Biomedical Imaging Group, Swiss Federal Institute of Technology, Lausanne (EPFL)

bigwww.epfl.ch

Summary

3-D Reconstruction of DNA Filaments from Stereo Cryo-Electron MicrographsMathews Jacob, Thierry Blu and Michael Unser

Stereo views separated by 30 degrees

Steerable filter implementation 3-D Reconstruction (Active contour algorithm)

We propose an algorithm for the 3-D reconstruction of DNA filaments from a pair of stereo cryo-electron micrographs. The underlying principle is to specify a 3-D model of a filament -- described as a spline curve -- and to fit it to the 2-D data using a snake-like algorithm. To drive the snake, we constructed a ridge-enhancing vector field for each of the images based on the maximum output of a bank of rotating matched filters. The magnitude of the field gives a confidence measure for the presence of a filament and the phase indicates its direction. We also propose a fast algorithm to perform the matched filtering. The snake algorithm starts with an initial curve (input by the user) and evolves it so that its projections on the viewing plane are in maximal agreement with the corresponding vector fields.

• 2-D Ridge Enhancing Vector Field• Rotational Matched Filtering• Confidence measure and direction • Steerable filter implementation

•Semi-automatic tracking - Snake Fit• 3-D curve model

• Cubic bspline representation• Projections matched with 2-D vector fields• Conjugate gradients optimization

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−15o

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15o

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30o

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0o

Conclusions

• Semi-automatic Tracking

• 3-D spline curve

• Implicit internal energy Easy optimization

• Projected onto image planes• Projection also spline curve

• Optimized to maximize the cost function

• Conjugate gradients optimization

• Distance map to enhance convergence

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r t( ) = ckk=0

M −1

∑ β p x − k( )

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where β p x( ) = β x − l M( )l=−∞

∞

∑

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E C( ) = uk rk( ) • drkCk∫

k=0

1

∑ Curve

projection onto image plane

Vector field on the kth image

Cubic Bspline

Ridge enhancing vector field

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u r( ) =hθ ∗ ∗ f( ) r( ) if h

θ ∗ ∗ f( ) r( ) ≥ 0

0 otherwise

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⎩ ⎪

Corresponding points

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θ∗ r( ) = arg maxθ

hθ ∗ f( ) r( ){ }

• Rotated Matched Filtering

Magnitude

Phase

• Extremely Noisy

• Ill posed due to few views• At least 2 possible curves exist

Challenges

Thresholded vector field

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h01 r( ) =

d2g r + α y( )dα 2 , where g r( ) = e

−r

2

σ 2

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u r( ) = λ 1 r( )

wθ = v1 r( )

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Minimum eigen value and the corresponding eigen

vector of

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Hg∗f r( )

Optimally elongated second order template

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h02

=gyy−gxx3

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u r( ) = λ 1 r( ) −λ 2 r( )

3;

wθ = v1 r( )

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f ∗hθ1 r( ) = wθ

THg∗f r( )wθ

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hθ*

1∗f ()

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θ

• Maximally flat along the axis of orientation

Visualization of 3-D reconstruction

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•h02,gxx=0

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h02x,0 ()=1+Ox

4()

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hθ*

2∗f ()

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h01

=gyy

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θ=0o

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θ=45o

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θ=90o

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θ=135o