THE EMERGENT STRUCTURE OF THE DROSOPHILA WING

A Dynamic Model Generator

Alberto Silletti

Department of Information Engineering, University of Padova, Via Gradenigo 6A, Padova, Italy

Angelo Cenedese

Department of Engineering and Management, University of Padova, Vicenza, Italy

Alessandro Abate

Department of Aeronautics and Astronautics, Stanford University, Stanford (CA), U.S.A.

Keywords:

Structure detection, Graphical model, Random walk, Drosophila wing, Morphogenesis.

Abstract:

Drosophila melanogaster is a model organism in genetics thanks to the compactness of its genome and its

relative simplicity. Recently, certain developmental patterns in Drosophila have been studied by mathematical

models, with the aim of gaining deeper and quantitative insight into the morphogenesis of this insect. There

is a need for accurate dynamical of the epithelial cell structure and organization within the ﬂy wing, to further

the understanding of a phenomenon known as planar cell polarity. The present study tackles the problem of

retrieving such a salient structure using classical tools of dynamical system theory embedded with network

and graph concepts. On the one hand the goal is to provide a visual detection and representation of the cell

packaging that is accurate and ﬁne. Particular care is also put in obtaining a model of this structure, whose

main features are the compactness and simplicity.

1 INTRODUCTION

Drosophila melanogaster is a model organism in de-

velopmental biology. Scientiﬁc interest in this organ-

ism can be traced back to some early work in genetics,

and more recently the understanding of this insect has

progressed with the investigation of its early devel-

opment and the sequencing of its relatively compact

genome. The relative simplicity of its morphogenesis

allows to edulcorate the study of its developmentfrom

many complications. Many human disease genes can

be investigated and understood through analogues in

the fruit ﬂy. An important issue for the ﬂy morpho-

genesis is the study of the polarization of cells on

its wing. A mechanistic approach to study this phe-

nomenon has led to some understanding of the un-

derlying structures and regulatory proteins in the ep-

ithelial cells of the ﬂy wing (Amonlirdviman et al.,

2005). The dynamical model hinges on the assump-

tion that the cell are regularly packed in a honeycomb

structure throughout the wing. In order to understand

how to extend these models, it makes sense to infer

the structure and motion of the network of epithelial

cells by looking at the phenotype of the ﬂy wing with

movies taken in the laboratory. This suggests to split

the approach into two sequential steps:

1. ﬁrst, given a single frame of the ﬂy wing, build a

network that closely represents the cell packing;

2. then, given a movie (a sequence of frames) of the

ﬂy wing, correlate the networks generated by con-

sidering single frames into a time-dependent dy-

namical model.

In this article we report on the former point, while the

extension of the study is currently under investigation.

The goal of this study is twofold. On the one

hand we try to detect the structure on the drosophila

wing and provide a visual representation. On the

other hand, we produce a model of the same struc-

ture, whose main features are the compactness and

the simplicity.

406

Silletti A., Cenedese A. and Abate A. (2009).

THE EMERGENT STRUCTURE OF THE DROSOPHILA WING - A Dynamic Model Generator.

In Proceedings of the Fourth International Conference on Computer Vision Theory and Applications, pages 406-410

DOI: 10.5220/0001795804060410

Copyright

c

SciTePress

Figure 1: Drosophila wing epithelial cells.

The following approach is thus developed both to

retrieve the network structure as a ﬁne representation

and to abstract the information related to the structure

into a manageable model. This is obtained by explor-

ing the visual data through a dynamic random walk

model, whose motion state summarizes the useful in-

formation and allows the full detailed reconstruction

of the network. To give a pictorial idea, the image

frame is considered as a landscape where clear paths

have to be discerned from darker areas. The agent

performing the detection advances by scanning the

neighborhood of his current position in search for ex-

plorable paths. When a bifurcation occurs the agent

generates one or more siblings that start to move in-

dependently, until the whole frame has been explored.

The remainder of the paper is organized as fol-

lows. After an overviewof the state of the art in struc-

ture detection (Sec. 2) and a description of the pro-

cedure for image preprocessing (Sec. 3), the adopted

model is introduced in Sec. 4 and the algorithm ex-

plained in Sec. 5. Then, the static analysis of the im-

age frame is discussed in Sec. 6 and, ﬁnally, in Sec. 7

some conclusions are drawn and insight on future de-

velopments is given.

2 THE STATE OF THE ART

The recognition of structures in digital images is a

crucial task in many automated algorithms used in

computer vision. For the task of edge detection,

several approaches based on derivatives have been

proposed, among which the seminal studies by So-

bel (Sobel, 1968) and Prewitt (Prewitt, 1970) and

Canny (Canny, 1986).

These approaches though are prone to failure,

since they do not incorporate any prior knowledge of

the object, nor do they include any geometrical model.

This can yield very fragmented edges and many false

classiﬁcations. They also do not return any com-

pact and light representation of the recognized pat-

tern. Such algorithms are good to represents patterns,

but not to model them. The Canny edge detector out-

puts a bitmap map, where each pixel is classiﬁed as

belonging to a border or not. Indeed, no structure is

returned, and no compact model is given to the user.

If there is a need for computing metrics or for analyti-

cally following a path around the edges, this approach

appears to be unsatisfactory.

Active Contours (Blake and Isard, 1998) based

approaches give better results, thanks to the elastic

model structure they incorporate. Active Contours

and Deformable Models (McInerney and Terzopou-

los, 1996) generally perform well in shape recogni-

tion. Still, they are very sensitive to noise, they need a

good initialization point to converge, and they require

hard-tuning of the parameters to make things really

work.

For thin linear structures such as vessels, mar-

ble veins, roads on terrain, better results can be

achieved using a model of the motion over the im-

age, which attempts to follow the structure of interest

(Grisan et al., 2003). This approaches is promising

and achieves near optimal results. A random motion

paradigm can also be used also for image segmen-

tation (Harel and Koren, 2001) and image enhance-

ment (Smolka and Wojciechowski, 2001).

3 IMAGE PREPROCESSING

The images retrieved from biological experiments

are particularly noisy and exhibit poor contrast, with

non-uniform background illumination, resulting in

structure boundaries not sufﬁciently sharp to be seg-

mented. The preprocessing stage presented consists

of four sequential steps (see Fig.2):

1. the image is ﬁltered with a low-pass gaussian ﬁlter

to soften high frequency noise;

2. an erosion ﬁlter suppresses the isolated bright pix-

els and decreases the intensity of cell edges, while

retaining all the signiﬁcant information;

3. an histogram stretch allows to partially recover the

color dynamic range;

4. an image intensity power enhances the contrast.

4 THE DYNAMICAL MODELS

In this section the principle underlying the algorithm

is brieﬂy described. To retrieve the salient structure in

the video sequence and to circumvent issues related

to disconnected edges and false recognition (such as

those mentioned in Sec. 2), the edge detection prob-

lem is re-interpreted as the problem of exploring a

digital frame in search of the connected tracks. Each

THE EMERGENT STRUCTURE OF THE DROSOPHILA WING - A Dynamic Model Generator

407

(a) (b)

Figure 2: Image preprocessing: original (a) and prepro-

cessed (b) frames.

digital picture is viewed as a landscape that has to

be explored and where bright pixels–belonging to the

edges of the cells–are thought of as roads, and dark

pixels–the interior of the cells–as unexplored loca-

tions.

The idea behind the algorithm for robust and fast

reconstruction of the interesting structure is based on

a two level model, referred to respectively as an ex-

plorer agent system and as a network agent system.

The explorer agent will retrieve the structure and pro-

vide a visual representation as ﬁne as possible, while

then network agent will abstract a compact model.

An explorer agent system A is deﬁned according

to a walk model, whose state equation is:

p(t + 1) = p(t) + g(θ(t)) , (1)

p(t) =

x(t)

y(t)

∈ L,

p(t) is the current point position on the discrete do-

main L ⊂ Z

2

of the image frame and g(·) is a motion

function. In the speciﬁc case,

g(θ(t)) = k

sin(θ(t))

cos(θ(t))

,

k being a design constant and θ(t) being the heading

direction, which assumes the role of the input to the

agent system.

The observation equation is the following:

y(t) =

x(t)

y(t)

θ(t)

=

1 0

0 1

0 0

p(t) +

0

0

1

θ(t).

Several instances of the explorer agent,

{A

i

, i = 1, . . . , N

A

}, are concurrently present in

the ﬁeld of vision, and each generates at any time one

or more directions viable to advance the exploration.

Precisely, for each A

i

the set of possible directions

θ

i, j

, j = 1, . . . , m

i

, originating new agents, are

collected in the vector Θ

i

∈ R

m

i

.

Globally, the observations y(t) of all the explorer

agents A

i

yield a graph model G = (N ,E) (being

N and E nodes and edges, respectively), where each

node n

i

∈ N is characterized by the state { x, y, Θ} of

the locations visited by one of the A

i

, and the edges

keep track of the path traveled by each agent (see

Fig. 3). This graph provides a good representation

of the retrieved structure, in the sense suggested in

Section 2.

(a) (b)

Figure 3: Representation and Model graphs G (a) G

r

(b).

For modeling purposes, the goal is to obtaining

compact-size models, ﬂexible and agile. In this re-

spect, only the subset N

r

of the nodes associated to

non-scalar Θ (and the correspondent edges E

r

) are in-

teresting in the description, that is:

G

r

= (N

r

, E

r

) N

r

⊂ N s.t. dim(Θ(n

i

)) > 1∀n

1

∈ N

r

,

where dim(Θ(n

i

)) indicates the cardinality of vector

Θ for node n

i

. If the border effects are neglected, the

resulting graph G

r

is a 3-connected graph, in which

the minimum vertex degree is d = 3.

4.1 Brightness Function

We now explain how to choose the input direction

θ(t). With reference to the pictorial interpretation,

the aim is to explore the surroundings of the current

position and to move from a bright location to an-

other bright location. Thus a good direction of move-

ment will be a direction that maximizes some sort of

brightness function. Local brightness information is

obtained by means of a function L(θ) : [0, 2π) → R

built with the speciﬁc purpose of ﬁnding good direc-

tions of exploration. Sectors centered on the current

position p are spanned, and for each sector the aver-

age brightness is computed. Formally, given a sector

Ω

i

, with any generic shape, we have

L(θ

i

) =

R

Ω

i

I(ω)dω

R

Ω

i

dω

(2)

A natural choice for Ω

i

is circular sector. Sectors

of different shapes can also be built: For the problem

of interest rectangular sectors also result to be quite

effective.

After smoothing and thresholding the function, lo-

cal maxima are collected as putative direction for the

motion. Let Θ = {θ

1

, θ

2

, ·, θ

n

} = arg max

θ

L(θ).

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

408

(a) (b)

Figure 4: A brightness function built on circular sectors is

a natural choice, since it mimics the ﬁeld of view of human

beings (a). Other shapes for sectors are also possible, de-

pending on the problem under study: in the speciﬁc case, a

rectangular shape efﬁciently serves the purpose (b).

Figure 5: Local maxima of the brightness function identify

candidate directions of expansion.

5 THE ALGORITHM

The initialization of the algorithm consists of a single

agent A

0

exploring the frame. At each motion step,

the arguments of local maxima of L provide the set Θ

of candidate directions. A Priority Queue Q is deﬁned

by inserting either the agent A

0

with its heading direc-

tion and its corresponding brightness value or, when

multiple directions m

i

are possible, instances of A

0

along with the direction θ

i

∈ Θ. Each of these agents

is then put in motion according to Eq. 1.

The priority queue is sorted according to the

brightness function associated with the current loca-

tion of each agent. At each iteration step, the agent A

i

with the highest brightness value is extracted and the

exploration process is iterated. The role of the prior-

ity queue is thus to keep in memory the front of the

expansion, and at the same time, to give the best pos-

sible expansion point, in the sense that the ﬁrst agent

in the queue will move to the best (i.e. brightest) area

of the image.

By doing so, the algorithm is building a graph G,

where each node is the location p

i

of an agent A

i

ex-

tracted from the priority queue, and each arc joins a

node and its nearest ancestor. Interpreting the under-

lying structure of the walk as a graph allows to resort

to graph theory (Cormen et al., 1990) to reﬁne the

procedure.

5.1 Graph Structure Reﬁnement

Small loops have to be avoided. Small loops are fre-

quent in extensive bright areas of the image, where

the brightness function L(θ) is not prone to differenti-

ate between redundant directions. In these situations,

a direction is as likely as any other, and the choice

is driven mainly by randomness. When this happens

we get |Θ| >> 1 and θ

i

≈ θ

i+1

∀ θ

i

∈ Θ. In other

words, all the values of θ

i

∈ Θ are very similar and

these directions spans uniformly the surrounding en-

vironment, so there is no particular reason to pick one

direction over another.

In order to avoid this, after extracting a point from

the queue, it is tested for loop creation in the graph.

Large loops are accepted because they correspond to

an actual closed path (for instance the perimeter of a

cell), while small loops are disregarded.

A joining procedure is then required to ﬁll small

gaps: If two valid agents A

i

and A

j

end up close to

each other, | {x, y}

i

− {x, y}

j

| < ε, they are joined to-

gether.

The algorithm also marks as boundary nodes

those nodes close to the boundary of the digital im-

age: The expansion process ends there (see Fig. 6(b))

and the agent A

i

stops.

(a) (b)

Figure 6: (a) Small loops are frequent and are mainly due to

the locality feature of the brightness function. (b) Raw re-

sults before postprocessing: many small spurious branches

span over the interior of the cells.

The expansion process could create dead-

branches, characterized by extremal non-border

nodes of degree 1. A polished graph is obtained it-

eratively removing all such nodes and related edge.

occurs.

The network agent system reconstruct the ﬁnal ge-

ometric model of the cells considering only nodes

with degree d > 2 and border nodes, considered the

corner of the cells. The new graph G

r

consists of such

corners and new edges created between them if in the

original graph there was a straight path (a path con-

THE EMERGENT STRUCTURE OF THE DROSOPHILA WING - A Dynamic Model Generator

409

necting them not passing through any other corner).

The model reconstruction thus prunes a lot of nodes

and edges and ends up with a light, compact represen-

tation of the cellular structure, achieving the second

goal of this paper.

6 STATIC ANALYSIS

Preliminary results show a good accuracy in retriev-

ing the salient structure on the ﬂy wing and robust-

ness in the detection procedure, for all the tested se-

quences. In Fig. 7 results obtained with the proposed

technique are shown. The two cases are indicative of

the critical issues that need to be faced, such as non-

uniform illumination, differentradii of the cells, noisy

images.

(a) (b) (c)

(d) (e) (f)

Figure 7: Example frames: a) -b) original; c) -d) represen-

tation G; e) -f) model G

r

.

It is interesting to show the dynamic evo-

lution of the exploration of the frame. A

video showing the full procedure can be found at

http://www.youtube.com/watch?v=I2HfKTCEd6w.

7 CONCLUSIONS AND FUTURE

DEVELOPMENTS

In this work a procedure to detect salient lattice fea-

tures from biological images has been developed,

with the twofold purpose of representation and mod-

eling. So far, the problem of the static reconstruction

from single frames has been addressed, with encour-

aging results both for the accuracy of the details and

for the robustness of the technique.

As a ﬁrst step towards the dynamic reconstruction

and tracking will be obtained by simply applying sev-

eral times, once per frame, the static analysis (Sec. 6).

A video obtained through this approach is available

at http://it.youtube.com/watch?v=WxWjZSvuoh4. A

fully dynamic tracking is still under study, and more

reﬁned dedicated approaches will be developedby ex-

ploiting temporal and local coherence.

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