Finding and exploiting correspondences in Drosophila
embryos
Charless Fowlkes and Jitendra Malik UC Berkeley Computer Science
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]
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]
Overview
• Finding Correspondences– Nuclear segmentation– Deformable matching
• Exploiting Correspondences– Preliminary results– Discussion
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
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
X
Y
Push average nuclear density forward thru correspondence function X
Building a nuclear density map
Current/Future Work
• Verifying the correspondences are biologically “correct”
• Analysis of variation in shapes of expression patterns
• Hybridization experiment 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