A short overview of electron tomography Ozan Oktem - - PowerPoint PPT Presentation

A short overview of electron tomography

Ozan ¨ Oktem

• zan.oktem@sidec.com

Sidec Technologies

January 17, 2006 IMA Workshop: New Mathematics and Algorithms for 3-D Image Analysis, January 9-12, 2006

Outline

1

The main application

2

The experimental setup

3

The forward model

4

Reconstruction methods

5

Open problems

6

Case studies

The structure determination problem

The structure determination problem

Recover the 3D structure of an individual molecule (e.g. a protein or a macromolecular assembly) at highest possible resolution in situ (in their cellular environment) or in vitro (in aqueous environment). X-ray crystallography and NMR are established methods.

1

Major advantage: Atomic resolution.

2

Major disadvantage: Inability to study individual molecules in their natural environment (in situ and in vitro).

Electron tomography (ET) is an emerging technology.

1

Major advantage: Enables study of individual molecules in their natural environment (in situ and in vitro).

2

Major disadvantage: Low resolution.

The structure determination problem

The structure determination problem

Recover the 3D structure of an individual molecule (e.g. a protein or a macromolecular assembly) at highest possible resolution in situ (in their cellular environment) or in vitro (in aqueous environment). X-ray crystallography and NMR are established methods.

1

Major advantage: Atomic resolution.

2

Major disadvantage: Inability to study individual molecules in their natural environment (in situ and in vitro).

Electron tomography (ET) is an emerging technology.

1

Major advantage: Enables study of individual molecules in their natural environment (in situ and in vitro).

2

Major disadvantage: Low resolution.

The structure determination problem

The structure determination problem

Recover the 3D structure of an individual molecule (e.g. a protein or a macromolecular assembly) at highest possible resolution in situ (in their cellular environment) or in vitro (in aqueous environment). X-ray crystallography and NMR are established methods.

1

Major advantage: Atomic resolution.

2

Major disadvantage: Inability to study individual molecules in their natural environment (in situ and in vitro).

Electron tomography (ET) is an emerging technology.

1

Major advantage: Enables study of individual molecules in their natural environment (in situ and in vitro).

2

Major disadvantage: Low resolution.

The transmission electron microscope (TEM)

Field Emisson Gun (FEG) First condenser lens Second condenser lens Condenser aperture Sample Objective lens Objective and selected area apertures First intermediate lens Second intermediate lens Projector lens Detector Electron beam

Sample preparation and data collection scheme

Sample preparation: Purpose is to enable thin (about 100 nm) fixed specimens while preserving the structure. In vitro samples: Flash-frozen in a millisecond. In situ samples: Chemically fixed, cryosectioned and immunolabeled (in order to find the molecule). Single axis tilting: The most common data collection scheme.

◮ Rotation around the tilt axis. The rotation angle is called the tilt angle

and the angular range is usually from [−60◦, 60◦].

◮ Can reduce 3D reconstruction problem to a stack of 2D reconstruction

problems.

Sample preparation and data collection scheme

Sample preparation: Purpose is to enable thin (about 100 nm) fixed specimens while preserving the structure. In vitro samples: Flash-frozen in a millisecond. In situ samples: Chemically fixed, cryosectioned and immunolabeled (in order to find the molecule). Single axis tilting: The most common data collection scheme.

◮ Rotation around the tilt axis. The rotation angle is called the tilt angle

and the angular range is usually from [−60◦, 60◦].

◮ Can reduce 3D reconstruction problem to a stack of 2D reconstruction

problems.

Properties of TEM images of biological specimens

Contrast depends on the atomic number. Biological specimens are composed of atoms of very low atomic number (carbon, hydrogen, nitrogen, phosphorus and sulphur). Main contrast mechanism is phase contrast which results from the quantum superposition of the electron wave and the interference caused by the optics rather than amplitude contrast. No quantum interference between components of the single electron wave function that originate from interaction with specimen in different quantum states, so a measured intensity is a superimposition

• f intensities.

Electron wavelength for 300 keV is about 0.0197 ˚ A, sample thickness about 1000 ˚ A, and atomic resolution is about 1-2 ˚

• A. Secondary

structures (near atomic resolution) are visible when resolution is about 5-8 ˚

• A. In any case, electron wavelength is not a limiting factor

for the resolution.

Properties of TEM images of biological specimens

Contrast depends on the atomic number. Biological specimens are composed of atoms of very low atomic number (carbon, hydrogen, nitrogen, phosphorus and sulphur). Main contrast mechanism is phase contrast which results from the quantum superposition of the electron wave and the interference caused by the optics rather than amplitude contrast. No quantum interference between components of the single electron wave function that originate from interaction with specimen in different quantum states, so a measured intensity is a superimposition

• f intensities.

Electron wavelength for 300 keV is about 0.0197 ˚ A, sample thickness about 1000 ˚ A, and atomic resolution is about 1-2 ˚

• A. Secondary

structures (near atomic resolution) are visible when resolution is about 5-8 ˚

• A. In any case, electron wavelength is not a limiting factor

for the resolution.

Properties of TEM images of biological specimens

Contrast depends on the atomic number. Biological specimens are composed of atoms of very low atomic number (carbon, hydrogen, nitrogen, phosphorus and sulphur). Main contrast mechanism is phase contrast which results from the quantum superposition of the electron wave and the interference caused by the optics rather than amplitude contrast. No quantum interference between components of the single electron wave function that originate from interaction with specimen in different quantum states, so a measured intensity is a superimposition

• f intensities.

Electron wavelength for 300 keV is about 0.0197 ˚ A, sample thickness about 1000 ˚ A, and atomic resolution is about 1-2 ˚

• A. Secondary

structures (near atomic resolution) are visible when resolution is about 5-8 ˚

• A. In any case, electron wavelength is not a limiting factor

for the resolution.

Properties of TEM images of biological specimens

Contrast depends on the atomic number. Biological specimens are composed of atoms of very low atomic number (carbon, hydrogen, nitrogen, phosphorus and sulphur). Main contrast mechanism is phase contrast which results from the quantum superposition of the electron wave and the interference caused by the optics rather than amplitude contrast. No quantum interference between components of the single electron wave function that originate from interaction with specimen in different quantum states, so a measured intensity is a superimposition

• f intensities.

Electron wavelength for 300 keV is about 0.0197 ˚ A, sample thickness about 1000 ˚ A, and atomic resolution is about 1-2 ˚

• A. Secondary

structures (near atomic resolution) are visible when resolution is about 5-8 ˚

• A. In any case, electron wavelength is not a limiting factor

for the resolution.

The forward model

Overview

The forward model can be divided into the following parts.

1 Electron-specimen interaction. 2 Optics of the TEM. 3 The intensity and detector response.

Electron-specimen interaction

Basic assumptions

One electron in the specimen at a time. Average distance between two successive electrons much greater than the specimen thickness. Interaction is governed by the scalar Schr¨

• dinger wave equation.

Coloumb potential models elastic interaction, absorption potential models decrease in the flux of the non-scattered and elastically scattered electrons. Incident wave Ψin is time harmonic and of the form Ψin(x, t) := e−ik2

2m tuin(x).

Electron-specimen interaction

Basic assumptions

One electron in the specimen at a time. Average distance between two successive electrons much greater than the specimen thickness. Interaction is governed by the scalar Schr¨

• dinger wave equation.

Coloumb potential models elastic interaction, absorption potential models decrease in the flux of the non-scattered and elastically scattered electrons. Incident wave Ψin is time harmonic and of the form Ψin(x, t) := e−ik2

2m tuin(x).

Electron-specimen interaction

Basic assumptions

One electron in the specimen at a time. Average distance between two successive electrons much greater than the specimen thickness. Interaction is governed by the scalar Schr¨

• dinger wave equation.

Coloumb potential models elastic interaction, absorption potential models decrease in the flux of the non-scattered and elastically scattered electrons. Incident wave Ψin is time harmonic and of the form Ψin(x, t) := e−ik2

2m tuin(x).

Electron-specimen interaction

Basic assumptions

One electron in the specimen at a time. Average distance between two successive electrons much greater than the specimen thickness. Interaction is governed by the scalar Schr¨

• dinger wave equation.

Coloumb potential models elastic interaction, absorption potential models decrease in the flux of the non-scattered and elastically scattered electrons. Incident wave Ψin is time harmonic and of the form Ψin(x, t) := e−ik2

2m tuin(x).

Electron-specimen interaction

Basic assumptions

One electron in the specimen at a time. Average distance between two successive electrons much greater than the specimen thickness. Interaction is governed by the scalar Schr¨

• dinger wave equation.

Coloumb potential models elastic interaction, absorption potential models decrease in the flux of the non-scattered and elastically scattered electrons. Incident wave Ψin is time harmonic and of the form Ψin(x, t) := e−ik2

2m tuin(x).

Electron wave number

Electron-specimen interaction

Basic assumptions

One electron in the specimen at a time. Average distance between two successive electrons much greater than the specimen thickness. Interaction is governed by the scalar Schr¨

• dinger wave equation.

Coloumb potential models elastic interaction, absorption potential models decrease in the flux of the non-scattered and elastically scattered electrons. Incident wave Ψin is time harmonic and of the form Ψin(x, t) := e−ik2

2m tuin(x).

Electron mass at rest

Electron-specimen interaction

Basic assumptions

One electron in the specimen at a time. Average distance between two successive electrons much greater than the specimen thickness. Interaction is governed by the scalar Schr¨

• dinger wave equation.

Coloumb potential models elastic interaction, absorption potential models decrease in the flux of the non-scattered and elastically scattered electrons. Incident wave Ψin is time harmonic and of the form Ψin(x, t) := e−ik2

2m tuin(x).

Amplitude term that fulfils the Helmholtz equation

Electron-specimen interaction

Basic assumptions

One electron in the specimen at a time. Average distance between two successive electrons much greater than the specimen thickness. Interaction is governed by the scalar Schr¨

• dinger wave equation.

Coloumb potential models elastic interaction, absorption potential models decrease in the flux of the non-scattered and elastically scattered electrons. Incident wave Ψin is time harmonic and of the form Ψin(x, t) := e−ik2

2m tuin(x).

Perfect coherent illumination, i.e. uin(x) := eikx·ω where ω is the beam direction.

Electron-specimen interaction

Basic assumptions

One electron in the specimen at a time. Average distance between two successive electrons much greater than the specimen thickness. Interaction is governed by the scalar Schr¨

• dinger wave equation.

Coloumb potential models elastic interaction, absorption potential models decrease in the flux of the non-scattered and elastically scattered electrons. Incident wave Ψin is time harmonic and of the form Ψin(x, t) := e−ik2

2m tuin(x).

Perfect coherent illumination, i.e. uin(x) := eikx·ω where ω is the beam direction. Specimen is weakly scattering so first order Born approximation is valid and we can linearise the intensity.

Optics of the TEM

The optical setup and assumptions

System is aligned w.r.t. optical axis ω

Optics of the TEM

The optical setup and assumptions

ω⊥ Ideal thin lens. Focal length f and spherical aberration Cs equals that of real objective lens, magnification equals that of the entire microscope.

Optics of the TEM

The optical setup and assumptions

ω⊥ Lens ω⊥ − qω Object plane

Optics of the TEM

The optical setup and assumptions

ω⊥ ω⊥ − qω Object plane ω⊥ + f ω Focal plane Aperture (rotation invariant)

Optics of the TEM

The optical setup and assumptions

ω⊥ Lens ω⊥ − qω Object plane ω⊥ + f ω Focal plane ω⊥ + rω Image plane

Optics of the TEM

The optical setup and assumptions

ω⊥ Lens ω⊥ − qω Object plane ω⊥ + f ω Focal plane ω⊥ + rω Image plane System is in focus, i.e. 1

r + 1 q = 1 f

The intensity

The standard model for image formation

Standard model for image formation in ET is based on taking the first term in the asymptotic expansion of the propagation operator as k → ∞. F(x) := −2m 2

• V (x) + iΛ(x)
• Coloumb potential

Absorption potential Ik(F)(ω, z) = 1 M2

• 1 − 2(2π)−2

The intensity

The standard model for image formation

Standard model for image formation in ET is based on taking the first term in the asymptotic expansion of the propagation operator as k → ∞. The intensity The magnification The unscattered wave Ik(F)(ω, z) = 1 M2

• 1 − 2(2π)−2

The intensity

The standard model for image formation

Standard model for image formation in ET is based on taking the first term in the asymptotic expansion of the propagation operator as k → ∞. Ik(F)(ω, z) = 1 M2

• 1 − 2(2π)−2
• PSFre

k (ω, ·) ⊛ ω⊥ P(F re)(ω, −·)

z M

• +

where P denotes the X-ray transform

The intensity

The standard model for image formation

Standard model for image formation in ET is based on taking the first term in the asymptotic expansion of the propagation operator as k → ∞. Ik(F)(ω, z) = 1 M2

• 1 − 2(2π)−2
• PSFre

k (ω, ·) ⊛ ω⊥ P(F re)(ω, −·)

z M

• +

where P denotes the X-ray transform and PSFre

k (ω, y) := Fω⊥

• χ

f k · +f ω

• sin
• γk
• | · |2

(y) The characteristic function for the aperture

The intensity

The standard model for image formation

Standard model for image formation in ET is based on taking the first term in the asymptotic expansion of the propagation operator as k → ∞. Ik(F)(ω, z) = 1 M2

• 1 − 2(2π)−2
• PSFre

k (ω, ·) ⊛ ω⊥ P(F re)(ω, −·)

z M

• +

where P denotes the X-ray transform and PSFre

k (ω, y) := Fω⊥

• χ

f k · +f ω

• sin
• γk
• | · |2

(y) γk(t) := −t 1

4k

• Csk−2t − 2△z

The intensity

The standard model for image formation

Standard model for image formation in ET is based on taking the first term in the asymptotic expansion of the propagation operator as k → ∞. Ik(F)(ω, z) = 1 M2

• 1 − 2(2π)−2
• PSFre

k (ω, ·) ⊛ ω⊥ P(F re)(ω, −·)

z M

• +

where P denotes the X-ray transform and PSFre

k (ω, y) := Fω⊥

• χ

f k · +f ω

• sin
• γk
• | · |2

(y) γk(t) := −t 1

4k

• Csk−2t − 2△z
• Spherical aberration

Defocus

The intensity

The standard model for image formation

Standard model for image formation in ET is based on taking the first term in the asymptotic expansion of the propagation operator as k → ∞. Ik(F)(ω, z) = 1 M2

• 1 − 2(2π)−2
• PSFre

k (ω, ·) ⊛ ω⊥ P(F re)(ω, −·)

z M

• +
• PSFim

k (ω, ·) ⊛ ω⊥ P(F im)(ω, −·)

z M

• k−1
• where P denotes the X-ray transform and

PSFim

k (ω, y) := Fω⊥

• χ

f k · +f ω

• cos
• γk
• | · |2

(y)

The forward operator and the measured data

A single measured data point is a sample from the random variable data[F](ω, z) :=

• F−1

ω⊥[ MTF](·) ⊛ ω⊥ c[F](ω, ·)

• (z) + E(ω, z)

The forward operator and the measured data

A single measured data point is a sample from the random variable data[F](ω, z) :=

• F−1

ω⊥[ MTF](·) ⊛ ω⊥ c[F](ω, ·)

• (z) + E(ω, z)

c[F](ω, z) is a random variable representing the stochastic- ity in the counting process. It has distribution c[F](ω, z) ∼ Poisson

• Dose(ω) Ik(F)(ω, z)
• with Dose(ω) representing the incoming dose of electrons

per pixel for the direction ω.

The forward operator and the measured data

A single measured data point is a sample from the random variable data[F](ω, z) :=

• F−1

ω⊥[ MTF](·) ⊛ ω⊥ c[F](ω, ·)

• (z) + E(ω, z)

E(ω, z) is a random variable that represents the additive measurement error from the detector. It is usually Gaus- sian or uniformly distributed and precise distribution is esti- mated from specific tests made on the detector.

The forward operator and the measured data

A single measured data point is a sample from the random variable data[F](ω, z) :=

• F−1

ω⊥[ MTF](·) ⊛ ω⊥ c[F](ω, ·)

• (z) + E(ω, z)

MTF is the modulation transfer function which essentially describes how the CCD camera attenuates different spatial frequencies present in the input signal.

The forward operator and the measured data

A single measured data point is a sample from the random variable data[F](ω, z) :=

• F−1

ω⊥[ MTF](·) ⊛ ω⊥ c[F](ω, ·)

• (z) + E(ω, z)

The measured data from an ET experiment is modelled as a sample of the mn2-dimensional random variable data[F] defined as data[F] :=

• data[F](ω1, z1,1), . . . , data[F](ωm, zm,n2)
• .

where {ω1, . . . , ωm} ⊂ S2 are the different directions and for each direction ωj we have n2 pixels with midpoints {zj,1, . . . , zj,n2} ⊂ ω⊥

j .

The forward operator and the measured data

A single measured data point is a sample from the random variable data[F](ω, z) :=

• F−1

ω⊥[ MTF](·) ⊛ ω⊥ c[F](ω, ·)

• (z) + E(ω, z)

The measured data from an ET experiment is modelled as a sample of the mn2-dimensional random variable data[F] defined as data[F] :=

• data[F](ω1, z1,1), . . . , data[F](ωm, zm,n2)
• .

where {ω1, . . . , ωm} ⊂ S2 are the different directions and for each direction ωj we have n2 pixels with midpoints {zj,1, . . . , zj,n2} ⊂ ω⊥

j .

The forward operator T (F)(ω, z) is given as the expected value of data[F](ω, z) and the forward data operator is defined as T(F) :=

• T (F)(ω1, z1,1), . . . , T (F)(ωm, zm,n2)
• .

Reconstruction methods

The inverse and forward problems

The inverse problem is to recover F from a single sample of data[F]. In some cases the measured data also depends on a number of parameters, represented by a vector α, that needs to be recovered alongside F. In such case we have the parametrised inverse problem where one seeks to recover F and α from a single sample of data[F, α]. The forward problem is to generate a single sample of data[F] from F.

Reconstruction methods

The inverse and forward problems

The inverse problem is to recover F from a single sample of data[F]. In some cases the measured data also depends on a number of parameters, represented by a vector α, that needs to be recovered alongside F. In such case we have the parametrised inverse problem where one seeks to recover F and α from a single sample of data[F, α]. The forward problem is to generate a single sample of data[F] from F.

Reconstruction methods

The inverse and forward problems

The inverse problem is to recover F from a single sample of data[F]. In some cases the measured data also depends on a number of parameters, represented by a vector α, that needs to be recovered alongside F. In such case we have the parametrised inverse problem where one seeks to recover F and α from a single sample of data[F, α]. The forward problem is to generate a single sample of data[F] from F.

Reconstruction methods

Basic assumptions and difficulties

All current reconstruction methods assume that the image formation is given by the standard model. Main difficulties stem from Incomplete data problems:

◮ The dose problem (limitations in the total number of images) ◮ Limited angle problem (limitations in the range of the tilt angle) ◮ Local tomography (only a subregion is exposed to electrons)

Alignment due to specimen drift. Parametrised inverse problem: Some parameters needs to be reconstructed along with the scattering potential:

◮ Instrument parameters, e.g. incoming dose, defocus and spherical

aberration (specimen independent)

◮ The amplitude contrast ratio (specimen dependent).

Reconstruction methods

Basic assumptions and difficulties

All current reconstruction methods assume that the image formation is given by the standard model. Main difficulties stem from Incomplete data problems:

◮ The dose problem (limitations in the total number of images) ◮ Limited angle problem (limitations in the range of the tilt angle) ◮ Local tomography (only a subregion is exposed to electrons)

Alignment due to specimen drift. Parametrised inverse problem: Some parameters needs to be reconstructed along with the scattering potential:

◮ Instrument parameters, e.g. incoming dose, defocus and spherical

aberration (specimen independent)

◮ The amplitude contrast ratio (specimen dependent).

Reconstruction methods

Basic assumptions and difficulties

All current reconstruction methods assume that the image formation is given by the standard model. Main difficulties stem from Incomplete data problems:

◮ The dose problem (limitations in the total number of images) ◮ Limited angle problem (limitations in the range of the tilt angle) ◮ Local tomography (only a subregion is exposed to electrons)

Alignment due to specimen drift. Parametrised inverse problem: Some parameters needs to be reconstructed along with the scattering potential:

◮ Instrument parameters, e.g. incoming dose, defocus and spherical

aberration (specimen independent)

◮ The amplitude contrast ratio (specimen dependent).

Reconstruction methods

Basic assumptions and difficulties

All current reconstruction methods assume that the image formation is given by the standard model. Main difficulties stem from Incomplete data problems:

◮ The dose problem (limitations in the total number of images) ◮ Limited angle problem (limitations in the range of the tilt angle) ◮ Local tomography (only a subregion is exposed to electrons)

Alignment due to specimen drift. Parametrised inverse problem: Some parameters needs to be reconstructed along with the scattering potential:

◮ Instrument parameters, e.g. incoming dose, defocus and spherical

aberration (specimen independent)

◮ The amplitude contrast ratio (specimen dependent).

Reconstruction methods

Algorithmic difficulties. The dose problem

This is the single most important incomplete data problem in ET. It limits the total number of images that can be taken and arises due to specimen damage during electron exposure. Total dose needs to be less than 20–70 e−/˚ A2 (depending on the type of sample). Usually one has about a total of 500–1250 e−/pixel (at 25000× magnification) distributed over 60 or 120 tilts, so each image is very noisy.

Reconstruction methods

Algorithmic difficulties. The dose problem

This is the single most important incomplete data problem in ET. It limits the total number of images that can be taken and arises due to specimen damage during electron exposure. Total dose needs to be less than 20–70 e−/˚ A2 (depending on the type of sample). Usually one has about a total of 500–1250 e−/pixel (at 25000× magnification) distributed over 60 or 120 tilts, so each image is very noisy.

Reconstruction methods

Algorithmic difficulties. The dose problem

This is the single most important incomplete data problem in ET. It limits the total number of images that can be taken and arises due to specimen damage during electron exposure. Total dose needs to be less than 20–70 e−/˚ A2 (depending on the type of sample). Usually one has about a total of 500–1250 e−/pixel (at 25000× magnification) distributed over 60 or 120 tilts, so each image is very noisy.

Reconstruction methods

Algorithmic difficulties. The dose problem. Example of an image

Figure: TEM image of an in vitro sample containing GroEl chaperone (a protein whose function is to assist other proteins in achieving proper folding). Length of sides are 980.1 nm. Left image is a low dose image used for reconstruction, right image is a high dose image of same area.

Reconstruction methods

Algorithmic difficulties. The limited angle problem and local tomography

Limited angle problem leads to unstable inversion of the X-ray transform. Local tomography leads to non-uniqueness, so one can only recover singularities (made precise by the concept of wavefront set) from local X-ray transform data. Theory of local tomography of the X-ray transform was extended by Quinto 1993 (also Katsevich 1997) to more general local limited data situations by introducing a precise concept of stability for the wavefront set.

“Non-mathematical” version of Quinto’s main result

Some singularities, that can be characterised, can be stably recovered and

• ne can obtain stability estimates of order 1/2 in Sobolev norms.

Reconstruction methods

Algorithmic difficulties. The limited angle problem and local tomography

Limited angle problem leads to unstable inversion of the X-ray transform. Local tomography leads to non-uniqueness, so one can only recover singularities (made precise by the concept of wavefront set) from local X-ray transform data. Theory of local tomography of the X-ray transform was extended by Quinto 1993 (also Katsevich 1997) to more general local limited data situations by introducing a precise concept of stability for the wavefront set.

“Non-mathematical” version of Quinto’s main result

Some singularities, that can be characterised, can be stably recovered and

• ne can obtain stability estimates of order 1/2 in Sobolev norms.

Reconstruction methods

Algorithmic difficulties. The limited angle problem and local tomography

Limited angle problem leads to unstable inversion of the X-ray transform. Local tomography leads to non-uniqueness, so one can only recover singularities (made precise by the concept of wavefront set) from local X-ray transform data. Theory of local tomography of the X-ray transform was extended by Quinto 1993 (also Katsevich 1997) to more general local limited data situations by introducing a precise concept of stability for the wavefront set.

“Non-mathematical” version of Quinto’s main result

Some singularities, that can be characterised, can be stably recovered and

• ne can obtain stability estimates of order 1/2 in Sobolev norms.

Reconstruction methods

Algorithmic difficulties. The limited angle problem and local tomography

Limited angle problem leads to unstable inversion of the X-ray transform. Local tomography leads to non-uniqueness, so one can only recover singularities (made precise by the concept of wavefront set) from local X-ray transform data. Theory of local tomography of the X-ray transform was extended by Quinto 1993 (also Katsevich 1997) to more general local limited data situations by introducing a precise concept of stability for the wavefront set.

“Non-mathematical” version of Quinto’s main result

Some singularities, that can be characterised, can be stably recovered and

• ne can obtain stability estimates of order 1/2 in Sobolev norms.

Reconstruction methods

Algorithmic difficulties. Visible singularities in single axis tilting

Consider the characteristic function of a ball in R3 so the set of singularities is the sphere S2.

Reconstruction methods

Algorithmic difficulties. Visible singularities in single axis tilting

Consider the characteristic function of a ball in R3 so the set of singularities is the sphere S2. Tilt axis (x-axis) Beam direction (z-axis)

Reconstruction methods

Algorithmic difficulties. Visible singularities in single axis tilting

Consider the characteristic function of a ball in R3 so the set of singularities is the sphere S2. Tilt axis (x-axis) Beam direction (z-axis)

Reconstruction methods

Approaches for solving the problems caused by incomplete data

The algorithmic development in ET has to a great extent been characterised by approaches to overcome the dose problem. Categorise approaches based on the assumptions that are imposed on the specimen:

◮ Helical reconstruction methods ◮ Electron crystallography ◮ Single particle methods ◮ Electron tomography

Reconstruction methods

Approaches for solving the problems caused by incomplete data

The algorithmic development in ET has to a great extent been characterised by approaches to overcome the dose problem. Categorise approaches based on the assumptions that are imposed on the specimen:

◮ Helical reconstruction methods ◮ Electron crystallography ◮ Single particle methods ◮ Electron tomography

Reconstruction methods

Electron Tomography methods. Overview

Assumption about specimen: None. Main advantage: Can study any specimen in situ and in vitro. Main disadvantage: Severely ill-posed reconstruction problem limits the resolution. Current status: Used very little.

Reconstruction methods

Electron Tomography methods. Current status

Despite this ill-posedness there are no systematic studies of regularisation methods applied to such reconstruction problems.

Reconstruction methods

Electron Tomography methods. Current status

Early approaches are Fourier based inversion. [de Rosier and Klug (1968), Vainshtein et.al. (1968), Hoppe et.al. (1968), Crowther et.al. (1970)]. (Low-pass) Filtered Filtered backprojection (Low-pass FBP). [Crowther et.al. (1970)]. Kaczmarz (ART) [Bender et.al. (1970), Gordon et.al. (1970), Herman and Rowland (1971)] Landweber iteration (SIRT, Simultaneous iterative reconstruction technique). [Gilber (1972)] Iterative Least-Squares Technique (ILST). [Goitein (1972)] Weighted backprojection (R-weighted backprojection algorithm) [Radermacher (1988)].

Reconstruction methods

Electron Tomography methods. Current status

More recent approaches are Projection onto Convex Sets (POCS). [Sezan and Stark (1982), Carazo and Carrascosa (1987)] Relative entropy regularisation with dynamically updated prior (COMET). [Skoglund et.al. (1996)] Strongly over-relaxed Kaczmarz (ART) with series representation using Kaiser-Bessel window functions (blobs). [Marabini et.al. (1997)] Limited angle Λ-CT. [Quinto (2005)] Adaptive resolution Tikhonov regularisation. [Rullg˚ ard (2005)]

Reconstruction methods

Tests of different reconstruction methods on real ET data

Low-pass FBP COMET Reconstruction of a Vesicle (part of a cell membrane consisting of lipids that folds as a spherical shell). Voxel size is 5.24 ˚ A, entire volume is 2563 and extracted region is a 124 × 124 × 105.

Reconstruction methods

Tests of different reconstruction methods on real ET data

Low-pass FBP COMET In vitro sample of monoclonal IgG antibodies (molecular weight about 150 kDa). Left part of image shows the entire sample, right part shows a single extracted IgG antibody. Total dose is 18.2 e−/˚ A2, voxel size is 5.24 ˚ A, entire volume is 2563 and extracted region is a 503 cube.

Reconstruction methods

Tests of different reconstruction methods on real ET data

Low-pass FBP Limited angle Λ-CT COMET Same isolated monoclonal IgG antibody extracted from larger reconstruc-

• tion. Limited angle Λ-CT reconstruction is based on local data.

Reconstruction methods

Tests of different reconstruction methods on real ET data

Low-pass FBP Limited angle Λ-CT COMET Same reconstructions as in previous slide. Threshold defining segmenta- tion is set much higher to remove more of the background.

Open problems

No proper theory for regularisation of parametrised inverse problems. Test different regularisation methods on the inverse problem in ET. Propose data discrepancy functionals and methods for choosing the regularisation parameter that reflects the probabilistic assumptions made on the ET data. Propose suitable regularising functionals that reflect prior assumptions about the sample. Construct an accurate quantum mechanical TEM simulator for amorphous specimens. Adopt a more accurate forward model in the reconstruction method. Full Helmholtz solver necessary? Extend Quinto’s microlocal analysis of the limited data problems for the local X-ray transform to the diffraction tomography case. Define a measure of resolution that is applicable to limited angle local X-ray transform reconstructions. Ridglets?

Open problems

No proper theory for regularisation of parametrised inverse problems. Test different regularisation methods on the inverse problem in ET. Propose data discrepancy functionals and methods for choosing the regularisation parameter that reflects the probabilistic assumptions made on the ET data. Propose suitable regularising functionals that reflect prior assumptions about the sample. Construct an accurate quantum mechanical TEM simulator for amorphous specimens. Adopt a more accurate forward model in the reconstruction method. Full Helmholtz solver necessary? Extend Quinto’s microlocal analysis of the limited data problems for the local X-ray transform to the diffraction tomography case. Define a measure of resolution that is applicable to limited angle local X-ray transform reconstructions. Ridglets?

Open problems

No proper theory for regularisation of parametrised inverse problems. Test different regularisation methods on the inverse problem in ET. Propose data discrepancy functionals and methods for choosing the regularisation parameter that reflects the probabilistic assumptions made on the ET data. Propose suitable regularising functionals that reflect prior assumptions about the sample. Construct an accurate quantum mechanical TEM simulator for amorphous specimens. Adopt a more accurate forward model in the reconstruction method. Full Helmholtz solver necessary? Extend Quinto’s microlocal analysis of the limited data problems for the local X-ray transform to the diffraction tomography case. Define a measure of resolution that is applicable to limited angle local X-ray transform reconstructions. Ridglets?

Open problems

No proper theory for regularisation of parametrised inverse problems. Test different regularisation methods on the inverse problem in ET. Propose data discrepancy functionals and methods for choosing the regularisation parameter that reflects the probabilistic assumptions made on the ET data. Propose suitable regularising functionals that reflect prior assumptions about the sample. Construct an accurate quantum mechanical TEM simulator for amorphous specimens. Adopt a more accurate forward model in the reconstruction method. Full Helmholtz solver necessary? Extend Quinto’s microlocal analysis of the limited data problems for the local X-ray transform to the diffraction tomography case. Define a measure of resolution that is applicable to limited angle local X-ray transform reconstructions. Ridglets?

Open problems

No proper theory for regularisation of parametrised inverse problems. Test different regularisation methods on the inverse problem in ET. Propose data discrepancy functionals and methods for choosing the regularisation parameter that reflects the probabilistic assumptions made on the ET data. Propose suitable regularising functionals that reflect prior assumptions about the sample. Construct an accurate quantum mechanical TEM simulator for amorphous specimens. Adopt a more accurate forward model in the reconstruction method. Full Helmholtz solver necessary? Extend Quinto’s microlocal analysis of the limited data problems for the local X-ray transform to the diffraction tomography case. Define a measure of resolution that is applicable to limited angle local X-ray transform reconstructions. Ridglets?

Open problems

No proper theory for regularisation of parametrised inverse problems. Test different regularisation methods on the inverse problem in ET. Propose data discrepancy functionals and methods for choosing the regularisation parameter that reflects the probabilistic assumptions made on the ET data. Propose suitable regularising functionals that reflect prior assumptions about the sample. Construct an accurate quantum mechanical TEM simulator for amorphous specimens. Adopt a more accurate forward model in the reconstruction method. Full Helmholtz solver necessary? Extend Quinto’s microlocal analysis of the limited data problems for the local X-ray transform to the diffraction tomography case. Define a measure of resolution that is applicable to limited angle local X-ray transform reconstructions. Ridglets?

Open problems

No proper theory for regularisation of parametrised inverse problems. Test different regularisation methods on the inverse problem in ET. Propose data discrepancy functionals and methods for choosing the regularisation parameter that reflects the probabilistic assumptions made on the ET data. Propose suitable regularising functionals that reflect prior assumptions about the sample. Construct an accurate quantum mechanical TEM simulator for amorphous specimens. Adopt a more accurate forward model in the reconstruction method. Full Helmholtz solver necessary? Extend Quinto’s microlocal analysis of the limited data problems for the local X-ray transform to the diffraction tomography case. Define a measure of resolution that is applicable to limited angle local X-ray transform reconstructions. Ridglets?

Open problems

No proper theory for regularisation of parametrised inverse problems. Test different regularisation methods on the inverse problem in ET. Propose data discrepancy functionals and methods for choosing the regularisation parameter that reflects the probabilistic assumptions made on the ET data. Propose suitable regularising functionals that reflect prior assumptions about the sample. Construct an accurate quantum mechanical TEM simulator for amorphous specimens. Adopt a more accurate forward model in the reconstruction method. Full Helmholtz solver necessary? Extend Quinto’s microlocal analysis of the limited data problems for the local X-ray transform to the diffraction tomography case. Define a measure of resolution that is applicable to limited angle local X-ray transform reconstructions. Ridglets?

Case 1: In situ sub-unit assembly of ion-channel complex

Reconstructions

Figure: COMET reconstruction of ion-channel dual dimers (in white, about 15–17 nm in height and 13–14 nm in extracellular width), presumably in the process of forming a tetrameric complex (in rat dorsal root ganglia cells).

Monomeric units (white) constituting the dimers are outlined in red

Case 1: In situ sub-unit assembly of ion-channel complex

Reconstructions

Figure: COMET reconstruction of ion-channel dual dimers (in white, about 15–17 nm in height and 13–14 nm in extracellular width), presumably in the process of forming a tetrameric complex (in rat dorsal root ganglia cells).

Protein complex (dark grey), possibly chaperone proteins

Case 1: In situ sub-unit assembly of ion-channel complex

Reconstructions

Figure: COMET reconstruction of ion-channel dual dimers (in white, about 15–17 nm in height and 13–14 nm in extracellular width), presumably in the process of forming a tetrameric complex (in rat dorsal root ganglia cells).

Primary antibody marking subunit (light blue)

Case 1: In situ sub-unit assembly of ion-channel complex

Reconstructions

Figure: COMET reconstruction of the ion-channel (in white, about 15–17 nm in height and 9–10 nm in width) expressed on the plasma membrane of RIN cells.

Subunits of the tetramer (encircled in red)

Case 1: In situ sub-unit assembly of ion-channel complex

Reconstructions

Figure: COMET reconstruction of the ion-channel (in white, about 15–17 nm in height and 9–10 nm in width) expressed on the plasma membrane of RIN cells.

Central pore opening (grey)

Case 1: In situ sub-unit assembly of ion-channel complex

Reconstructions

Figure: COMET reconstruction of the ion-channel (in white, about 15–17 nm in height and 9–10 nm in width) expressed on the plasma membrane of RIN cells.

Dual dimers that are fused at dashed line