTION

A

contour is a curve that evolves from an initial position, which

is usually specified by a user, toward the boundary of an object.

The evolution of the curve is formulated as a minimization

problem. The associated cost function is called snake energy.

Snakes have become popular because it is possible for the

user to interact with them, not only when specifying its initial

position, but also during the segmentation process.

Research in this area has been fruitful and has resulted in

many snake variants [1], [2]. They differ in the type of curve

representation and in the choice of the energy term [3]. Snakes

can be broadly categorized in terms of curve representation as

• point-snakes, where the curve is described in a discrete

fashion by a set of points [4], [5], [6];

• parametric snakes, where the curve is described continu-

ously by some coefficients using basis functions [7], [8],

[9], [10], [11];

• implicit snakes, where the representation of the curve is

implicit and described as the level-set of a surface [12],

[13], [14], [15].

Point-snakes can be viewed as a special case of parametric

snakes where a large number of coefficients is used [10].

Parametric snakes require fewer parameters and result in

faster optimization. It can be shown that the computation

complexity of the snake energy, and, therefore, the speed

of the optimization algorithms is related to the size of the

CTIVE contours, and snakes in particular, are effective

tools for image segmentation. Within an image, an active

Copyright (c) 2010 IEEE. Personal use of this material is permitted.

However, permission to use this material for any other purposes must be

obtained from the IEEE by sending a request to pubs-permissions@ieee.org.

R. Delgado-Gonzalo, P. Th´ evenaz, and M. Unser are with the Biomedical

Imaging Group,´Ecole polytechnique f´ ed´ erale de Lausanne (EPFL), Switzer-

land. C.S. Seelamantula is with the Department of Electrical Engineering,

Indian Institute of Science (IISc), Bangalore, India.

This work was funded by the Swiss SystemsX.ch initiative under Grant

2008/005 and the Swiss National Science Foundation under Grant 200020-

121763.

Fig. 1. Approximation capabilities of the proposed parametric snake. The thin

solid line corresponds to an elliptical fit. The dashed thick line corresponds

to a generalized shape.

support of the basis functions [3]. It is therefore critical to

minimize this support while designing parametric snakes. The

curve of parametric snakes is represented explicitly, so that it

is easy to introduce smoothness and shape constraints [7]. It

is also straightforward to accommodate user interaction. This

is often achieved by allowing the user to specify some anchor

points the curve should go through [4]. The downsize of the

method is that the topology of the curve is imposed by the

parameterization. This makes parametric snakes less suitable

for handling topological changes, although solutions have been

proposed for specific cases [16], [17].

Implicit approaches offer great flexibility as far as the

curve topology is considered [18]. However, they tend to

be computationally more expensive since they evolve a 2-D

surface rather than a 1-D curve.

In this paper, we design fast parametric snakes capable of

perfectly outlining elliptic objects and yet versatile enough to

provide a close approximation of any closed curve in the plane.

We illustrate in Figure 1 how our snake can adopt the shape of

a perfect ellipse (i.e. reproduces the ellipse) as well as more

refined shapes. Segmenting circles and ellipses in images is

a problem that arises in many fields, for example biomedical

engineering [19], [20], [21], [22] or computer graphics [23],

[24]. In medical imaging in particular, it is usually necessary

to segment arteries and veins within tomographic slices [25].

Because those objects are physiological tubes, their section

show up as ellipses in the image. Ellipse-like objects are also

present at microscopic scales. For instance, cell nuclei are

known to be nearly circular [26] and water drops are similarly

spherical thanks to surface-tension forces [27]. However, these

elements deform and become elliptical when they are subject

to stress forces.

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SNAKES WITH ELLIPSE-REPRODUCING PROPERTY2

In order to efficiently segment elliptical objects, a paramet-

ric snake named the Ovuscule was proposed in [28]. It is a

minimalistic elliptical snake defined by three control points.

Its main drawback was that it was unable to represent shapes

different from circles and ellipses. Our goal here is to create

a more versatile parametric snake whose basis functions are

short, perfectly reproduce ellipses, and have good approxima-

tion properties. Our main contribution in this paper is to fulfill

this goal by selecting a special kind of exponential B-splines.

We are actually able to prove that our basis functions are the

ones with the shortest support among all admissible functions.

Since the computational cost of spline snakes is determined in

part by the size of the support of the basis function, our use

of the shortest possible support favors optimal performance.

The paper is organized as follows: In Section II we review

the general parametric snake model and formalize our design

constraints. Our main contribution is described in Section III,

where we build an explicit expression for the underlying basis

functions that fulfill our requirements, and we analyze in detail

its reproduction and approximation properties. Implementation

details such as energy functionals and discretization issues are

addressed in Section IV. Finally, we perform report evaluations

in Section V.

II. PARAMETRIC SNAKES

A. Parametric Representation of Closed Curves

A curve r(t) on the plane can be described by a pair of

Cartesian coordinate functions x1(t) and x2(t), where t ∈ R

is a continuous parameter. The one-dimensional functions x1

and x2are efficiently parameterized by linear combinations of

suitable basis functions. Among all possible bases, we focus

on those derived from a compactly supported generator ϕ

and its integer shifts {ϕ(· − k)}k∈Z. This allows us to take

advantage of the availability of fast and stable interpolation

algorithms [29].

We are interested in close curves specified by an M-periodic

sequence of control points {c[k]}k∈Z, with c[k] = c[k + M].

The parametric representation of the curve is then given by

the vectorial equation

r(t) =

∞

?

k=−∞

c[k]ϕ(M t − k).

(1)

The number of control points M determines the degrees of

freedom in the model (1). Small numbers lead to constrained

shapes, and large numbers lead to additional flexibility and

more general shapes.

Since the curve r is closed, each coordinate function is

periodic, and the period is common for both. For simplicity,

in (1) we normalized this period to be unity. Under these

conditions, we can reduce the infinite summation in (1) to

a finite one involving periodized basis functions as

r(t)=

M−1

?

k=0

∞

?

n=−∞

c[M n + k]ϕ(M (t − n) − k)

=

M−1

?

k=0

c[k]

∞

?

n=−∞

?

ϕ(M (t − n) − k)

???

ϕM(M t−k)

,

(2)

where ϕM is the M-periodization of the basis function ϕ.

This kind of curve parameterization is general. Using this

model, we can approximate any closed curve as accurately

as desired by using a higher number of vector coefficients

M2> M, provided that ϕ satisfies some mild conditions [30].

B. Desirable Properties for the Basis Functions

We now enumerate the conditions that our parametric snake

model should satisfy and introduce the corresponding mathe-

matical formalism.

1) Unique and Stable Representation. We want our para-

metric curve to be defined in terms of the coefficients in

such a way that unicity of representation is satisfied.

Furthermore, for computational purposes, we ask the

interpolation procedure to be numerically stable.

A generating function ϕ is said to satisfy the Riesz basis

condition if and only if there exist two constants 0 <

A ≤ B < ∞ such that

√M

A ?c??2≤

?????

∞

?

?2.

k=−∞

c[k]ϕ(M · −k)

?????

L2

≤ B ?c??2

(3)

of for

the

?∞

are linearly independent and every function is uniquely

specified by its coefficients. The upper inequality

ensures the stability of the interpolation process [29].

It has been shown in [31] that, due to the integer-

shift-invariant structure of the representation, the Riesz

condition has the following equivalent expression in the

Fourier domain:

∞

?

where ˆ ϕ(ω) =

transform of ϕ. Once expressed in the Fourier domain,

the Riesz condition provides a practical way to verify if

a given generating function ϕ satisfies (3).

2) Affine Invariance. Since we are interested in outlining

shapes irrespective of their position and orientation,

we would like our model to be invariant to affine

transformations, which we formalize as

∞

?

where A is a (2 × 2) matrix and b is a two-dimensional

vector. From (4), it is easy to show that affine invariance

is ensured if and only if

?

all

lower

k=−∞c[k]ϕ(M t − k) = 0 for all t ∈ R implies

that c[k] = 0 for all k ∈ Z. Thus, the basis functions

c

∈

inequality

A direct

that

consequence

theis condition

A ≤

k=−∞

?

|ˆ ϕ(· + 2πk)|2≤ B,

Rϕ(x)e−jω xdx denotes the Fourier

Ar(t) + b =

k=−∞

(Ac[k] + b) ϕ(M t − k),

(4)

∀t ∈ R :

∞

k=−∞

ϕ(M t − k) = 1.

(5)

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SNAKES WITH ELLIPSE-REPRODUCING PROPERTY3

Fig. 2.

functions (b) with exponential B-splines and M = 10. The dashed lines in

(b) indicate the corresponding basis functions.

Parametric representation of the unit circle (a) and its coordinate

In the literature, this constraint is often named the

partition-of-unity condition [29].

3) Well-Defined Curvature. The curvature of a parametric

curve at a point (x1(t),x2(t)) is given by

κ(x1,x2) =˙ x1¨ x2− ¨ x1˙ x2

(˙ x2

1+ ˙ x2

2)3/2,

where the dot denotes the derivative with respect to

t. We would like to be able to compute κ for every

point on the snake. To do so, each coordinate function

(or, equivalently, the basis ϕ) must be at least twice

differentiable and its second derivative must be bounded.

III. REPRODUCTION OF ELLIPSES

Since every ellipse can be obtained by applying an affine

transformation to the unit circle, we focus on the reproduction

of this simpler shape. This simplification is allowed whenever

the affine-invariance requirement stated in Section II-B is

satisfied.

A parametric snake defined by M vectorial coefficients

and by a generating function ϕ is said to reproduce the unit

circle if there exist two M-periodic sequences {c1[k]}k∈Zand

{c2[k]}k∈Zsuch that

∞

?

∞

?

That is, we need to be able to reproduce sinusoids of unit

period for each component of the parametric snake, as il-

lustrated in Figure 2. Note that, when (6) and (7) hold, it

is possible to represent any sinusoid of unit period for an

arbitrary initial phase using linear combinations of the two

sequences of coefficients.

cos(2πt) =

k=−∞

c1[k]ϕ(M t − k)

(6)

sin(2πt) =

k=−∞

c2[k]ϕ(M t − k).

(7)

A. Minimum-Support Ellipse-Reproducing Basis

We now present and prove our main result. We provide an

explicit expression for the minimum-support basis functions

that reproduce sinusoids.

Theorem 1: The centered generating function with minimal

support that satisfies all conditions in Section II-B and repro-

duces sinusoids of unit period with M coefficients is

In order to prove Theorem 1, we refer to the Distributional

Decomposition Theorem detailed in [32]. This Decomposition

Theorem provides a complete characterization of the fam-

ily of basis functions with minimum-support that reproduce

exponential polynomials. It states that every minimum sup-

port function ϕ that reproduces exponentials eαnt, for all

n ∈ [0...N − 1] with αi− αj / ∈ 2πjZ, can be written

as

N−1

?

where a is an arbitrary shift parameter that corresponds to

the lower extremity of the support of ϕ, and where βαis the

appropriate exponential B-spline

ϕ(t) =

cos2 π |t|

M

cos

π

M−cos2 π

M

M

1−cos2 π

M

M)

0 ≤ |t| <1

1

2≤ |t| <3

3

2≤ |t|.

2

1−cos2 π (3/2−|t|)

2(1−cos2 π

0

2

(8)

ϕ(t) =

n=0

λn

dn

dtnβα(t − a),

(9)

ˆβα(ω) =

N

?

n=1

1 − eαn−jω

jω − αn

.

(10)

Note that exponential B-splines are entirely specified by the

collection α = (α1,...,αN). The ordering of the poles αnis

irrelevant. A complete survey of the properties of exponential

B-splines can be found in [33].

We finally have the mathematical tools to justify our choice

for the generating function in (8).

Proof: Using (9), we see that ϕ needs to be constructed

from combinations of exponential B-splines with parameters

α = (0,j2π

M,−j2π

M) and N = 3. Therefore, we have

ϕ(t) =

2

?

n=0

λn

dn

dtnβα(t − a).

(11)

This ensures that ϕ is the shortest generating function that

reproduces constants and all sinusoids of unit period with

M coefficients. The constant-reproduction property is a direct

consequence of using α0= 0, and the sinusoids-reproduction

property can be proved by using α1= j2π

Euler’s identity.

Using properties of exponential B-splines, we know that

βα is twice differentiable. Moreover, the second derivative

is bounded but may be discontinuous. Therefore, λ1 and

λ2 in (11) must vanish to ensure that the curvature of the

snake is well-defined. Since ϕ reproduces constants, λ0 can

be computed by imposing the partition-of-unity condition.

From (5), we have that

?2π

Exponential B-splines parameterized by α form a Riesz

basis if and only if αm1− αm2/ ∈ 2πjZ for all pairs such

that m1?= m2. In our case, it is important to realize that this

condition is satisfied if and only if M ≥ 3. In other words, at

M, α2= −j2π

M, and

λ0=

M

?2

2?1 − cos2π

M

?.

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SNAKES WITH ELLIPSE-REPRODUCING PROPERTY4

Fig. 3. Plot of a quadratic B-spline β2and the resulting generating functions

given in (8) for M = 3, 4, 5, and 6. The function with the lowest peak at

t = 0 corresponds to M = 3, and as M increases, the height of the central

peak increases as well.

least three control points are needed to define our parametric

snake.

Finally, a closed form for ϕ is obtained by applying the

inverse Fourier transform to (11), which yields

ˆ ϕ(ω) = λ0ej3 ω

2

1 − e−jω

jω

1 − ej2 π

jω − j2π

2in order to ensure that the basis

M−jω

M

1 − e−j2 π

jω + j2π

M−jω

M

,

where we have set a = −3

function is centered.

We show in Figure 3 some members of this family of

functions for several values of M. We observe that they share

with the quadratic B-spline a finite support of length W = 3.

Likewise, they are one-time continuously differentiable and

have a similar bump-like appearance.

B. Approximation Properties of ϕ

Not only are we interested in reproducing ellipses, but

we would like our snake to be able to approximate any

other shape s. This is achieved by increasing the number

of degrees of freedom afforded by the number M of nodes.

In the Fourier domain, it is easy to see that ϕ converges to

a quadratic B-spline as M increases. Therefore, we expect

similar approximation properties for large values of M.

While ϕ leads to integer-shift invariance, the space spanned

by the generating function ϕ is not shift-invariant in general.

Hence, the approximation error using M vector coefficients is

dependent upon a shift in the continuous parameter t of the 1-

periodic function s. The minimum-mean-square approximation

error for a shifted function is given by

?1

=

?s(· − τ) − r(·)?2

where

r

is thebestapproximation

{ϕ(M · −k)}k∈Z. Since τ is usually unknown, we measure

the error averaged over all possible shifts as

??1

γ(τ,M)=

0

?s(t − τ) − r(t)?2dt

L2([0,1]),

withinthespan

η(M) =

0

γ(τ,M)dτ

?1

2

.

(12)

We give in Section III-C the decay of η as M → ∞,

following the method described in [30]. As expected, we find

that the best averaged-quadratic-mean error decays as 1/M3

when the number of vector coefficients M increases—the

same rate as the quadratic B-spline [34].

C. Approximation Order of ϕ

In this section, we introduce the necessary formalism to

compute the order of the approximation error associated to

the best-possible approximation of a periodic vector function

s within the span of the basis {ϕ(M · −k)}k∈Z, where ϕ is

given by (8).

As explained in Section III-B about the approximation

properties of ϕ, the space spanned by the generating function

ϕ is not shift-invariant in general. Hence, as a metric of

dissimilarity between shapes, we use the averaged minimum-

mean-square approximation error η.

Using the main result of [30], we obtain the asymptotic

behavior of η as

η2(M)=C2

+ C2

???

1(M) ?˙ s?2

2(M)?¨ s?2

L2([0,1])M−2

L2([0,1])M−4+ O?M−6?,

??ˆ ϕ(L)(2πk)??2?

where CL =

L-th derivative of the Fourier transform of ϕ. Following

lengthy calculations, we get (13) and (14), where we defined

M0= π cotπ

C2(M) = O(M−2). Since the curve s does not depend on

M, we can also write that

η(M) =?O?M−6??1

which shows that the averaged quadratic mean error decays as

M−3.

1

L!

k?=0

and ˆ ϕ(L)is the

M. It can be shown that C1(M) = O(M−2) and

2= O?M−3?,

D. Best Constant and Ellipse Fitting

Since our snakes have the capability of perfectly repro-

ducing ellipses, it is natural to ask which is the best ellipse

that approximates the parametric curve r defined by the M-

periodic sequence {c[k]}k∈Z. In other words, we are interested

in finding the ellipse rethat minimizes

?1

Since r is continuous and 1-periodic, we can expand it in a

Fourier series as

∞

?

The Fourier-series vector coefficients R in (15) are given by

?1

1

Mˆ ϕ(2πn

M

?r − re?2

L2([0,1])=

0

?r(t) − re(t)?2dt.

r(t) =

n=−∞

R[n]ej2πnt.

(15)

R[n]=

0

r(t)e−j2πntdt

=

)

M−1

?

k=0

c[k]e−j2 π

Mnk,

(16)

where the parametric expression of r has been used in the

second equality.

Page 5

SNAKES WITH ELLIPSE-REPRODUCING PROPERTY5

C1(M)=

1

12π

?

18 (M0− M) (M0+ 4M) + 30π2

?

(13)

C2(M)=

1

120π2

225 (2M4

0− 7M2M2

0− 15M3M0+ 20M4) + 75 (8M2

0− 29M2) π2+ 170π4

(14)

From the classical theory of harmonic analysis, we know

that the best ellipse approximation (component-wise sinusoids)

of r, in the L2([0,1]) sense, is the first-order truncation of the

series (15), where only the terms n = −1, n = 0, and n = 1

are kept. Therefore, we have that

re(t)=R[0] + (R[1] + R[−1]) cos(2πt)

+ j (R[1] − R[−1]) sin(2πt),

(17)

where R[0] is the center of gravity of the snake. The Fourier

coefficients in (17) can easily be obtained from (16) as

R[0]=

1

M

M−1

?

k=0

c[k]

R[1] + R[−1]=

M−1

?

M−1

?

k=0

hc[k]c[k]

j (R[1] − R[−1])=

k=0

hs[k]c[k],

where

hc[k]=

2

Mcosπ

2

Mcosπ

Mcos2πk

Msin2πk

M

hs[k]=

M

.

Since all sinusoids of unit period can be reproduced by

the generating function ϕ and the appropriate M-periodic

sequence of coefficients c, the curve re belongs to the span

of ϕ. For the sake of completeness, we provide in the next

section an explicit expansion of sinusoids in terms of ϕ.

E. Expansion of Sinusoids with ϕ

Here, we explicitly find the sequence of M vector co-

efficients that reproduce sinusoids of unit period using the

generating function ϕ given in (8). We start by recalling the

exponential-reproducing property of the exponential B-splines

as

∞

?

Setting α = j2π

complex exponential ej2 π

convolve both sides of (18) with β(0,−j2 π

?

=ej2 π

?

eαt=

k=−∞

M, we see that β(j2 π

Mt, which is M-periodic. If we now

M), we get that

M·?

Mk(β(j2 π

??

(2 π

eαkβ(α)(t − k).

(18)

M)reproduces the

β(0,−j2 π

M)∗ ej2 π

∞

?

(t)

k=−∞

M)∗ β(0,−j2 π

M))(t − k)

?

2(1−cos2 π

M)

M)

2

ϕ(t−3

2−k)

,

where we have used the definition of ϕ from (8), along

with the fact that the convolution operator commutes with

the shift operator. To simplify the left-hand side, we invoke

an important property of linear shift-invariant (LSI) systems:

complex exponentials are eigenfunctions of LSI operators. By

virtue of this property, if the complex exponential ejαtis

presented at the input of a system specified by the impulse

response h, then its output is given byˆh(α)ejαt, whereˆh

denotes the Fourier transform of h. If we consider β(0,−j2 π

as the impulse response of a LSI system, then

?

?

Therefore, we have that

∞

?

By flipping the sign of α we can easily obtain an analogous

result for the reproduction of e−j2 π

results, we have that

?

?

where

2?1 − cos2π

2?1 − cos2π

Note that the sequences c1and c2are M-periodic and that the

summations in (19) and (20) can be reduced to finite ones if

we make use of the periodized basis functions.

We have expressed in (19) and (20) how to compute the

vector coefficients for reproducing sinusoids of unit period

and initial phase of

of c1and c2then allows one to reproduce sinusoids of arbitrary

shape.

M)

β(0,−j2 π

M)∗ ej2 π

M·?

(t) =ˆβ(0,−j2 π

M)(ω)

??

???

ω=2 π

M

?

λ

ej2 π

Mt.

ej2 π

Mt=

k=−∞

ej2 π

Mk21 − cos2π

λ?2π

M

?2ϕ(t −3

Mt. Finally, by using both

M

2− k).

cos2π

?

?

t +

3

2M

??

??

=

∞

?

∞

?

k=−∞

c1[k]ϕ(M t − k)(19)

sin2π

t +

3

2M

=

k=−∞

c2[k]ϕ(M t − k), (20)

c1[k]=

M

?cosπ(2k+3)

M

?sinπ(2k+3)

M

M

cosπ

M− cos3π

c2[k]=

MM

cosπ

M− cos3π

.

3π

M. The appropriate linear combination

IV. IMPLEMENTATION

Since the presented parametric active contour is a spline

snake, it is capable of handling all traditional energies ap-

plicable to point-snakes and parametric snakes. However,

to illustrate the behavior of our parameterization in a real

implementation, we performed our experiments with a specific

snake energy that we designed to be versatile.

Page 6

SNAKES WITH ELLIPSE-REPRODUCING PROPERTY6

In this section, we first introduce the snake energy that

drives the optimization process, and then we provide a de-

scription of the implementation details for the proposed snake.

We construct the energy functional to detect dark objects on

a brighter background.

A. Snake Energy

The active-contour algorithm is always driven by a chosen

energy function. Thus, the quality of the segmentation depends

on the choice of this energy term. There are many construction

strategies which can be categorized in two main families: 1)

edge-based schemes, which use gradient information to detect

contours [4], [7], [10] and 2) region-based methods, which use

statistical information to distinguish different homogeneous

regions [9], [35]. In order to benefit from the advantages of

both strategies, a unified energy was proposed in [3]. In our

case, we are going to follow a similar approach by using a

convex combination of gradient and region energies, like in

E = αEedge+ (1 − α) Eregion

where α ∈ [0,1]. The tradeoff parameter α balances the contri-

bution of the edge-based energy and the region-based energy.

Its value depends on the characteristics of each particular

application.

For the gradient-based (or edge) energy, we consider the

one described in [35] since it has the advantage of penalizing

the snake when the orientation is inconsistent with the object

to segment. Let r be our parametric snake. The contour energy

term is then given by

?

where dx denotes the tangent vector of the curve in the three-

dimensional space formed by the image plane and its or-

thogonal dimension, where k = (0,0,1) denotes the outward

vector orthonormal to the image plane, where ∇f(x1,x2) =

?

image f at (x1,x2) on the curve, and where × is the 3D

cross product. In Figure 4, we present the configuration of

the various quantities involved. The chirality of the system

of coordinates will determine the sign of the integrand, as

discussed in [3], [35]. Using Green’s theorem, the edge energy

can be also expressed as the surface integral

?

where x = (x1,x2), ∆f is the Laplacian of the image f, and

Ω is the region enclosed by r.

For the region-based energy, we adopt a strategy similar

to [28]. More precisely, our region-based energy discriminates

an object from its background by building an ellipse rλaround

the snake and maximizing the contrast between the intensity

of the data averaged within the curve, and the intensity of the

data averaged over the elliptical shell Ωλ. When Ω ⊂ Ωλ, the

(21)

Eedge= −

r

kT(∇f(x1,x2) × dx),

∂f(x1,x2)

∂x1

,∂f(x1,x2)

∂x2

,0

?

is the within-plane gradient of the

Eedge= −

Ω

∆f(x)dx1dx2,

(22)

Fig. 4.

its interaction with an object constituted by a gray semicircle (representing

low pixel values), of the vector dx tangent to the curve, and of the gradient

vector ∇f of the image. The vector k, which is mentioned in the text, is

perpendicular to the image plane and points outwards, towards the reader

.

Schematic representation of a parametric snake r (dashed line), of

region energy term can be expressed as

Eregion

=

1

|Ω|

??

?

Ω

f(x)dx1dx2

−

Ωλ\Ω

f(x)dx1dx2

?

,

(23)

where |Ω| is given by

M−1

?

The normalization factor |Ω| can be interpreted as the signed

area, defined as |Ω| = −?

on the curve r. In this paper, we follow the usual convention

whereby an anti-clockwise path leads to a positive sign. We

enforce our criterion to remain neutral (Eregion = 0) when

f takes a constant value, for instance in flat regions of the

image. To achieve this we set |Ωλ| = 2 |Ω|.

The construction of the elliptic shell is performed using the

best ellipse re given in (17), and magnifying its axes by a

factor λ to achieve

|Ω| = −

k=0

M−1

?

n=0

c1[k]c2[n]

?M

0

ϕM(t − n) ˙ ϕM(t − k)dt.

rx2dx1. The sign of the quantity

|Ω| depends on the clockwise or anti-clockwise path followed

rλ(t)=R[0] + λ (R[1] + R[−1]) cos(2πt)

+ jλ (R[1] − R[−1]) sin(2πt),

where λ =?2|Ω|/|Ωe| and |Ωe| is the signed area enclosed

|Ωe| = −4π

M

k=0

n=0

In Figure 5, we illustrate how we take advantage of the

ideas presented in Section III-D to build the best ellipse

approximation reof an arbitrary snake r. Using the constraint

|Ωλ| = 2 |Ω|, we can determine the contour rλ of the

enclosing shell Ωλ.

by the curve re, with

M2cosπ

M−1

?

M−1

?

c1[k]c2[n] sin2π(n − k)

M

.

B. Accelerated Implementation

The computational cost is dominated by the evaluation of

the surface integrals in (22) and (23). An efficient way to

implement these operations is the use of pre-integrated images.

Let g be the function we are integrating (∆f, f, or −f,

Page 7

SNAKES WITH ELLIPSE-REPRODUCING PROPERTY7

Fig. 5.

mation re, and the corresponding enclosing shell rλused in Eregion.

Representation of the parametric snake r, the best ellipse approxi-

respectively) and let Γ be the domain of integration (Ω or Ωλ).

Then, by Green’s theorem, we rewrite the surface integrals as

the line integrals

?

=

Γ

g(x)dx1dx2

=

−

?

?

∂Γ

g2(x1,x2)dx1

∂Γ

g1(x1,x2)dx2,

where ∂Γ is the boundary of Γ, and

g2(x1,x2) =

?x2

?x1

−∞

g(x1,ξ2)dξ2

(24)

g1(x1,x2) =

−∞

g(ξ1,x2)dξ1.

(25)

The use of Green’s theorem to rewrite the surface integrals

as line integrals reduces dramatically the computational load.

This can only be achieved if the curve is defined continuously,

like with the curves of Section II-A. By contrast, this acceler-

ation would not be available to methods such as point-snakes

and level-sets, because their implementation ultimately relies

on discretization.

C. Sampling

Despite the fact that we are assuming a continuously defined

model for our functions, in a real-world implementation we

only have at our disposal a sampled version of the functions we

want to pre-integrate. To solve this inconsistency, we perform

a bilinear interpolation of the sampled data and we store in

lookup tables the values of (24) and (25) at integer locations.

Then, the energies are obtained using the first approximation

given by the lookup tables. In our implementation, we have

corrected them by supplying a residual that allows us to get

the exact result. We were able to determine this residual

analytically but, in the interest of space, we do not provide

it here.

D. Optimization

The optimization of the snake can be efficiently carried

out by Powell-like line-search methods [36]. These methods

Fig. 6. Mean time of one iteration in the snake evolution.

require the derivatives of the energy function with respect

to the parameters (i.e., the knot coefficients), and converge

quadratically to the solution. The algorithm proceeds as fol-

lows: firstly, one direction within the parameter space is

chosen depending on the partial derivatives of the energy.

Secondly, a one-dimensional minimization is performed within

the selected direction. Finally, a new direction is chosen using

the partial derivatives of the energy function once more, while

enforcing conjugation properties. This scheme is repeated

till convergence. Assuming a bilinear interpolation of the

original function f, we were able to derive exact and closed

expressions for these derivatives that take the residual of the

lookup table into account.

For spline snakes it has been shown that the evaluation of

the partial derivatives of the energy of the form (21) depends

quadratically on the number of parameters [3]. In Figure 6,

we compare the computational cost of the snake during line

minimization (simple update), and when the energy gradient

is required to chose a new direction (gradient update). For the

latter case, we contrast the computation time of an analytical

computation of the gradient to that of a centered finite differ-

ences approach. For low values of M, the simple update and

the gradient update using analytical energy gradient lead to a

similar computational load. As the value of M increases, the

quadratic behavior of the computation of the gradient makes

the update cost increase. This quadratic behavior can be easily

discerned in the topmost curve of Figure 6.

V. EXPERIMENTS

We present in this section four experimental setups. In the

first one, we compare our choice in (8) against the classical

quadratic B-spline when representing sinusoids. We move

away from sinusoids in the second experiment, where we work

with synthetic data and perform an objective validation of the

segmentation properties of our snake in noiseless and noisy

environments. In the fourth setup, we also perform a quantita-

tive evaluation by segmenting real cardiac MRI data. Finally,

in the last experiment, we illustrate some real applications of

our snake where the ground truth is not available.

A. Approximation of Sinusoids

By design, our basis function ϕ has the property of repro-

ducing sinusoids exactly. By contrast, the classical polynomial

Page 8

SNAKES WITH ELLIPSE-REPRODUCING PROPERTY8

Fig. 7.

representation (solid line) using ϕ3 (dashed lines). (b) Best parametric

approximation (solid line) using β2(dashed lines).

Approximations of a sin function with unit period. (a) Parametric

B-splines do not enjoy this property. In this section, we

are focusing on this aspect and exhibit the amount of error

committed by B-splines when attempting to reproduce a sin

function.

We start with exact reproduction by our basis. Using the

result of Section III-E, we determine the coefficients for the

case M = 3 (smallest possible M). They are given by

√3 (ϕ3(3t − 1) − ϕ3(3t + 1)),

where ϕ3 corresponds to the 3-periodization of the basis

function (8), as in (2).

We continue with approximate reproduction by B-splines.

For fairness, we choose a quadratic B-spline β2so that the

size of the support of β2and ϕ is the same. The reproduction

will be approximate, not because of the limited size of the

support, but because the sin function does not lie in the span

of polynomial B-splines of any degree. Nevertheless, we can

compute the coefficients that best adjust the sinusoid with unit

period in the least-squares sense. This yields

sin2πt ≈1215

26π2

where

periodized basis function as in (2).

We observe in Figure 7 that both constructions result in

sine-like functions. However, the reproduction is exact in the

left part of Figure 7, while it is only approximate in the right

part. This happens even though the support of β2is identical

to the support of ϕ, even though the asymptotic approximation

properties of β2and ϕ are identical, and even though β2and ϕ

have the same degree of differentiability. We show in Figure 8

the amount of error committed by the parabolic approximation.

We determine that MSE =1

sin(2πt) =

?β2

3(3t − 1) − β2

3(3t + 1)?,

β2(t) =

3

4− |t|2

1

2

0

0 ≤ |t| <1

1

2≤ |t| <3

3

2≤ |t|

2

?3

2− |t|?2

2

(26)

is the quadratic B-spline and the subscript 3 indicates a 3-

2−98415

208π6.

B. Accuracy and Robustness to Noise

In this section, two experiments are carried out. The first

one consists in outlining different synthetic blob-like shapes

Fig. 8. Sinusoid of period 3, its representation with our basis function (solid

line), and its best quadratic B-spline approximation (dashed line).

in a noise-free environment. The second experiment consists in

outlining one specific target within an image, this time, in the

presence of noise. In both experiments we set α = 0, that is,

we make use of the region energy only. This particular choice

ensures that the snake is not mislead by noisy boundaries in

the presence of excessive of noise.

In the first experiment, we generate 10 test images of size

(512 × 512) by pixel-wise sampling of our shape of interest,

which is built by intersecting or making the union of two

circles of radius 50 pixel units. We illustrate these shapes in

the header of Table I. They are parameterized with the distance

d, in pixel units, between the centers of the circles. For d < 0,

the shape is built by the intersection of the two circles. For

d ≥ 0, they are parameterized by their union. The grayscale

values of the images are 255 for the shape, and 0 for the

background.

We used the Jaccard distance J = 1 − |Θ ∩ Ω|/|Θ ∪ Ω|

to measure as a percentage the dissimilarity between the two

sets. There, Θ corresponds to the ground-truth region, and Ω

corresponds to the region enclosed by the snake. We computed

J with a pixel-wise discretization of the images.

In the simulations of Table I, we investigated the depen-

dence of J on the number M of coefficients and the distance

d between the circles. We denoted with a dash (−) when the

snake did not converge, and therefore, we could not compute

the Jaccard distance. We initialized every snake as a circle with

a radius of 75 pixel and a center that lay in the middle of the

shape. We observe that the results in Table I tend to improve

as the number M of control points is increased, especially for

the non-elliptical shapes. However, the increase in the number

of control points does not bring any further improvement when

the shape to segment is a perfect circle. This result is expected

since the circular shape is reproduced exactly for any M ≥ 3.

The residual error seen in Table I for d = 0 can be attributed

to the discretization of Θ and Ω. We also observe that for

d = −80 and d = −64 the Jaccard distance starts increasing

severely for M ≥ 7 and for M ≥ 9, respectively. This is due to

the fact that the sharp corners of the shape lead to loops in the

curve during the optimization process. Such self-intersections

violate the conditions of Green’s theorem in Section IV-A.

In the second experiment, we investigated the sensitivity

Page 9

SNAKES WITH ELLIPSE-REPRODUCING PROPERTY9

TABLE I

ERROR PERCENTAGE OF OUR SNAKE FOR NOISELESS SYNTHETIC DATA.

TABLE II

PERCENTAGE OF SUCCESS RATE OF OUR SNAKE FOR NOISY SYNTHETIC DATA.

to noise of our snake depending on the number of snake

coefficients M. We generated 100 noisy realizations of a circle

of radius 50 pixel units for different signal-to-noise ratios. We

computed the power of the noise over a region of interest of

size (200 × 200). We illustrate a realization of the resulting

images in the header of Table II.

We show the percentage of success in Table II. We consid-

ered that our snake succeeded in segmenting the circle when

the optimization process led to a segmentation with J < 1%.

This criterion is very conservative as shown in Figure 9. We

observe from the results that our snake is robust against noise

since it is capable of giving a proper segmentation even for low

signal-to-noise ratios. Furthermore, the increased sensitivity

to noise as we increase the number of vector coefficients M

corresponds to the appearance of additional noise-related local

minima in the energy of the snake. Therefore, M should be

chosen as small as possible in order to avoid over-fitting of

the noise, but big enough to be able to approximate the shape

of interest.

C. Medical Data

Now, we move away from synthetic data. We compare our

snake against other snake variants in terms of accuracy and

speed. We quantify their accuracy at outlining the endocardial

wall of the left ventricle within slices of 3D cardiac MR image

sequences.

The data we used are short-axis cardiac MR image se-

quences from 33 subjects acquired in the Department of Imag-

ing of the Hospital for Sick Children in Toronto, Canada [37].

Fig. 9.

Barely accepted with J= 0.853%. (b) Barely rejected with J= 1.001%. (a)

Rejected with J= 81.065%.

Segmentation results for noisy synthetic data with SNR= −5dB. (a)

For each subject, data consist of a time-series of 20 volumes.

For each volume, the number of slices varies from 8 to 15.

Each slice is a (256 × 256) image with a pixel spacing be-

tween 0.93 mm and 1.64 mm. The ground truth was obtained

by manual annotation. In each segmented image 1,000 points

(named landmark points) define a closed polygon outlining

the endocardial wall.

1) Accuracy: For each subject, we selected one slice guided

by its anatomical structures along the long axis and its timing

in the cardiac cycle. Since the region of interest is nearly

elliptical, we used the minimalistic elliptical active contour

named Ovuscule to provide a first estimate of the location

and orientation of the left ventricle [28]. Then, we refined

the segmentation of the endocardial wall using the general

parametric active contour model (1) for different values of M

and several basis functions. More specifically, we used linear

Page 10

SNAKES WITH ELLIPSE-REPRODUCING PROPERTY10

??

??

??

?

?

?

?

?

???????????????????????????

?????????

?

?

?????????

?

??????????

?

?????????????????

?

???????????????????????????????

?

????????????????????????????????

Fig. 10. Mean and variance of the landmark error across all 33 patients.

?

?

?

?

?

?

?

??????????????????????????

?????????

?

?

?????????

?

??????????

?

?????????????????

?

???????????????????????????????

?

????????????????????????????????

Fig. 11. Median of the landmark error across all 33 patients.

??

??

??

??

??

??

??

??

?

???????????????????????????

?????????

?

?

?????????

?

??????????

?

?????????????????

?

???????????????????????????????

?

????????????????????????????????

Fig. 12. Maximum landmark error among all 33 patients.

and quadratic B-splines, our function (8) that we refer to as

third-order exponential spline, and an extended version of (8)

that we refer to as fourth-order exponential spline. The linear

B-spline basis function has a smaller support than our function

(8). However, it can only adopt the form of polygons. The

quadratic B-spline basis function has the same support and

regularity than (8). However, it is unable to reproduce ellipses.

Finally, the fourth-order exponential spline is an extended

version of (8), with one more degree of regularity, but with

a support one unit larger. The initialization provided by the

Ovuscule could be carried over to (8) and to the fourth-order

exponential spline. In the case of the other types of snake,

the perfect ellipse of the Ovuscule cannot be reproduced but

must be approximated. This approximation was achieved by

sampling the outline of the Ovuscule.

In a preprocessing step, the images were magnified four

times horizontally and vertically. Firstly, we evolved the

Ovuscule on the magnified image. Secondly, we evolved more

refined snakes, guided exclusively by the edge energy on a

smoothed version of the magnified image. The smoothing

was Gaussian, with a kernel of variance σ2= 102. We then

measured the landmark error. We computed this error as the

mean distance of the snake to the landmark points given by

the ground truth, as was done in [37].

In Figures 10, 11, and 12, we show the mean, median,

and maximum values of the landmark error, respectively.

From these graphs, we validate that the Ovuscule provides

a good and robust starting point to be refined by the snakes

investigated in this paper. The polygonal snake does not reach

the accuracy of the Ovuscule till M = 7, and exhibits a

high variance across subject. The quadratic-spline snake and

the third-order exponential-spline snake converge to similar

accuracy starting with M = 4. This was expected, since we

showed in Section III-B that our function does converge to a

quadratic B-spline when M increases. However, for low values

of M, the difference is noticeable, and the quadratic-spline

snakes produce shapes that are not compatible with the region

of interest. Finally, the fourth-order exponential-spline snakes

produce equivalent results in terms of accuracy and stability

than the third-order exponential-spline snake, at a price of a

larger support, and therefore, of a slower convergence.

In Figure 13b, we illustrate the initialization provided to

the Ovuscule, and in Figure 13c the outcome of optimizing

the Ovuscule, which will provide the initialization for further

processing. We also show the result of several more elaborated

snake variants, and how they compare with the ground truth.

The fourth-order exponential-spline snake results in an outline

that is visually indistinguishable from that of the third-order

one, but comes at an increased computational cost.

2) Speed: In terms of speed, we compared our proposed

snake to some classic traditional snakes such as a Kass-like

snake [38] and a traditional Geodesic Active Contour (GAC)

model [13].

In this analysis, we used the anatomical structures of the

33 patients found in Section V-C1. However, we modified our

initialization procedure to accommodate for the GAC model,

since it fails unless the initial contour lies totally inside or out-

side of the boundary of interest. Therefore, we scaled down the

initialization that was provided by the outcome of optimizing

an Ovuscule in Section V-C1. By doing so, we guarantee that

all initial contours lay inside the endocardial wall to segment.

Unfortunately, neither the Kass-like snake nor the GAC model

are able to reproduce the initial ellipse perfectly and their

initialization must be approximated. This approximation was

achieved by sampling the outline of the Ovuscule. Finally, we

refined the segmentation of the endocardial wall either using

our snake model for different values of M, the Kass-like snake,

or the GAC.

This experiment was performed on a MacPro 3.1 with two

Quad-Core Intel Xeon processors and 8GB of RAM memory

running Mac OS X 10.6.8. The implementation of the Kass-

like active contour was taken from [38], and the one of GAC

model from the free open-source image processing package

Page 11

SNAKES WITH ELLIPSE-REPRODUCING PROPERTY 11

Fig. 13. Outline of the endocardial wall in the first frame and fourth slice of

the second patient. (a) Raw data. (b) Initialization. (c) Ovuscule. (d) Ground

truth. (e) Polygonal snake with M = 3. (f) Quadratic-spline snake with M =

3. (g) Third-order exponential-spline snake with M = 3. (h) Fourth-order

exponential-spline snake with M = 4.

?

?

?

?

?

???????????????????????????

?????????

?????????????

????????????????????????????????

?

????? ????

Fig. 14. Temporal evolution of the Jaccard distance. During the 2 seconds of

snake evolution, the proposed method with M = 3 performed 1479 iterations,

with M = 5 it performed 1406 iterations, and with M = 3 it performed 889

iterations. The Kass snake performed 17 iterations, the first of which took

370ms, and the GAC performed 34 iterations.

Fiji1implementing the algorithm described in [13].

In Figure 14, we show the mean temporal evolution of

the improvement of the Jaccard distance during the snake

evolution process for the 33 patients. We can clearly see

that the proposed snake reaches its optimum earlier than the

classical Kass-like snake and the GAC model. The Kass-like

snake has a very costly first step, and then it cannot escape

a local minimum. The GAC is executed with an advection

value of 2.20, and a propagation value of 1. These parameters

make the GAC succeed in overcoming the local minimum, but

the convergence rate is still slower than that of the parametric

case. It is important to notice that, for our proposed model,

an increase in the number M of control points slows the

convergence. As pointed out in Section IV-D, this is due to the

fact that larger values of M increase the computational load

per iteration of the snake.

D. Real Data

Here, we illustrate the behavior of our snake and provide

further insights into its capabilities. In the context of this

section, the ground truth is missing, so we must relinquish

quantitative assessments in favor of qualitative ones.

1) HeLa Nuclei: We want to evaluate the success of our

snake model at outlining ellipse-like targets in the context of

automated time-lapse microscopy. We use (434 × 434) images

of HeLa nuclei that express fluorescent core histone 2B on an

RNAi live cell array. We show in Figure 15 the result of the

optimization process with (8) and M = 5. This number of

points is high enough to capture small departures from an

elliptic shape.

We initialized every snake as a circle of radius of 25

pixel units, as shown in Figure 15. These initial circles were

centered on the locations given by a maxima detector applied

over a version of the image that was smoothed with a Gaussian

kernel of variance σ2= 122pixel. A total number of 23

maxima were detected. We then proceed with an inverted

version of the original, unsmoothed image to optimize the

snakes. The optimization process converged in 22 cases. We

1http://fiji.sc/

Page 12

SNAKES WITH ELLIPSE-REPRODUCING PROPERTY 12

Fig. 15.

parametric snakes were built with M = 5. (a) The initial contour of the snake.

(b) Result provided by our snake.

Outline of HeLa nuclei in a fluorescence microscopy image. The

show in Figure 15 the result of the outlining process. We

observe that our snakes were successful in most of the cases.

2) Droplets: As a second example, we show the outline of

sprayed and deformed water droplets hitting a surface. The

flight and the impact of the droplet was captured by a high-

speed camera (Photron Fastcam) at a rate of 10,000 images/s.

The shape of the droplet is changing during flight, at impact,

and while bouncing. After cropping, the size of the image was

(663 × 663) pixels.

We analyzed two frames. One was an image taken before

the collision took place, the other was taken after the impact.

In both cases, we initialized the snake as a circle with a

position and size that we chose manually. These initializations

are shown in Figure 16. In the image prior to the impact, which

we show in the left part of Figure 16, a snake with M = 5 was

used. We selected a small value for M because the droplet

is nearly circular. In the image after the impact, which we

show in the right part of Figure 16, five control points did

not provide enough freedom to cope with the discontinuity

created by the attachment to the surface. However, the outline

Fig. 16.

snake is is represented with a black dashed line. (b) After the impact: The

initial contour of the snake is is represented with a black dashed line. (c) Prior

to the impact: The outline of our snake with M = 5 is represented with a

white dashed line. (d) After the impact: The outline of the successful snake

is is represented with a white dashed line (M = 8), while the configuration

with M = 5 is represented with a gray solid line. The droplet edges are

partially out of focus, making them blurry and noisy.

Sprayed droplets. (a) Prior to the impact: The initial contour of the

was successfully retrieved when slightly increasing the number

of nodes to M = 8.

The method described in this article has been programmed

as a plugin for ImageJ, which is a free open-source multi-

platform Java image-processing software2. Our plugin3is

independent of any imaging hardware and, thanks to ImageJ,

any common file format may be used.

VI. CONCLUSIONS

Our contribution in this paper is a new family of basis

functions that we use to describe parametric contours in terms

of a set of control points. We were able to single out the

basis of shortest support that allows one to reproduce circles

and ellipses. Those can be characterized exactly by as few as

three control points but, by considering additional ones, our

parametric contours can reproduce with arbitrary precision any

planar closed curve. In particular, we have shown that the mean

error of approximation decays in inverse proportion of the cube

of the number of control points. We have used our ellipse-

reproducing parametric curves to build snakes driven by a

combination of contour and region-based energies. In the latter

case, the energy depends on the contrast between two regions,

one being delineated by the curve itself, and the other by an

ellipse of double area. To determine this ellipse, we showed

2http://rsb.info.nih.gov/ij/

3http://bigwww.epfl.ch/algorithms/esnake/

Page 13

SNAKES WITH ELLIPSE-REPRODUCING PROPERTY 13

how to compute the best elliptical approximation, in a least-

squares sense, of a contour described by an arbitrary number of

control points. We were able to accelerate the implementation

of our snakes by taking advantage of Green’s theorem, which

was facilitated by the availability of the explicit expressions of

our basis. We have applied our snakes to a variety of problems

that involve synthetic simulations and real data. We achieved

excellent objective and subjective performance.

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Page 14

SNAKES WITH ELLIPSE-REPRODUCING PROPERTY 14

Ricard Delgado-Gonzalo was born in 1983 in

Barcelona, Spain. He received two diplomas in

Telecommunications Engineering and in Mathemat-

ics from the Universitat Polit` ecnica de Catalunya

(UPC) in 2006 and 2007, respectively. In 2008,

he joined the Biomedical Imaging Group at the

´Ecole polytechnique f´ ed´ erale de Lausanne (EPFL),

Switzerland, as a Ph.D. student and assistant. He

currently works on applied problems related to im-

age reconstruction, segmentation, and tracking, as

well as on the mathematical foundations of signal

processing and spline theory. His other important professional interests include

technology management and transfer.

Philippe Th´ evenaz was born in 1962 in Lausanne,

Switzerland. He graduated in January 1986 from the

´Ecole polytechnique f´ ed´ erale de Lausanne (EPFL),

Switzerland, with a diploma in microengineering.

He then joined the Institute of Microtechnology

(IMT) of the University of Neuchˆ atel, Switzerland,

where he worked in the domain of image processing

(optical flow) and in the domain of speech process-

ing (speech coding and speaker recognition). He

obtained his Ph.D. in June 1993, with a thesis on

the use of the linear prediction residue for text-

independent speaker recognition. He then worked as a Visiting Fellow with the

Biomedical Engineering and Instrumentation Program, National Institutes of

Health (NIH), Bethesda MD, USA, where he developed research interests that

include splines and multiresolution signal representations, geometric image

transformations, and biomedical image registration. Since 1998, he is with

the EPFL as senior researcher.

Chandra Sekhar Seelamantula (M’99) was born

in 1976 in Gollepalem, Andhra Pradesh, India. He

obtained a Bachelor of Engineering (B.E.) degree in

1999 with a Gold medal from the University College

of Engineering, Osmania University, India, with a

specialization in Electronics and Communication

Engineering. He obtained a direct Ph.D. degree in

2005 from the Indian Institute of Science, Depart-

ment of Electrical Communication Engineering, with

a thesis titled, Time- varying Signal Models: Enve-

lope and Frequency Estimation with Applications to

Speech and Music Signal Compression. During his Ph.D., he also specialized

in the development of auditory-motivated signal processing models for speech

and audio applications. During April 2005 - March 2006, he worked as a

Technology Consultant for M/s. ESQUBE Communication Solutions Private

Limited, Bangalore, and developed proprietary audio coding solutions. In

April 2006, he joined the Biomedical Imaging Group,´Ecole polytechnique

f´ ed´ erale de Lausanne, Switzerland, as postdoctoral fellow and specialized in

the field of Image Processing, Optical-Coherence Tomography, Holography,

Splines, and Sampling Theories. Since July 2009, he is Assistant Professor

at the Department of Electrical Engineering, Indian Institute of Science,

Bangalore.

Michael Unser (M’89-SM’94-F’99) received the

M.S. (summa cum laude) and Ph.D. degrees in

Electrical Engineering in 1981 and 1984, respec-

tively, from the ´Ecole polytechnique f´ ed´ erale de

Lausanne (EPFL), Switzerland. From 1985 to 1997,

he worked as a scientist with the National Institutes

of Health, Bethesda USA. He is now full professor

and Director of the Biomedical Imaging Group at the

EPFL. His main research area is biomedical image

processing. He has a strong interest in sampling

theories, multiresolution algorithms, wavelets, and

the use of splines for image processing. He has published 200 journal papers

on those topics, and is one of ISI’s Highly Cited authors in Engineering

(http://isihighlycited.com). Dr. Unser has held the position of associate Editor-

in-Chief (2003-2005) for the IEEE Transactions on Medical Imaging and has

served as Associate Editor for the same journal (1999-2002; 2006-2007), the

IEEE Transactions on Image Processing (1992-1995), and the IEEE Signal

Processing Letters (1994-1998). He is currently member of the editorial boards

of Foundations and Trends in Signal Processing, and Sampling Theory in

Signal and Image Processing. He co-organized the first IEEE International

Symposium on Biomedical Imaging (ISBI2002) and was the founding chair

of the technical committee of the IEEE-SP Society on Bio Imaging and Signal

Processing (BISP). Dr. Unser received the 1995 and 2003 Best Paper Awards,

the 2000 Magazine Award, and two IEEE Technical Achievement Awards

(2008 SPS and 2010 EMBS). He is an EURASIP Fellow and a member of

the Swiss Academy of Engineering Sciences.