viewing direction estimation in cryo-em using …saharon/bigdata2015/yoel.pdfviewing direction...
Post on 20-Apr-2020
9 Views
Preview:
TRANSCRIPT
Viewing direction estimation in cryo-EM usingsynchronization
Yoel Shkolnisky
Tel-Aviv University
Joint work with Amit Singer (Princeton)
July 08, 2014
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 1 / 48
Structural Biology
GoalUnderstand structure and function of molecular structures.
ReasonUnderstand biological processes, understand diseases, find cures.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 2 / 48
Orders of Magnitude
Optical microscopy → cannot see too small (wavelength ≈ 550nm).
Resolving power of optical microscope ≈ 100nm. Magnification < ×2000.
Magnification of an electron microscope is hundreds of thousands.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 3 / 48
Electron microscope
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 4 / 48
Electron microscope
SEM 20, 000×Toumey, Nature Nanotechnology
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 4 / 48
Electron microscope
Light microscope 200×Toumey, Nature Nanotechnology
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 4 / 48
Electron microscope
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 4 / 48
The Experiment
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 5 / 48
The Experiment (cont.)
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 6 / 48
Cryo-Electron Microscopy
Tomographic image 3D structure
?
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 7 / 48
Physical setting – CryoEM imaging
Electron source
Copy 1
Projection 1
Electron source
Copy 2
Projection 2
Electron source
Copy 3
Projection 3
g1∈SO(3) g2∈SO(3) g3∈SO(3)
CryoEM: Put sample in liquidmedium, freeze, take images usingan Electron Microscope.
Many images: each imagecorresponds to a different copy ofthe molecule in a different spatialorientation.
Orientations are random andunknown: there might be apreferred orientation!
Goal: Reconstruct the 3D molecule from its projections taken atrandom unknown directions
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 8 / 48
Workflow
1 Particle selection – manual or automatic image segmentation.
2 Preprocessing – centering, masking, normalization, CTF correction,. . .
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 9 / 48
Workflow (cont.)
3 Class averaging – find images with similar viewing direction, register andaverage to improve their SNR.
4 Orientations assignment.
5 Reconstructing an initial model – generate a 3D volume by a tomographicinversion algorithm.
6 Iterative refinement.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 10 / 48
Iterative refinement
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 11 / 48
EM imaging – mathematical modeling
Projection Pi
Molecule φ
Electronsource
Ri =
| | |
R(1)i
R(2)i
R(3)i
| | |
φ(r) is the electric potential of the molecule(r = (x , y , z)T ∈ R
3).
The molecule is rotated byRi ∈ SO(3): φi (r) = φ(Ri r).
A projection image is
Pi (x,y) =
∫ ∞
−∞
φ (Ri r) dz
=
∫ ∞
−∞
φ(
xR(1)i + yR
(2)i + zR
(3)i
)
dz .
(known as Radon transform)
Each projection corresponds to an unknownrotation Ri ∈ SO(3).
Goal: Given n images P1, . . . ,Pn, estimate R1, . . . ,Rn.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 12 / 48
Projection Images: Toy Example
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 13 / 48
Projection Images: Toy Example
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 13 / 48
Projection Images: Real Data
Courtesy of Dr. Joachim Frank
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 14 / 48
The Fourier projection-slice theorem
The Radon transform and the Fourier transform are related by theprojection-slice theorem.
The 2D FT of the projection image is the double integral
P̂i (ωx , ωy) =
∫
R2
e−ı(xωx+yωy )Pi(x , y) dx dy .
The 3D FT of the molecule is the triple integral
φ̂(ωx , ωy , ωz) =
∫
R3
e−ı(xωx+yωy+zωz )φ(x , y , z) dx dy dz .
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 15 / 48
Fourier projection-slice theorem (cont.)
P̂i (ωx , ωy) = φ̂(
ωxR(1)i + ωyR
(2)i
)
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 16 / 48
Fourier projection-slice theorem (cont.)
P̂i (ωx , ωy) = φ̂(
ωxR(1)i + ωyR
(2)i
)
The Fourier transform P̂i of the projection Pi equals φ̂ on the plane
ω = ωxR(1)i + ωyR
(2)i (recall: R
(1)i , R
(2)i , R
(3)i are the columns of Ri).
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 16 / 48
Fourier projection-slice theorem (cont.)
P̂i (ωx , ωy) = φ̂(
ωxR(1)i + ωyR
(2)i
)
The Fourier transform P̂i of the projection Pi equals φ̂ on the plane
ω = ωxR(1)i + ωyR
(2)i (recall: R
(1)i , R
(2)i , R
(3)i are the columns of Ri).
Projection Fourier transform 3D Fourierspace
3D Fourierspace
Fourier transformProjection
P0 P̂0
P1 P̂1
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 16 / 48
Fourier projection-slice theorem (cont.)
P̂i (ωx , ωy) = φ̂(
ωxR(1)i + ωyR
(2)i
)
The Fourier transform P̂i of the projection Pi equals φ̂ on the plane
ω = ωxR(1)i + ωyR
(2)i (recall: R
(1)i , R
(2)i , R
(3)i are the columns of Ri).
Projection Fourier transform 3D Fourierspace
3D Fourierspace
Fourier transformProjection
P0 P̂0
P1 P̂1
Any two planes intersect at a common line ⇒ any two images have acommon line.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 16 / 48
Angular Reconstitution (Van Heel 1987, Vainshtein & Goncharov 1986)
Why is reconstruction possible at all?
Problems: requires accurate detection of common lines (noise), sensitivity toerrors (depends on the three initial images).
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 17 / 48
Detection rates
Each projection is 129x129 pixels. Additive Gaussian white noise.
SNR =Var(Signal)
Var(Noise),
(a) Clean (b) SNR=1 (c) SNR=1/2 (d) SNR=1/4 (e) SNR=1/8
(f) SNR=1/16 (g) SNR=1/32 (h) SNR=1/64 (i) SNR=1/128 (j) SNR=1/256
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 18 / 48
Detection rates (cont.)
Correctly identified common line – within 10◦ of its true value (known fromsimulation).
(a) n = 100
SNR p
clean 0.9971 0.968
1/2 0.9301/4 0.8281/8 0.6531/16 0.4441/32 0.2471/64 0.1081/128 0.0461/256 0.0231/512 0.017
(b) n = 500
SNR p
clean 0.9971 0.967
1/2 0.9221/4 0.8171/8 0.6391/16 0.4331/32 0.2481/64 0.1131/128 0.0461/256 0.0231/512 0.015
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 19 / 48
Main idea (but slightly wrong)
Use common lines information to compute all matrices R−1i Rj ,
i , j = 1, . . . ,N .Define the 3N × 3N matrix S , whose (i , j) block of size 3× 3 is R−1
i Rj .S is given by
S =
...· · · R−1
i Rj · · ·...
=
RT1
...
RTN
R1 · · · RN
Extract rotations from SVD of S .Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 20 / 48
Resolving handedness ambiguity
Given only common lines between three images, we can recover either R−1i Rj
or JR−1i RjJ – No way to distinguish between the two! J = diag(1, 1,−1)
The sum R−1i Rj + JR−1
i RjJ is invariant under this ambiguity: if we haverecovered JR−1
i RjJ then (J = J−1, J2 = I )
JR−1i RjJ + J
(JR−1
i RjJ)J = JR−1
i RjJ + R−1i Rj .
No matter if we recover the product R−1i Rj or JR
−1i RjJ, the sum
R−1i Rj + JR−1
i RjJ remains the same.
Write the rotation matrix R−1i Rj explicitly as
R−1i Rj=
r(ij)11 r
(ij)12 r
(ij)13
r(ij)21 r
(ij)22 r
(ij)23
r(ij)31 r
(ij)32 r
(ij)33
⇒ R−1i Rj+JR
−1i RjJ=
2r(ij)11 2r
(ij)12 0
2r(ij)21 2r
(ij)22 0
0 0 2r(ij)33
.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 21 / 48
Synchronization matrix
Take the matrix B(k)ij to be the upper-left 2× 2 block of
(R−1i Rj + JR−1
i RjJ)/2
B(k)ij =
(
r(ij)11 r
(ij)12
r(ij)21 r
(ij)22
)
=
R
(1)i
T
R(1)j R
(1)i
T
R(2)j
R(2)i
TR
(1)j R
(2)i
TR
(2)j
=
(
— R(1)i
T—
— R(2)i
T—
)
| |
R(1)j R
(2)j
| |
.
(R−1i Rj)
T = R−1j Ri ⇒ B
(k)ji =
(
B(k)ij
)T
.
Due to common line misidentifications, each block B(k)ij might contain some
error – improve estimate by averaging over all k
Bij =1
N − 2
∑
k 6=i ,j
B(k)ij .
Define Bii = I2 (I2 is the 2× 2 identity matrix).
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 22 / 48
Synchronization matrix (cont.)
The synchronization matrix S is defined as the 2N × 2N matrix whose (i , j)block of size 2× 2 is Bij
Sij = Bij , i , j = 1, . . . ,N .
S is symmetric and admits the decomposition
S =
...
· · · Bij · · ·
...
=
HT
︷ ︸︸ ︷
— R(1)1
T—
— R(2)1
T—
...
— R(1)N
T—
— R(2)N
T—
H︷ ︸︸ ︷
| | | |
R(1)1 R
(2)1 · · · R
(1)N R
(2)N
| | | |
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 23 / 48
Synchronization matrix (cont.)
Reason: The (i , j) 2× 2 block of HTH is give by
...
— R(1)i
T—
— R(2)i
T—
...
| |
· · · R(1)j R
(2)j · · ·
| |
=
...
· · · Bij · · ·
...
.
S is of rank 3.
Three nontrivial eigenvectors of S are linear combinations the columns of HT
(the column space of S).
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 24 / 48
Unmixing
The nontrivial eigenvectors of S , denoted V = (v1, v2, v3), vi ∈ R2N , span
the same space as the columns of HT .
Need to unmix the columns of HT from these eigenvectors – find A ∈ R3×3
such thatVAT = HT ,
or equivalently,
A
— vT1 —
— vT2 —
— vT3 —
︸ ︷︷ ︸
VT
=
| | | |
R(1)1 R
(2)1 · · · R
(1)N R
(2)N
| | | |
︸ ︷︷ ︸
H
.
Gives first two columns of the rotation matrices.
The third column of each rotation matrix is then given by the cross productof the first two columns.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 25 / 48
Unmixing (cont.)
Split V T into two sets
A
v(1)1 v
(3)1 · · · v
(2N−1)1
v(1)2 v
(3)2 · · · v
(2N−1)2
v(1)3 v
(3)3 · · · v
(2N−1)3
︸ ︷︷ ︸
V1
=
| | | |
R(1)1 R
(1)2 · · · R
(1)N
| | | |
.
A
v(2)1 v
(4)1 · · · v
(2N)1
v(2)2 v
(4)2 · · · v
(2N)2
v(2)3 v
(4)3 · · · v
(2N)3
︸ ︷︷ ︸
V2
=
| | | |
R(2)1 R
(2)2 · · · R
(2)N
| | | |
.
From the orthogonality conditions, for i = 1, . . . ,N,(
R(1)i
)T
R(1)i = 1,
(
R(2)i
)T
R(2)i = 1,
(
R(1)i
)T
R(2)i = 0,
we get the 3N linear equations
(V T1 A
TAV1)ii = 1, (V T
2 ATAV2)ii = 1, (V T
1 ATAV2)ii = 0,
for the 6 unknowns of ATA.
Get A from ATA using SVD.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 26 / 48
Algorithm
Require: Common line matrix C = (cij) of size N × N .
1: for i , j = 1, . . . ,N do
2: for k = 1, . . . ,N do
3: if i 6= j 6= k then
4: Gijk =
(1 〈cij ,cik〉 〈cji ,cjk〉
〈cik ,cij 〉 1 〈cki ,ckj〉〈cjk ,cji〉 〈ckj ,cki〉 1
)
.
5: Gijk = QTijkQijk , Qijk = (qij qik qjk ) ∈ R
3×3.6: Ci = (cij cik cij × cik) , Qi = (qij qik qij × qik).7: Cj = (cji cjk cji × cjk ) , Qj = (qji qjk qji × qjk ) .8: Rij = CiQ
−1i QjC
−1j . //Now Rij = R−1
i Rj or Rij = JR−1i RjJ.
9: B(k)ij = upper-left 2× 2 block of Rij .
10: end if
11: end for
12: Bij =1
N−2
∑
k 6=i ,j B(k)ij , Bii = I2, Sij = Bij .
13: end for
14: Compute the to three eigenvectors of S – v1, v2, v3.
15: Unmix rotation matrices from v1, v2, v3.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 27 / 48
The limit of S
Take a vector f = (f1, . . . , f2N)T, viewed as the samples of some function
f : SO(3) → R2, that is f (Ri) = (f2i−1, f2i )
T , i = 1, . . . ,N .The operation of S on f is given by
Sf =
...· · · Bij · · ·
...
f1...
f2N
,
or equivalently, the ith 2× 1 block of the product is
Sf (Ri ) =N∑
j=1
Bij
(f2j−1
f2j
)
=N∑
j=1
Bij f (Rj).
Bij is the upper-left 2× 2 block of R−1i Rj , thus
S(Ri ,Rj) = Bij =
(1 0 00 1 0
)
R−1i Rj
(1 0 00 1 0
)T
, for i 6= j ,
S(Ri ,Ri) = Bii =
(1 00 1
)
, for i = j .
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 28 / 48
The limit of S (cont.)
Using the kernel S(Ri ,Rj), S is given by
Sf (Ri ) =
N∑
j=1
S(Ri ,Rj)f (Rj).
Assuming Ri are sampled from SO(3) uniformly at random
limN→∞
1
N(Sf ) (R) = E [S(R ,U)f (U)] =
∫
SO(3)
S(R ,U)f (U)dU ,
where dU is the Haar measure.
We will compute the eigenvalues of the operator
(Sf ) (R) =
∫
SO(3)
S(R ,U)f (U)dU .
Those are approximations to the eigenvalues of the matrix S .
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 29 / 48
The limit of S (cont.)
In fact, we will show that the functions g : SO(3) → R2×3 given by
g(R) =
(1 0 00 1 0
)
R−1.
are eigenfunctions corresponding to the nontrivial eigenvalue.
g(R) extracts the first two rows of the rotation matrix R−1, that is, the firsttwo columns of R .
In other words, the first two columns of all rotation matrices areeigenfunctions of S.
We have constructed an operator whose eigenfunctions are the unknownrotations.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 30 / 48
The limit of S (cont.)
By direct evaluation
S(R ,U)g(U) =
(1 0 00 1 0
)
R−1U
(1 0 00 1 0
)T (1 0 00 1 0
)
U−1
=
(1 0 00 1 0
)
R−1U
1 0 00 1 00 0 0
U−1
=
(1 0 00 1 0
)
R−1(
I − U(3)U(3)T)
,
where U(3) is the third column of the matrix U .
Write U(3) in coordinates as
U(3) = (x , y , z)T, x2 + y2 + z2 = 1,
and so
U(3)U(3)T =
x2 xy xz
xy y2 yz
xz yz z2
.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 31 / 48
The limit of S (cont.)
Substitute in S
(Sg) (R) =
∫
SO(3)
(1 0 00 1 0
)
R−1(
I − U(3)U(3)T)
dU
=
(1 0 00 1 0
)
R−1
∫
SO(3)
(
I − U(3)U(3)T)
dU
=
(1 0 00 1 0
)
R−1
∫
SO(3)
(1−x2 −xy −xz
−xy 1−y2 −yz
−xz −yz 1−z2
)
dU .
Parameterize SO(3) as an axis of rotation and a rotation around that axis –SO(3) as S2 × [0, 2π) ⇒ the last integral depends only on the axis ofrotation; the integral collapses to an integral over S2.
From symmetry of S2
∫
S2
x2dµ =
∫
S2
y2dµ =
∫
S2
z2dµ.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 32 / 48
The limit of S (cont.)
Since∫
S2
x2dµ+
∫
S2
y2dµ+
∫
S2
z2dµ =
∫
S2
(x2 + y2 + z2
)dµ =
∫
S2
dµ = 1,
we get that ∫
S2
x2dµ =
∫
S2
y2dµ =
∫
S2
z2dµ = 1/3.
Again from symmetry of S2
∫
S2
xy dµ =
∫
S2
xz dµ =
∫
S2
yz dµ = 0,
We conclude that∫
SO(3)
(1−x2 −xy −xz
−xy 1−y2 −yz
−xz −yz 1−z2
)
dU =
(2/3
2/32/3
)
,
and so,
(Sg) (R) =2
3
(1 0 00 1 0
)
R−1 =2
3g(R),
The eigenvalues of the matrix S converge to 23N .
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 33 / 48
Examples: noiseless projections – data set
N = 10, 100, 1000 projections from the density map of the 50S ribosomalsubunit. Each of size 129× 129.
Rotations drawn uniformly at random from the uniform distribution on SO(3).
Compute L = 360 Fourier rays in each projection.
Find common lines using correlation.
Construct the matrix S .
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 34 / 48
Examples: noiseless projections – spectrum of S
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
8
9
i
λi
N = 10
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
i
λi
N = 100
1 2 3 4 5 6 7 8 9 100
100
200
300
400
500
600
700
i
λi
N = 1000
Three dominant eigenvalues.
Fourth and higher eigenvalues are not exactly zero due to the discretizationof Fourier space using L = 360 (but much smaller than the first three ones).
Top eigenvalues approximately equal 2N/3 (688, 656, and 655 for N = 1000).
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 35 / 48
Examples: noiseless projections – estimation error
Estimated rotations – R̃1, . . . , R̃N .
True rotations – R1, . . . ,RN .
Split each projection Pi into L = 360 rays c(l)i = (cos 2πl/L, sin 2πl/L, 0),
l = 0, . . . , L− 1.
Projection Pi
Projection Pj
P̂i
P̂j
3D Fourier space
3D Fourier space
(cosαij , sinαij )
(cosαji , sinαji )
Ricij = qij
Ricij = Rjcji = qij
Estimation error for ray l of projection i : 〈Ric(l)i , R̃ic
(l)i 〉.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 36 / 48
Examples: noiseless projections – estimation error (cont.)
Histogram of estimation errors (after registration) in degrees:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
50
100
150
200
250
N = 100 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0
500
1000
1500
2000
2500
N = 1000 0.02 0.04 0.06 0.08 0.1 0.12
0
0.5
1
1.5
2
2.5x 10
4
N = 1000
Error is very small even for N = 10, and decreases as N increases.
Error is not zero even in the noiseless case due to discretization of Fourierspace into L = 360.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 37 / 48
Examples: noisy projections – data set
500 projections from the density map of the 50S ribosomal subunit.
corrupted them with additive Gaussian white noise with SNR = 1/16 andSNR = 1/32.
Each image of size 129× 129.
Clean
SNR = 1/16
SNR = 1/32
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 38 / 48
Examples: noisy projections – spectrum of S
SNR = 1/16
1 2 3 4 5 6 7 8 9 100
50
100
150
200
250
300
i
λi
−50 0 50 100 150 200 250 300 3500
50
100
150
200
250
300
350
SNR = 1/32
1 2 3 4 5 6 7 8 9 100
50
100
150
200
250
i
λi
−50 0 50 100 150 200 2500
20
40
60
80
100
120
140
160
180
200
S still has three dominant eigenvalues.
Remaining eigenvalues are no longer small due to misidentifications ofcommon lines.
Still, there is a significant gap between the third and the fourth eigenvalues.
Gap gets smaller as the SNR decreases.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 39 / 48
Examples: noisy projections – estimation error
Angle estimation errors (in degrees):
0 5 10 15 20 25 300
2000
4000
6000
8000
10000
12000
14000
SNR = 1/16Mean error 5.73◦
90% less than 10◦
0 10 20 30 40 50 60 70 80 900
0.5
1
1.5
2
2.5x 10
4
SNR = 1/32Mean error 10.64◦
60% less than 10◦
For SNR = 1/16 only 28% of the common lines were identified correctly(deviates from true common line by up to 5 degrees).
At least 72% of the blocks Bij are completely wrong (the ones for which thecommon line between Pi and Pj was misidentified).
For SNR = 1/32 only 14% of the common lines were identified correctly.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 40 / 48
Examples: noisy projections – reconstructions
Reference SNR = 1/16 SNR = 1/32
Reconstructed from noisy images and estimated orientations
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 41 / 48
Examples: real data
40, 338 images of 70S split randomly into two group (20, 169 images each).
Each image 247× 247 pixels.
Class averaging - average each image with its 5 nearest neighbors.
Choose 1450 best class averages.
Pipeline implemented with ASPIRE (http://spr.math.princeton.edu/).
Raw images (provided by Dr. Joachim Frank)
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 42 / 48
Examples: real data
40, 338 images of 70S split randomly into two group (20, 169 images each).
Each image 247× 247 pixels.
Class averaging - average each image with its 5 nearest neighbors.
Choose 1450 best class averages.
Pipeline implemented with ASPIRE (http://spr.math.princeton.edu/).
Class averages
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 42 / 48
Examples: real data (cont.)
Group 1 Group 2
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 43 / 48
Examples: real data (cont.)
18.3 9.2 6.1 4.6 3.7 3.1
Resolution=24.90A
0.055 0.109 0.164 0.219 0.273 0.3280
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1/A
Fourier shell correlation
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 44 / 48
What’s next?
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 45 / 48
What’s next?
Details inY. Shkolnisky, A. Singer, Viewing Direction Estimation in Cryo-EM UsingSynchronization, SIAM Journal on Imaging Sciences, 5 (3), pp. 1088-1110(2012).
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 45 / 48
What’s next?
Details inY. Shkolnisky, A. Singer, Viewing Direction Estimation in Cryo-EM UsingSynchronization, SIAM Journal on Imaging Sciences, 5 (3), pp. 1088-1110(2012).
3N × 3N
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 45 / 48
What’s next?
Details inY. Shkolnisky, A. Singer, Viewing Direction Estimation in Cryo-EM UsingSynchronization, SIAM Journal on Imaging Sciences, 5 (3), pp. 1088-1110(2012).
3N × 3N
Improved common lines detection
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 45 / 48
What’s next?
Details inY. Shkolnisky, A. Singer, Viewing Direction Estimation in Cryo-EM UsingSynchronization, SIAM Journal on Imaging Sciences, 5 (3), pp. 1088-1110(2012).
3N × 3N
Improved common lines detection
Multiple common line candidates
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 45 / 48
What’s next?
Details inY. Shkolnisky, A. Singer, Viewing Direction Estimation in Cryo-EM UsingSynchronization, SIAM Journal on Imaging Sciences, 5 (3), pp. 1088-1110(2012).
3N × 3N
Improved common lines detection
Multiple common line candidates
common line algorithms for molecules with symmetries
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 45 / 48
What else (1)?
Common-Lines Based Approaches for 3D Ab-Initio Modeling
◮ R. R. Coifman, Y. Shkolnisky, F. J. Sigworth, A. Singer, Reference FreeStructure Determination through Eigenvectors of Center of Mass Operators,Applied and Computational Harmonic Analysis, 28 (3), pp. 296-312 (2010).
◮ A. Singer, R. R. Coifman, F. J. Sigworth, D. W. Chester, Y. Shkolnisky,Detecting Consistent Common Lines in Cryo-EM by Voting, Journal ofStructural Biology, 169 (3), pp. 312-322 (2010).
◮ A. Singer, Y. Shkolnisky, Three-Dimensional Structure Determination fromCommon Lines in Cryo-EM by Eigenvectors and Semidefinite Programming,SIAM Journal on Imaging Sciences, 4 (2), pp. 543-572 (2011).
◮ Y. Shkolnisky, A. Singer, Viewing Direction Estimation in Cryo-EM UsingSynchronization, SIAM Journal on Imaging Sciences, 5 (3), pp. 1088-1110(2012).
◮ L. Wang, A. Singer, Z. Wen, Orientation Determination from Cryo-EM imagesusing Least Unsquared Deviations, SIAM Journal on Imaging Sciences, 6 (4),pp. 2450-2483 (2013).
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 46 / 48
What else (2)?
2D Class Averaging
◮ A. Singer, Z. Zhao, Y. Shkolnisky, R. Hadani, Viewing Angle Classification ofCryo-Electron Microscopy Images using Eigenvectors, SIAM Journal onImaging Sciences, 4 (2), pp. 543-572 (2011).
◮ Z. Zhao, A. Singer, Rotationally Invariant Image Representation for ViewingDirection Classification in Cryo-EM, Journal of Structural Biology, 186 (1), pp.153-166 (2014).
◮ A. Singer, H.-T. Wu, Vector Diffusion Maps and the Connection Laplacian,Communications on Pure and Applied Mathematics (CPAM), 65 (8), pp.1067-1144 (2012).
Code available in ASPIRE (http://spr.math.princeton.edu/).
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 47 / 48
Thank You!
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj
qij – unit vector in the direction of intersection of the planes.Since Ricij = Rjcji = qij and Ri are orthogonal
Gijk =
— qTij —
— qTik —
— qTjk —
| | |qij qik qjk| | |
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj
qij – unit vector in the direction of intersection of the planes.Since Ricij = Rjcji = qij and Ri are orthogonal
Gijk =
— qTij —
— qTik —
— qTjk —
| | |qij qik qjk| | |
=
1 〈qij , qik〉 〈qij , qjk〉〈qik , qij〉 1 〈qik , qjk〉〈qjk , qij〉 〈qjk , qik〉 1
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj
qij – unit vector in the direction of intersection of the planes.Since Ricij = Rjcji = qij and Ri are orthogonal
Gijk =
— qTij —
— qTik —
— qTjk —
| | |qij qik qjk| | |
=
1 〈qij , qik〉 〈qij , qjk〉〈qik , qij〉 1 〈qik , qjk〉〈qjk , qij〉 〈qjk , qik〉 1
=
1 〈Ricij ,Ri cik〉 〈Rj cji ,Rjcjk 〉〈Ricik ,Ri cij 〉 1 〈Rkcki , Rkckj〉〈Rjcjk ,Rjcji 〉 〈Rkckj ,Rkcki 〉 1
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj
qij – unit vector in the direction of intersection of the planes.Since Ricij = Rjcji = qij and Ri are orthogonal
Gijk =
— qTij —
— qTik —
— qTjk —
| | |qij qik qjk| | |
=
1 〈qij , qik〉 〈qij , qjk〉〈qik , qij〉 1 〈qik , qjk〉〈qjk , qij〉 〈qjk , qik〉 1
=
1 〈Ricij ,Ri cik〉 〈Rj cji ,Rjcjk 〉〈Ricik ,Ri cij 〉 1 〈Rkcki , Rkckj〉〈Rjcjk ,Rjcji 〉 〈Rkckj ,Rkcki 〉 1
=
1 〈cij , cik 〉 〈cji , cjk 〉〈cik , cij〉 1 〈cki , ckj 〉〈cjk , cji 〉 〈ckj , cki〉 1
.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
Gijk =
— qTij —
— qTik —
— qTjk —
︸ ︷︷ ︸
QTijk
| | |qij qik qjk| | |
︸ ︷︷ ︸
Qijk
=
1 〈cij , cik〉 〈cji , cjk 〉〈cik , cij〉 1 〈cki , ckj 〉〈cjk , cji 〉 〈ckj , cki〉 1
.
Extract Qijk = (qij qik qjk) from Gijk by SVD.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
Gijk =
— qTij —
— qTik —
— qTjk —
︸ ︷︷ ︸
QTijk
| | |qij qik qjk| | |
︸ ︷︷ ︸
Qijk
=
1 〈cij , cik〉 〈cji , cjk 〉〈cik , cij〉 1 〈cki , ckj 〉〈cjk , cji 〉 〈ckj , cki〉 1
.
Extract Qijk = (qij qik qjk) from Gijk by SVD.
Actually extract Q̃ijk = OijkQijk , for some arbitrary orthogonal matrix Oijk
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
Gijk =
— qTij —
— qTik —
— qTjk —
︸ ︷︷ ︸
QTijk
| | |qij qik qjk| | |
︸ ︷︷ ︸
Qijk
=
1 〈cij , cik〉 〈cji , cjk 〉〈cik , cij〉 1 〈cki , ckj 〉〈cjk , cji 〉 〈ckj , cki〉 1
.
Extract Qijk = (qij qik qjk) from Gijk by SVD.
Actually extract Q̃ijk = OijkQijk , for some arbitrary orthogonal matrix Oijk
(since Q̃Tijk Q̃ijk = (OijkQijk)
TOijkQijk = QT
ijkOTijkOijkQijk = QT
ijkQijk = Gijk).
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
From the triplet of images Pi , Pj , Pk we have computed the matrixQijk = (qij qik qjk ).
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
From the triplet of images Pi , Pj , Pk we have computed the matrixQijk = (qij qik qjk ).Construct the 3× 3 matrices (using the computed Qijk)
Ci = (cij cik cij × cik) , Qi = (qij qik qij × qik) ,
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
From the triplet of images Pi , Pj , Pk we have computed the matrixQijk = (qij qik qjk).Construct the 3× 3 matrices (using the computed Qijk)
Ci = (cij cik cij × cik) , Qi = (qij qik qij × qik) ,
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
From the triplet of images Pi , Pj , Pk we have computed the matrixQijk = (qij qik qjk).Construct the 3× 3 matrices (using the computed Qijk)
Ci = (cij cik cij × cik) , Qi = (qij qik qij × qik) ,
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
From the triplet of images Pi , Pj , Pk we have computed the matrixQijk = (qij qik qjk ).Construct the 3× 3 matrices (using the computed Qijk)
Ci = (cij cik cij × cik) , Qi = (qij qik qij × qik) ,
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
From the triplet of images Pi , Pj , Pk we have computed the matrixQijk = (qij qik qjk).Construct the 3× 3 matrices (using the computed Qijk)
Ci = (cij cik cij × cik) , Qi = (qij qik qij × qik) ,
Cj = (cji cjk cji × cjk) , Qj = (qji qjk qji × qjk) .
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
From the triplet of images Pi , Pj , Pk we have computed the matrixQijk = (qij qik qjk ).Construct the 3× 3 matrices (using the computed Qijk)
Ci = (cij cik cij × cik) , Qi = (qij qik qij × qik) ,
Cj = (cji cjk cji × cjk) , Qj = (qji qjk qji × qjk) .
Common lines equations Ricij = qij give that
RiCi = Ri
(| | |cij cik cij×cik| | |
)
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
From the triplet of images Pi , Pj , Pk we have computed the matrixQijk = (qij qik qjk ).Construct the 3× 3 matrices (using the computed Qijk)
Ci = (cij cik cij × cik) , Qi = (qij qik qij × qik) ,
Cj = (cji cjk cji × cjk) , Qj = (qji qjk qji × qjk) .
Common lines equations Ricij = qij give that
RiCi = Ri
(| | |cij cik cij×cik| | |
)
=
(| | |
Ri cij Ri cik Ricij×Ri cik| | |
)
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
From the triplet of images Pi , Pj , Pk we have computed the matrixQijk = (qij qik qjk ).Construct the 3× 3 matrices (using the computed Qijk)
Ci = (cij cik cij × cik) , Qi = (qij qik qij × qik) ,
Cj = (cji cjk cji × cjk) , Qj = (qji qjk qji × qjk) .
Common lines equations Ricij = qij give that
RiCi = Ri
(| | |cij cik cij×cik| | |
)
=
(| | |
Ri cij Ri cik Ricij×Ri cik| | |
)
=
(| | |qij qik qij×qik| | |
)
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
From the triplet of images Pi , Pj , Pk we have computed the matrixQijk = (qij qik qjk ).Construct the 3× 3 matrices (using the computed Qijk)
Ci = (cij cik cij × cik) , Qi = (qij qik qij × qik) ,
Cj = (cji cjk cji × cjk) , Qj = (qji qjk qji × qjk) .
Common lines equations Ricij = qij give that
RiCi = Ri
(| | |cij cik cij×cik| | |
)
=
(| | |
Ri cij Ri cik Ricij×Ri cik| | |
)
=
(| | |qij qik qij×qik| | |
)
= Qi .
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
From the triplet of images Pi , Pj , Pk we have computed the matrixQijk = (qij qik qjk ).Construct the 3× 3 matrices (using the computed Qijk)
Ci = (cij cik cij × cik) , Qi = (qij qik qij × qik) ,
Cj = (cji cjk cji × cjk) , Qj = (qji qjk qji × qjk) .
Common lines equations Ricij = qij give that
RiCi = Ri
(| | |cij cik cij×cik| | |
)
=
(| | |
Ri cij Ri cik Ricij×Ri cik| | |
)
=
(| | |qij qik qij×qik| | |
)
= Qi .
Similarly, RjCj = Qj .
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
From the triplet of images Pi , Pj , Pk we have computed the matrixQijk = (qij qik qjk ).Construct the 3× 3 matrices (using the computed Qijk)
Ci = (cij cik cij × cik) , Qi = (qij qik qij × qik) ,
Cj = (cji cjk cji × cjk) , Qj = (qji qjk qji × qjk) .
Common lines equations Ricij = qij give that
RiCi = Ri
(| | |cij cik cij×cik| | |
)
=
(| | |
Ri cij Ri cik Ricij×Ri cik| | |
)
=
(| | |qij qik qij×qik| | |
)
= Qi .
Similarly, RjCj = Qj .
w�
Ri = QiC−1i , Rj = QjC
−1j
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
From the triplet of images Pi , Pj , Pk we have computed the matrixQijk = (qij qik qjk ).Construct the 3× 3 matrices (using the computed Qijk)
Ci = (cij cik cij × cik) , Qi = (qij qik qij × qik) ,
Cj = (cji cjk cji × cjk) , Qj = (qji qjk qji × qjk) .
Common lines equations Ricij = qij give that
RiCi = Ri
(| | |cij cik cij×cik| | |
)
=
(| | |
Ri cij Ri cik Ricij×Ri cik| | |
)
=
(| | |qij qik qij×qik| | |
)
= Qi .
Similarly, RjCj = Qj .
w�
Ri = QiC−1i , Rj = QjC
−1j
w�
R−1i Rj =
(QiC
−1i
)−1QjC
−1j = CiQ
−1i QjC
−1j .
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
We actually have Q̃ijk = OijkQijk for some arbitrary orthogonal Oijk .
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
We actually have Q̃ijk = OijkQijk for some arbitrary orthogonal Oijk .
If detOijk = 1, we get the matrices
Qi = ( qij , qik , qij × qik)
Qj = ( qji , qjk , qji × qjk)
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
We actually have Q̃ijk = OijkQijk for some arbitrary orthogonal Oijk .
If detOijk = 1, we get the matrices
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik)
Q̃j = (Oijkqji ,Oijkqjk ,Oijkqji × Oijkqjk)
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
We actually have Q̃ijk = OijkQijk for some arbitrary orthogonal Oijk .
If detOijk = 1, we get the matrices
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik) = Oijk (qij , qik , qij × qik)
Q̃j = (Oijkqji ,Oijkqjk ,Oijkqji × Oijkqjk) = Oijk (qji , qjk , qji × qjk )
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
We actually have Q̃ijk = OijkQijk for some arbitrary orthogonal Oijk .
If detOijk = 1, we get the matrices
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik) = Oijk (qij , qik , qij × qik) = OijkQi ,
Q̃j = (Oijkqji ,Oijkqjk ,Oijkqji × Oijkqjk) = Oijk (qji , qjk , qji × qjk ) = OijkQj .
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
We actually have Q̃ijk = OijkQijk for some arbitrary orthogonal Oijk .
If detOijk = 1, we get the matrices
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik) = Oijk (qij , qik , qij × qik) = OijkQi ,
Q̃j = (Oijkqji ,Oijkqjk ,Oijkqji × Oijkqjk) = Oijk (qji , qjk , qji × qjk ) = OijkQj .
Then, instead of Ri = QiC−1i and Rj = QjC
−1j we recover
R̃i = Q̃iC−1i = OijkQiC
−1i , R̃j = Q̃jC
−1j = OijkQjC
−1j ,
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
We actually have Q̃ijk = OijkQijk for some arbitrary orthogonal Oijk .
If detOijk = 1, we get the matrices
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik) = Oijk (qij , qik , qij × qik) = OijkQi ,
Q̃j = (Oijkqji ,Oijkqjk ,Oijkqji × Oijkqjk) = Oijk (qji , qjk , qji × qjk ) = OijkQj .
Then, instead of Ri = QiC−1i and Rj = QjC
−1j we recover
R̃i = Q̃iC−1i = OijkQiC
−1i , R̃j = Q̃jC
−1j = OijkQjC
−1j ,
and
R̃−1i R̃j =
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
We actually have Q̃ijk = OijkQijk for some arbitrary orthogonal Oijk .
If detOijk = 1, we get the matrices
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik) = Oijk (qij , qik , qij × qik) = OijkQi ,
Q̃j = (Oijkqji ,Oijkqjk ,Oijkqji × Oijkqjk) = Oijk (qji , qjk , qji × qjk ) = OijkQj .
Then, instead of Ri = QiC−1i and Rj = QjC
−1j we recover
R̃i = Q̃iC−1i = OijkQiC
−1i , R̃j = Q̃jC
−1j = OijkQjC
−1j ,
and
R̃−1i R̃j =
(OijkQiC
−1i
)−1OijkQjC
−1j
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
We actually have Q̃ijk = OijkQijk for some arbitrary orthogonal Oijk .
If detOijk = 1, we get the matrices
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik) = Oijk (qij , qik , qij × qik) = OijkQi ,
Q̃j = (Oijkqji ,Oijkqjk ,Oijkqji × Oijkqjk) = Oijk (qji , qjk , qji × qjk ) = OijkQj .
Then, instead of Ri = QiC−1i and Rj = QjC
−1j we recover
R̃i = Q̃iC−1i = OijkQiC
−1i , R̃j = Q̃jC
−1j = OijkQjC
−1j ,
and
R̃−1i R̃j =
(OijkQiC
−1i
)−1OijkQjC
−1j = CiQ
−1i O−1
ijk OijkQjC−1j
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
We actually have Q̃ijk = OijkQijk for some arbitrary orthogonal Oijk .
If detOijk = 1, we get the matrices
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik) = Oijk (qij , qik , qij × qik) = OijkQi ,
Q̃j = (Oijkqji ,Oijkqjk ,Oijkqji × Oijkqjk) = Oijk (qji , qjk , qji × qjk ) = OijkQj .
Then, instead of Ri = QiC−1i and Rj = QjC
−1j we recover
R̃i = Q̃iC−1i = OijkQiC
−1i , R̃j = Q̃jC
−1j = OijkQjC
−1j ,
and
R̃−1i R̃j =
(OijkQiC
−1i
)−1OijkQjC
−1j = CiQ
−1i O−1
ijk OijkQjC−1j
= CiQ−1i QjC
−1j
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
We actually have Q̃ijk = OijkQijk for some arbitrary orthogonal Oijk .
If detOijk = 1, we get the matrices
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik) = Oijk (qij , qik , qij × qik) = OijkQi ,
Q̃j = (Oijkqji ,Oijkqjk ,Oijkqji × Oijkqjk) = Oijk (qji , qjk , qji × qjk ) = OijkQj .
Then, instead of Ri = QiC−1i and Rj = QjC
−1j we recover
R̃i = Q̃iC−1i = OijkQiC
−1i , R̃j = Q̃jC
−1j = OijkQjC
−1j ,
and
R̃−1i R̃j =
(OijkQiC
−1i
)−1OijkQjC
−1j = CiQ
−1i O−1
ijk OijkQjC−1j
= CiQ−1i QjC
−1j = R−1
i Rj .
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
We actually have Q̃ijk = OijkQijk for some arbitrary orthogonal Oijk .
If detOijk = 1, we get the matrices
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik) = Oijk (qij , qik , qij × qik) = OijkQi ,
Q̃j = (Oijkqji ,Oijkqjk ,Oijkqji × Oijkqjk) = Oijk (qji , qjk , qji × qjk ) = OijkQj .
Then, instead of Ri = QiC−1i and Rj = QjC
−1j we recover
R̃i = Q̃iC−1i = OijkQiC
−1i , R̃j = Q̃jC
−1j = OijkQjC
−1j ,
and
R̃−1i R̃j =
(OijkQiC
−1i
)−1OijkQjC
−1j = CiQ
−1i O−1
ijk OijkQjC−1j
= CiQ−1i QjC
−1j = R−1
i Rj .
If detOijk = 1 then R−1i Rj is independent of Oijk .
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
If detOijk = −1, we get
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik)
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
If detOijk = −1, we get
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik)
= (Oijkqij ,Oijkqik ,−Oijk (qij × qik))
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
If detOijk = −1, we get
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik)
= (Oijkqij ,Oijkqik ,−Oijk (qij × qik))
= Oijk (qij , qik ,− (qij × qik))
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
If detOijk = −1, we get
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik)
= (Oijkqij ,Oijkqik ,−Oijk (qij × qik))
= Oijk (qij , qik ,− (qij × qik))
= OijkQiJ, J = diag(1, 1,−1).
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
If detOijk = −1, we get
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik)
= (Oijkqij ,Oijkqik ,−Oijk (qij × qik))
= Oijk (qij , qik ,− (qij × qik))
= OijkQiJ, J = diag(1, 1,−1).
Similarly Q̃j = OijkQjJ.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
If detOijk = −1, we get
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik)
= (Oijkqij ,Oijkqik ,−Oijk (qij × qik))
= Oijk (qij , qik ,− (qij × qik))
= OijkQiJ, J = diag(1, 1,−1).
Similarly Q̃j = OijkQjJ.
The matrices R̃i and R̃j are given by
R̃i = OijkQiJC−1i , R̃j = OijkQjJC
−1j .
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
If detOijk = −1, we get
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik)
= (Oijkqij ,Oijkqik ,−Oijk (qij × qik))
= Oijk (qij , qik ,− (qij × qik))
= OijkQiJ, J = diag(1, 1,−1).
Similarly Q̃j = OijkQjJ.
The matrices R̃i and R̃j are given by
R̃i = OijkQiJC−1i , R̃j = OijkQjJC
−1j .
Since the third coordinate in all vectors cij is zero, by a direct calculation weget that CiJ = JCi , and JC−1
i = C−1i J.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
If detOijk = −1, we get
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik)
= (Oijkqij ,Oijkqik ,−Oijk (qij × qik))
= Oijk (qij , qik ,− (qij × qik))
= OijkQiJ, J = diag(1, 1,−1).
Similarly Q̃j = OijkQjJ.
The matrices R̃i and R̃j are given by
R̃i = OijkQiJC−1i , R̃j = OijkQjJC
−1j .
Since the third coordinate in all vectors cij is zero, by a direct calculation weget that CiJ = JCi , and JC−1
i = C−1i J. Thus,
R̃−1i R̃j =
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
If detOijk = −1, we get
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik)
= (Oijkqij ,Oijkqik ,−Oijk (qij × qik))
= Oijk (qij , qik ,− (qij × qik))
= OijkQiJ, J = diag(1, 1,−1).
Similarly Q̃j = OijkQjJ.
The matrices R̃i and R̃j are given by
R̃i = OijkQiJC−1i , R̃j = OijkQjJC
−1j .
Since the third coordinate in all vectors cij is zero, by a direct calculation weget that CiJ = JCi , and JC−1
i = C−1i J. Thus,
R̃−1i R̃j =
(OijkQiJC
−1i
)−1OijkQjJC
−1j
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
If detOijk = −1, we get
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik)
= (Oijkqij ,Oijkqik ,−Oijk (qij × qik))
= Oijk (qij , qik ,− (qij × qik))
= OijkQiJ, J = diag(1, 1,−1).
Similarly Q̃j = OijkQjJ.
The matrices R̃i and R̃j are given by
R̃i = OijkQiJC−1i , R̃j = OijkQjJC
−1j .
Since the third coordinate in all vectors cij is zero, by a direct calculation weget that CiJ = JCi , and JC−1
i = C−1i J. Thus,
R̃−1i R̃j =
(OijkQiJC
−1i
)−1OijkQjJC
−1j
= CiJQ−1i QjJCj
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
If detOijk = −1, we get
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik)
= (Oijkqij ,Oijkqik ,−Oijk (qij × qik))
= Oijk (qij , qik ,− (qij × qik))
= OijkQiJ, J = diag(1, 1,−1).
Similarly Q̃j = OijkQjJ.
The matrices R̃i and R̃j are given by
R̃i = OijkQiJC−1i , R̃j = OijkQjJC
−1j .
Since the third coordinate in all vectors cij is zero, by a direct calculation weget that CiJ = JCi , and JC−1
i = C−1i J. Thus,
R̃−1i R̃j =
(OijkQiJC
−1i
)−1OijkQjJC
−1j
= CiJQ−1i QjJCj = JCiQ
−1i QjC
−1j J (since CiJ = JCi )
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
Computing the ratios R−1i Rj (cont.)
If detOijk = −1, we get
Q̃i = (Oijkqij ,Oijkqik ,Oijkqij × Oijkqik)
= (Oijkqij ,Oijkqik ,−Oijk (qij × qik))
= Oijk (qij , qik ,− (qij × qik))
= OijkQiJ, J = diag(1, 1,−1).
Similarly Q̃j = OijkQjJ.
The matrices R̃i and R̃j are given by
R̃i = OijkQiJC−1i , R̃j = OijkQjJC
−1j .
Since the third coordinate in all vectors cij is zero, by a direct calculation weget that CiJ = JCi , and JC−1
i = C−1i J. Thus,
R̃−1i R̃j =
(OijkQiJC
−1i
)−1OijkQjJC
−1j
= CiJQ−1i QjJCj = JCiQ
−1i QjC
−1j J (since CiJ = JCi )
= JR−1i RjJ.
Yoel Shkolnisky (Tel-Aviv University) Synchronization in Cryo-EM July 08, 2014 48 / 48
top related