Finding and exploiting correspondences in Drosophila

embryos

Charless Fowlkes and Jitendra Malik UC Berkeley Computer Science

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Motivation for combining measurements

• Average noisy flouresence data over multiple embryos

• High throughput– N versus N2 hybridizations to capture

colocation of N gene products

• Visualization of composite expression map

• Study shape of expression patterns

Sources of Variation

• Not so interesting:– Staining– Shrinking– Spinning– Squashing– Staging

• Interesting:– Biological Variation

Overview

• Finding Correspondences– Nuclear segmentation– Deformable matching

• Exploiting Correspondences– Preliminary results– Discussion

Segmenting Nuclei

~500µm

~20

0µm

x-y

x-y

x-z

Embryo is approximately 500µm by 200µm and contains about 5000 to 6000 nuclei

[C. Luengo, D. Knowles]

Segmentation output

Mesh generation• Point cloud doesn’t capture the blastoderm topology.

Locally, it is a 2D sheet of cells• Utilize off the shelf tools from computational geometry

[Kolluri et al, 2004]

Clyindrical Projection

Clyindrical Projection

Ventral

Dorsal

Dorsal

Anterior Posterior

Overview

• Finding Correspondences– Nuclear segmentation– Deformable matching

• Exploiting Correspondences– Preliminary results– Discussion

FTZ expression

FTZ Edge Points

Two Coarsely Registered Embryos

“Shape Context Descriptor”

“Shape Context Descriptor”

Xij = 1 if point i is matched to point j 0 otherwise

Correspondence as optimization

Cij = disimilarity of local descriptor for points i and j

Dij = distance between points i and j

minimize : Σij (Cij + λDij) • Xij

subject to : Σi Xij = 1

Σj Xij = 1

λ sets the relative importance of distance versus shape context match

j

i

1. Find correspondence by optimizing Xij

2. Smoothly warp source embryo to bring into alignment with corresponding points

3. Repeat…

Problem: correspondence may not be smooth

Solution: iteratively correspond and warp

Deformable Matching

Overview

• Finding Correspondences– Nuclear segmentation– Deformable matching

• Exploiting Correspondences– Preliminary results

• Composite Expression Map• Nuclear Density Map• Shape

– Discussion

Preliminary Results

• 34 embryos stained for ftz and one other gene product

• Choose a target embryo

• Find correspondences with remaining embryos and “transfer” measurements

X

Y

Push expression levels forward thru correspondence function X

Building a composite expression map

Source Embryos

Target Embryo

FTZ average aftercoarse alignment

FTZ average afterdetailed matching

ftzevesnailknihb

Composite Map: View #1

ftzevesnailknihb

Composite Map: View #2

X

Y

Push average nuclear density forward thru correspondence function X

Building a nuclear density map

Nuclear Density

X-1

Y-1

Pull back selected region thru inverse correspondence function.

Shape Analysis

Current/Future Work

• Verifying the correspondences are biologically “correct”

• Analysis of variation in shapes of expression patterns

• Hybridization experiment design

Eve Slp

Kni

Sna

Hb

Ftz

Hybridization Design

Eve

SlpKniSna

Hb Ftz

Hybridization Design

EveSlp

Kni

Sna

Hb

Ftz

Eve

Sna

Hb

Ftz

1. Can build composite map from any connected graph 2. Error accumulates so diameter should be small3. Some genes provide more powerful constraints than others

Future Work