Computerized Classiﬁcation of Surface Spikes in Three ... · 8/26/2014 · Motivation • Inﬂuenza is a rapidly changing virus that appears seasonally in the human population.

Computerized Classification of SurfaceSpikes in Three-Dimensional Electron Microscopic

Reconstructions of Viruses

Younes Benkarroum

The Graduate Center of City University of New York

08/26/2014

Motivation• Influenza is a rapidly changing virus that appears

seasonally in the human population.

• Every year a new strain of the influenza virus appearswith the potential to cause a serious global pandemic.

• During mixed infections, RNPs are reassorted resulting in new combination of HA and NA.

• Knowledge of the structure and density of the surface proteins is of critical importance in avaccine candidate.

• ART with blobs provides 3D reconstructions of viruses from tomographic tilt series that allow thedesired reliable quantification of the surface proteins.

name of author Reconstructions and Proteins Classification of Viruses Current: 2, total: 88

Series expansion methods

Series Expansion Methods

• Assume that the reconstructed object can be described by a linear combination of finite set offixed basis functions

f ∗ (x ,y ,z)=N∑

j=1cjb

(x −xj ,y −yj ,z −zj

)

• Estimate the unknown coefficients of the linear combination based on projection images

p∗i =

N∑j=1

ai ,jcj

where ai ,j is the line integral, along the line i , of the shifted basis function centered at(xj ,yj ,zj

)name of author Reconstructions and Proteins Classification of Viruses Current: 3, total: 88

Series expansion methods

Algebraic Reconstruction Technique

• ART calculates and uses the difference between the measured data and the calculated forwardprojection to update the image

c(k+1)j = c(k)j +λ

pi(k) −∑h

ai(k),hc(k)h∑h

a2i(k),h

ai(k),j

λ is the relaxation parameter and i(k) = (k mod M)+1

• The algorithm attempts to find a vector c that is an approximate solution to the linear systemp =Ac, where p is the data vector


VoxelsVoxel Basis Function

• The choice of the set of basis functions greatly influences the result of the reconstructionalgorithm.

• The conventional choice is the voxel basis function which has the value 1 inside the j th voxel andthe value 0 otherwise.

• In such case, the coefficient cj becomesthe average value of ƒ∗ inside the j th

voxel.

• Reconstructions using cubic voxels haveundesirable artificial sharp edges.


BlobsBlob Basis Function

• Basis functions with spherical symmetry and a smooth transition from one to zero.

b (r)=

I2

(α

√1−( r

a )2)

I2(α)

(1− ( r

a)2

), if 0≤ x ≤ a,

0, otherwise.

where I2 denotes the modified Besselfunction of order 2, a determines thesupport of the blob (radius), and α is aparameter controlling the blob shape.

α= 6, a= 1


f

f (x)


b

b(x −xj

)


Approximation

Approximation of Constant-Valued Functions Using Blobs

f ∗ (x)=N∑

j=1cjb

(x −xj

), where cj = f

(xj

).


Approximation


f ∗ (x)=N∑

j=1cjb

(x −xj

), where cj = f

(xj

).

Mathematically, the function f ∗ can be written as : f ∗ = (f ×XG∆)∗b.


Approximation


f ∗ (x)=N∑

j=1cjb

(x −xj

), where cj = f

(xj

).


f (x)={

1, if |x | ≤ ρ,0, otherwise.


Approximation


f ∗ (x)=N∑

j=1cjb

(x −xj

), where cj = f

(xj

).


f (x)={


XG4(x)=∑

pεG4δ(x −p),

where G∆ is the set of integers ×4.


Approximation


f ∗ (x)=N∑

j=1cjb

(x −xj

), where cj = f

(xj

).


f (x)={


XG4(x)=∑

pεG4δ(x −p),

where G∆ is the set of integers ×4.

b (x)=

I2

(α

√1−( x

a )2)

I2(α)

(1− ( x

a)2

), if 0≤ x ≤ a,

0, otherwise.


Sha

(f×XG4)∗b

4= 0.5name of author Reconstructions and Proteins Classification of Viruses Current: 10, total: 88

f

(f×XG4)∗b


f x Sha

(f ×XG4)∗b


f star

f ∗ = (f ×XG4)∗b


f Vs f star

f vs f ∗


Fourier Space

Fourier Space Analysis

f ∗ = (f ×XG4)∗b ⇐⇒ f̂ ∗ =(f̂ ∗ �XG4

)× b̂


Fourier Space


f ∗ = (f ×XG4)∗b ⇐⇒ f̂ ∗ =(f̂ ∗ �XG4

)× b̂

f̂ (X )= 2ρ sin(ρX)ρX


Fourier Space


f ∗ = (f ×XG4)∗b ⇐⇒ f̂ ∗ =(f̂ ∗ �XG4

)× b̂


�XG4 = 143 XG1/4


Fourier Space


f ∗ = (f ×XG4)∗b ⇐⇒ f̂ ∗ =(f̂ ∗ �XG4

)× b̂


�XG4 = 143 XG1/4

b̂ (X )=

(2π)32 a3α2

I2(α)

I7/2

(√α2−(2πaX)2

)(√

α2−(2πaX)2)7/2 , if 2πaX ≤α,

(2π)32 a3α2

I2(α)

J7/2

(√(2πaX)2−α2

)(√

(2πaX)2−α2)7/2 , otherwise.


f hat

(f̂∗�XG4)× b̂


f hat x Sha hat

(f̂ ∗ �XG4)× b̂


b hat - random variables(f̂∗�XG4)× b̂

b̂ with random parameters:a= 0.8 and α= 11


Final product - random variables(f̂ ∗ �XG4)× b̂



Final product 2 - random variables(f̂ ∗ �XG4)× b̂



f Vs f star - random variables

f vs f ∗



b hat - crossing N1

(f̂∗�XG4)× b̂

b̂ crossing N1:a= 0.6 and α= 4.861220


b hat - Comparison 1

(f̂∗�XG4)× b̂

Random Crossing N1


Final product - Crossing N1

(f̂ ∗ �XG4)× b̂



Final product - Comparison 1

f̂ ∗ = (f̂ ∗ �XG4)× b̂

Random Crossing N1


f star - Crossing N1

f ∗ = (f ×XG4)∗b



f Vs f star - Crossing N1

f vs f ∗



f Vs f - Comparison 1

f vs f ∗

Random Crossing N1


b hat - Crossing N1 and N2

(f̂∗�XG4)× b̂

b̂ crossing N1 and N2:a= 0.500427 and α= 2.515620


Parameters

Blob Parameters

b̂ (X )=

(2π)32 a3α2

I2(α)

I7/2

(√α2−(2πaX)2

)(√

α2−(2πaX)2)7/2 , if 2πaX ≤α,

(2π)32 a3α2

I2(α)

J7/2

(√(2πaX)2−α2

)(√



Parameters

Blob Parameters

b̂ (X )=

(2π)32 a3α2

I2(α)

I7/2

(√α2−(2πaX)2

)(√

α2−(2πaX)2)7/2 , if 2πaX ≤α,

(2π)32 a3α2

I2(α)

J7/2

(√(2πaX)2−α2

)(√


√(2πaX1)

2 −α2 = j1√(2πaX2)

2 −α2 = j2


Parameters

Blob Parameters

b̂ (X )=

(2π)32 a3α2

I2(α)

I7/2

(√α2−(2πaX)2

)(√

α2−(2πaX)2)7/2 , if 2πaX ≤α,

(2π)32 a3α2

I2(α)

J7/2

(√(2πaX)2−α2

)(√


√(2πaX1)

2 −α2 = j1

a= 12π

√j22−j21

X22 −X2

1

√(2πaX2)

2 −α2 = j2

α=√

X21 j22−X2

2 j21X2

2 −X21


Final product - Crossing N1 and N2

f̂ ∗ = (f̂ ∗ �XG4)× b̂



Final product - Comparison 2

f̂ ∗ = (f̂ ∗ �XG4)× b̂

Crossing N1 Crossing N1 & N2


Final product - Comparison 2b

f̂ ∗ = (f̂ ∗ �XG4)× b̂



f star - Crossing N1 and N2

f ∗ = (f ×XG4)∗b



f Vs f star - Crossing N1 and N2

f vs f ∗




f vs f ∗



b hat - 3rd zero crossing at N2

b̂ with 3rd zero crossing at N2

Blob parameters:a= 0.500427 and α= 2.515620


b hat - 4th zero crossing at N2

b̂ with 4th zero crossing at N2



b hat - 9th zero crossing at N2

b̂ with 9th zero crossing at N2



Bessel

Bessel Function

√(2πaX1)

2 −α2 = j1√(2πaX2)

2 −α2 = j2name of author Reconstructions and Proteins Classification of Viruses Current: 40, total: 88

f Vs f star - 9th zero crossing at N2

f vs f ∗

b̂ with 9th zero crossing at N2:a= 1.414401 and α= 16.813501



f vs f ∗

3rd zero crossing at N2 9th zero crossing at N2


f Vs f - Comparison 3b

f vs f ∗

3rd zero crossing at N2 9th zero crossing at N2


f̂ Vs b̂3 and b̂9

f̂ vs (b̂3 & b̂9)


f̂ x b̂3 and b̂9

f̂ × (b̂3 & b̂9)


3D Grids

3D Grids

�XBβ= 1

4β3 XF1/2β


Approximation


f ∗ = (f ×XBβ)∗b

f (~x)={

1, if∥∥~x∥∥≤ ρ,

0, otherwise.

b (~x)=

I2

(α

√1−( r

a )2)

I2(α)

(1− ( r

a)2

), if 0≤ r ≤ a,

0, otherwise,

where r denotes the norm∥∥~x∥∥ of the vector ~x .


Fourier Space


f ∗ = (f ×XBβ)∗b ⇐⇒ f̂ ∗ =

(f̂ ∗ �XBβ

)× b̂


Fourier Space


f ∗ = (f ×XBβ)∗b ⇐⇒ f̂ ∗ =

(f̂ ∗ �XBβ

)× b̂

f̂ (R)= sin(2πρR)−2πρcos(2πρR)2π2R3

�XBβ= 1

4β3 XF1/2β

b̂ (X )=

(2π)32 a3α2

I2(α)

I7/2

(√α2−(2πaX)2

)(√

α2−(2πaX)2)7/2 , if 2πaX ≤α,

(2π)32 a3α2

I2(α)

J7/2

(√(2πaX)2−α2

)(√



f hat

(f̂∗�XBβ)× b̂



f̂ vs (b̂4 & b̂9)



f̂ vs (b̂4 & b̂9)


b hat

(f̂∗�XBβ)× b̂


f hat

(f̂∗�XBβ)× b̂


f hat x Sha hat 1(f̂ ∗ �XBβ

)× b̂


f hat x Sha hat 2

(f̂ ∗ �XBβ)× b̂


Final product

(f̂ ∗ �XBβ)× b̂


Recommended Blob Parameters


a= β

πp

2

√j2r2 − j2r1 α=

√j2r2 −2j2r1

jy , jz a α

j1, j2 β×1.738875 3.294537j1, j3 β×2.651778 9.485434j1, j4 β×3.469269 13.738507j1, j5 β×4.247117 17.527826j1, j6 β×5.003932 21.105107j1, j7 β×5.748062 24.563319j1, j8 β×6.483890 27.946764j1, j9 β×7.213964 31.279745


Approximations

Central slice of the phantom and its blob representations. Display window: 0.999 - 1.001

Approximations Edges

Error

Errors

f̂ vs (b̂1,4 & b̂1,9)

Eβ,r1,r2 =ˆ

∣∣∣~X ∣∣∣<1/2p

2β

1−b̂r1,r2

(~X

)b̂r1,r2

(~0

)2

d~X

+ˆ

∣∣∣~X ∣∣∣>1/2p

2β

b̂r1,r2

(~X

)b̂r1,r2

(~0

)2

d~X .



Errors

jr1� jr2 j4 j5 j6 j7 j8 j9j1 0.211697 0.255772 0.294584 0.327072 0.353848 0.375856j2 0.218827 0.257221 0.294687 0.326843 0.353571 0.375609j3 0.313518 0.307784 0.331719 0.355704 0.376623j4 0.362973 0.366868 0.381674j5 0.424956 0.400413

Values of the error Eβ,r1,r2 for the (r1,r2) pairs (1≤ r1 < r2 ≤ 9) with β= 1/p

2


Blob Parameters


a α

j1, j4 β×3.469269 13.738507


Blob Parameters


a α

j1, j4 β×3.469269 13.738507

Standard Blob Parameters

α=√

2π2(

aβ

)2 −6.9879322


Blob Parameters


a α

j1, j4 β×3.469269 13.738507

β= 1/p

2 a= 2.453144 α= 13.738507

Standard Blob Parameters

α=√

2π2(

aβ

)2 −6.9879322

β= 1/p

2 a= 2.0 α= 10.4


Approximations

Approximations Using Blobs

Display Window: 0.999 - 1.001

RecommendedParametersa= 2.453144α= 13.738507

StandardParametersa= 2.0α= 10.4

Modified StandardParametersa= 2.0α= 10.444256


ApproximationsApproximation Using Blobs


Central slice of the FORBILD abdomen phantom and its blob representation


ApproximationsApproximation Using Blobs


Central slice of the FORBILD thorax phantom and its blob representation


ReconstructionsReconstructions Using Blobs

RecommendedParametersa= 2.453144α= 13.738507

StandardParametersa= 2.0α= 10.4

ReconstructionsART Reconstruction


Central slice of the reconstruction of the FORBILD abdomen phantom

ReconstructionsART Reconstruction


Central slice of the reconstruction of the FORBILD thorax phantom

Projection images

Projection Images


Coulomb

Data ProcessingTheory requires that the object to bereconstructed can be represented by afunction of finite suppor.

pi =´ b

a γdl +´ cb v (l) dl +´ d

c γdl ,

pi ′ =´ d ′

a′ γdl = ´ da γdl .


Coulomb

Data ProcessingTheory requires that the object to bereconstructed can be represented by afunction of finite suppor.

pi =´ b

a γdl +´ cb v (l) dl +´ d

c γdl ,

pi ′ =´ d ′

a′ γdl = ´ da γdl .

pi −pi ′ =´ c

b [v(l)−γ] dl .

´∞−∞ f (l) dl = ´ c

b [v(l)−γ] dl .


Projection images


Elevation

C1u =√

x21 +x2

3 ×cos(αu +arctan

(x3x1

)),


Least sq solution

Least Squares Estimator

C1u (x1,x3)=√

x21 +x2

3×cos(αu +arctan

(x3x1

)),

Let d (x1,x3)=∑u

(C1u (x1,x3)−C

′1u

)2,

C′1us are the manually determined values.


Least sq solution


C1u (x1,x3)=√

x21 +x2

3×cos(αu +arctan

(x3x1

)),

Let d (x1,x3)=∑u

(C1u (x1,x3)−C

′1u

)2,


d (x1d ,x3d )= minx1,x3

(d (x1,x3)) ,

C1u =√

x21d +x2

3d ×cos(αu +arctan

(x3dx1d

)).


Least sq solution


C1u (x1,x3)=√

x21 +x2

3×cos(αu +arctan

(x3x1

)),

Let d (x1,x3)=∑u

(C1u (x1,x3)−C

′1u

)2,


d (x1d ,x3d )= minx1,x3

(d (x1,x3)) ,

C1u =√

x21d +x2

3d ×cos(αu +arctan

(x3dx1d

)).


Software


Data processing

Data Processing


Processed data

Processed Data

Three projections of the isolated virus after processing


ART reconstruction

ART Reconstruction

Three different near-central cross-sections


Optimal reconstruction 1

Standard reconstruction 1

Optimal reconstruction 2

Standard reconstruction 2

Sikes 1

Docked atomic models of HA (yellow) and NA (red)


Al slices

Circles

Identification of a Region that Contains All Spikes


Segmentation

Segmentation

- Thresholding Segmentation.

Fisher’s linear discriminant:

J = |m1−m2|2s2

1+s22

,

where mi and s2i represent the mean

and the variance of the class i .

- Simultaneous fuzzy segmentation intomultiple objects.

- Segmentation based on the use ofcomponent trees.


Features

Feature Extraction & Classification


Evaluation

Evaluation Methodology

• Create 3D phantoms of viruses having HA and NA spikes.

• Generate projection images by simulating the behavior of cryogenic electron microscopy.

• Calculate 3D reconstructions from projection images.

• Apply the full classification procedure to the outputs of the reconstructions.

• Test the classifier massively on many test phantoms with different HA and NA conformations.

• Evaluate the performance of the classifier using a simple and transparent figure of merit calledclassification purity.


Evaluation

Evaluation Methodology

c1 c2HA 890 110NA 70 930

Table: Example of an array created to evaluate the classification purityt.

• The classification purity is 91%

(100× (890+930)) / (890+110+70+930) %

• An efficacious classification procedure should result in a high value of classification purity(above 95 %).


Timeline

Timeline

Item Execution timeTheoretical Development Sep 2014 - Feb 2015

Experiments with simulated data Mar 2015 - Apr 2015Dissertation writing May 2015 - Jul 2015

Review by the committee and defense Aug 2015


Computerized Classiﬁcation of Surface Spikes in Three ... · 8/26/2014 · Motivation • Inﬂuenza is a rapidly changing virus that appears seasonally in the human population.

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