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Chapter 7 Cryoelectron Tomography (CET) 575

tives and hybrids of these three different acquisition models, especially

modiﬁ ed to serve speciﬁ c purposes and applications. The full-tracking

full-focusing scheme (described above) is the most common, due to the

fact that it can be literally used universally, e.g., with almost any kind

of sample and microscope and holder system. However, the time

required for the recording of a tilt series can be very long, up to several

hours, depending on the acquisition parameters.

The precalibration model was developed originally for side-entry

systems. It mainly includes the calibration of the tilting stage synchro-

nized with the specimen holder. The basic idea is that the image move-

ment is determined prior to data collection. The movement of the stage

is measured with a calibration sample (typically ﬁ ducial markers on a

carbon foil) both in the xy plane (image shifts) and the z direction

(defocus change) for the range of tilt angles needed to acquire a tilt

series. Typically, the calibration is started at a high tilt angle (e.g., either

−65° or +65°, depending on the characteristics of the stage) and contin-

ues to the outermost tilt on the opposite side. For every tilt angle an

image is acquired and the image feature is recentered automatically

(as described above) while the defocus change is measured (not cor-

rected) only with the beam-tilt method. In this way, the absolute image

shifts in xy and z (defocus) can be determined and stored in a calibra-

tion ﬁ le, which then provides the basis for the xy and z shift compensa-

tion during actual data acquisition. Moreover, it is also possible to

mathematically model (least-squares ﬁ t) the displacements during

tilting, based on the calibration curves now available, to predict the

behavior of the stage. However, there are some restraints on this course

of action: the calibration strongly depends on the initial setup of the

microscope (illumination settings, image shift settings, distance from

eucentricity, initial xy position of the stage), sample being used, and

thermal stability of the cryoholder utilized. Thermal drift of the speci-

men is not included in the calibration. For higher magniﬁ cations, espe-

cially, it has to be addressed separately, for example, with an additional

tracking step by cross-correlation. Additionally, a signiﬁ cant misalign-

ment of the optic axis relative to the tilt axis can necessitate large cor-

rections in both translation and focus, which in turn can lead to image

rotation and magniﬁ cation changes. Since the image rotation and a

possible change in magniﬁ cation are not covered by the pre-calibration

procedure, the acquisition is severely hampered. However, prealigning

the optic axis to the tilt axis by invoking an appropriate amount of

image shift can circumvent this possible source of error (Ziese et al.,

2002). Based on the measured calibration curve this “optimized posi-

tion” (where the optic axis coincides with the tilt axis) can be deter-

mined and adjusted. Overall, it can be said that even in case of thermal

instabilities, which will lead to rather small displacements, not every

change in tilt angle will require a tracking step and thus the acquisition

of a tilt series by precalibration is deﬁ nitely sped up enormously (ﬁ ve

times faster than the full-tracking method).

The prediction-based scheme assumes that the sample follows a

simple geometric rotation and that the optical system can be character-

ized in terms of an offset between the optic and mechanical axes of

576 J.M. Plitzko and W. Baumeister

the microscope (Zheng et al., 2004). In contrast to the precalibration

and full-tracking procedures where the acquisition begins at one end

of the tilt range, the prediction scheme starts at zero tilt. In this way, it

is possible to model and compensate for any errors in sample geometry

simultaneously. The image movement in the xy and z direction due to

a stage tilt can be dynamically predicted without precalibration.

Mathematically, one data point, characterized by the tilt angle and the

x translation, is sufﬁ cient to calculate the offset between the object and

tilt axis (z

). However, least-squares ﬁ tting from multiple data points

is typically used to minimize the consequences of errors introduced

when measuring the x translation. In this way, it is possible to estimate

and reﬁ ne “on-the-ﬂ y” the focus offset z

during the course of data

collection, with an accuracy of 100 nm.

The microscope optical system (beam/image shift and focus) is auto-

matically adjusted to compensate for the predicted and reﬁ ned image

movement prior to taking the projected image at each tilt angle. As a

consequence, it is not necessary either to record additional images for

tracking and focusing during the course of data collection or to spend

valuable setup time in a lengthy precalibration of stage motions.

However, thermal drift is not taken into account in the dynamic predic-

tion procedure, which will affect the precision of the prediction and

thus the quality of the recorded tilt series.

Nevertheless, the acquisition of tilt series of cryosamples even

with the given advances in automation is not yet performed on a high

throughput basis. This is mainly due to the fact that side-entry cryo-

holders are typically used, which are still subject to thermal instabili-

ties, and that the accuracy and reproducibility of mechanical tilting

still require compensation with elaborate automation procedures (see

above). However, recent advances in instrumentation, like the integra-

tion of the cooling system into the microscope column, the simpliﬁ ca-

tion of the sample transfer and mounting procedure, multispecimen

holders, and the development of piezo-controlled stages will widen the

prospects of high-throughput approaches in the near future.

3.5 Alignment, Reconstruction, and Visualization

3.5.1 Alignment

To obtain the 3D image from a set of acquired projections two initial

steps have to be carried out: ﬁ rst the individual micrographs need to

be aligned to a common coordinate system. The second step is the

actual 3D reconstruction of the tomographic volume. The compensa-

tion of the specimen movement during the data acquisition process

will keep the feature focused in most cases but the compensation for

the xy displacement is not sufﬁ cient for a subsequent reconstruction,

which makes a second, more precise alignment necessary. Primarily,

the alignment procedure has to determine the angle of the tilt axis and

the lateral shifts. Other changes, such as magniﬁ cation changes or

image rotation, due to large defocus changes during the acquisition,

have to be determined and compensated for as well, if present in the

recorded tilt series.

Chapter 7 Cryoelectron Tomography (CET) 577

Alignment of the individual micrographs in cellular tomography is

normally done by high-density markers, so-called ﬁ ducial markers.

Typically, spherical gold beads with a diameter of ∼10 nm are added to

the sample solution or directly onto the carbon foil prior to the vitriﬁ ca-

tion process. They can be easily recognized within a single 2D micro-

graph as “black dots,” due to their high Z number and thus their

pronounced elastic scattering. The coordinates of the markers on each

projection are determined manually or automatically. To minimize the

alignment error as a function of the lateral translations and the tilt axis

angle, an alignment model can be calculated based on least-squares

procedures (Lawrence, 1992). Their locations are then adjusted to a 3D

coordinate system. Apart from translations, this procedure for the

determination of a common origin often accounts for possible image

rotations or magniﬁ cation changes (Lawrence, 1992; Luther, 1988).

However, the larger the number of determined parameters, the more

gold beads need to be located to achieve a sufﬁ cient signiﬁ cance level

during the minimization of the residual. Therefore, another alignment

approach is by means of cross-correlating the entire image (Taylor et

al., 1997) or part of the images. Sequentially the images are compared

and compensated for image shifts throughout the series. This proce-

dure can be repeated iteratively to achieve a higher accuracy of the

alignment. However, this approach inevitably requires some approxi-

mation of the shape of the objects to interpolate data from different tilt

angles. Moreover, distinct features will dominate the outcome of the

cross-correlation, which might not be suitable for proper alignment.

Therefore it is often useful to apply image ﬁ lters (Fourier ﬁ lter, Sobel

ﬁ lter, etc.) prior to the cross-correlation, to enhance features that

are expected to correlate well. The major problem in using cross-

correlation-based alignment procedures in combination with low-dose

cryoimages of vitriﬁ ed samples is the very low SNR of these images

and the very weak contrast, since cross-correlation methods are typi-

cally very noise sensitive. Therefore they work well only for a limited

class of specimens, e.g., high-contrast or paracrystalline objects, or in

cases where ﬁ ducial markers are not present, or difﬁ cult to add, as seen

in tilt series from cryosections. The marker alignment method is deﬁ -

nitely more general because it can even cope with very low SNRs and

is therefore more common in cryo-ET applications.

3.5.2 Reconstruction

Although the ﬁ rst practical formulation for applied tomography was

achieved in the 1950s (Bracewell, 1956), Johan Radon ﬁ rst outlined the

mathematical principles behind the technique in 1917 (Radon, 1917;

English translation in Deans, 1983). The paper deﬁ nes the Radon trans-

form R as the mapping of a function f(x, y), describing a real space object

D, by the projection, or line integral, through f along all possible lines L:

Rf f x y ds

()

∫

where ds is the unit length of L. A discrete sampling of the Radon

transform is geometrically equivalent to the sampling of an experimen-

578 J.M. Plitzko and W. Baumeister

tal object by some form of transmitted signal: a projection. The conse-

quence of such equivalence is that the reconstruction of an object

f(x, y) from projections Rf can be achieved by implementation of the

inverse Radon transform. All conventional reconstruction algorithms

are approximations, with varying accuracy, to this inverse transform.

It is mathematically convenient to use polar coordinates (r, φ), related

to Cartesian axes for an explicit description of the Radon transform.

The Radon transform operation converts the coordinates of the experi-

mental data into “Radon space,” where l is a line perpendicular to the

projection direction and θ denotes the angle of the projection. Thus a

point in real space (r, φ) becomes a line in Radon space (l, θ) with the

relation l = r ⋅ cos (θ − φ). This relationship between real space and

Radon space allows a more explicit examination of the experimental

situation. A single projection of a one-dimensional (1D) object, for

example, a point in real space (a discrete sampling of the Radon trans-

form) is a 2D line at constant θ in Radon space. Thus a series of projec-

tions at different angles will provide an increased sampling of the

Radon space. Given a sufﬁ cient number of (n − 1)-dimensional projec-

tions (of an n-dimensional object) from different views, an inverse

Radon transform should reconstruct the n-dimensional object. However,

in ET the experimental sampling (l, θ) is discrete (limited amount of

projections in a restricted angular range), therefore the inversion will

be imperfect and the problem of an undistorted and complete recon-

struction, especially in ET, becomes evident: to achieve the best recon-

struction of the object it is necessary to obtain as many projections as

possible in an angular regime as large as possible.

In practice the reconstruction from projections is aided by the under-

standing of the relationship of a projection in real space and Fourier

space. The “projection-slice theorem” states that a 2D projection at a

given angle is a central section through the 3D Fourier transform of

this object (Figure 7–19). If a series of projections is acquired at different

tilt angles, each projection will correspond to part of an object’s Fourier

transform, thus sampling the object over the full range of frequencies

in a central section. The shape of most objects will be described only

partially by the frequencies in one section, but by taking multiple pro-

jections at different angles many sections will be sampled in Fourier

space. This will describe the Fourier transform of an object in many

directions, allowing a fuller description of an object in real space. In

principle a sufﬁ ciently large number of projections taken over all angles

will provide a complete description of the object. Therefore tomo-

graphic reconstruction is possible from an inverse Fourier transform

of the superposition of a set of Fourier transformed projections: an

approach known as direct Fourier reconstruction. This was the

approach formulated by Bracewell (1956) and was used for the ﬁ rst

tomographic reconstruction from electron micrographs (DeRosier and

Klug, 1968). It is also known as “Fourier synthesis” and used for atomic

structure determination by electron (Henderson et al., 1990) and X-ray

crystallography (Perutz et al., 1960).

The direct interpretation of the projection-slice theorem would

suggest a reconstruction algorithm based in Fourier space (Hoppe and

Chapter 7 Cryoelectron Tomography (CET) 579

Hegerl, 1980). Despite the fact that the ﬁ rst 3D reconstruction by

DeRosier and Klug (1968) was carried out in Fourier space, it is common

to use real-space-based reconstruction algorithms, because the practi-

cal implementation of Fourier-space reconstruction is not as simple as

an inverse transform. The projection data are always sampled at dis-

crete angles leaving regular gaps in Fourier space (Figure 7–20). But

the inverse transform intrinsically requires a continuous function and

therefore interpolation is required to ﬁ ll the gaps in Fourier space

(Crowther et al., 1970). However, the quality of a Fourier-space recon-

struction is greatly affected by the type of interpolation implemented,

Figure 7–19. ‘Projection-slice theorem’. a) An object is shown in real space (x, y) at the origin, and one

of its projection images is presented fromed by tilted parallel beams. b) The Fourier transform of the

projection is a section through the origin of the Fourier space (k

, k

) tilted by θ.

Figure 7–20. Illustration of the effect of sampling at discrete angles in a 2 dimensional example. a)

For 18 projections over the full angular range at 10° increment. b) For 36 projections at 5° increment

and c) for 72 projections at 2.5° increment which is almost identical to the original image.

580 J.M. Plitzko and W. Baumeister

as examined by Smith et al. (1973). Although elegant, Fourier recon-

struction methods have the disadvantage of being computationally

intensive and rather difﬁ cult to implement. Therefore they have been

superseded by faster real-space alternatives since interpolations are

easier to implement there.

By far the most widely utilized algorithm is weighted backprojection

(Radermacher, 1992). Algebraic reconstruction techniques (ART) are

also well established, despite the fact that they were subject to criticism

in their beginnings (Crowther and Klug, 1971; for a detailed review of

reconstruction algorithms we refer to Frank, 1996). The theory of back-

projection relies on a simple principle: a point in space may be uniquely

described by any three “rays” passing through that point. However, as

the object increases in complexity more “rays” are required to describe

it uniquely. A projection of an object is the inverse of such a “ray,” and

will describe some of the complexity of the object at hand. Therefore by

inversing the projection, smearing out the projection into an object space

at the angle of projection, a “ray” is generated that will uniquely describe

an object in the projection direction, a process known as backprojection.

With sufﬁ cient projections, from different angles, the superposition of

all the backprojected “rays” will reproduce the shape of the original

object—a reconstruction technique known as direct backprojection.

Direct backprojection is used for reconstruction in classical computer-

assisted tomography (CT; Herman, 1980), and it was the technique used

by Hart (1968) for the “polytropic montage.” It was also mentioned as

an alternative to Fourier methods by Crowther et al. (1970).

However, reconstructions made by direct backprojection are excep-

tionally blurred, showing distinct enhancement of low frequencies,

while ﬁ ne spatial details are reconstructed poorly. This problem is an

effect of the uneven sampling of spatial frequencies in the series of

original projections (Figure 7–21). In 2D each of the acquired projec-

tions is a line intersecting the center of the Fourier space. Assuming a

regular sampling of Fourier space in each projection, far more sam-

pling points are located in the center of Fourier space than in the

periphery. The outcome of this is an “undersampling” of the high

spatial frequencies and an “oversampling” of the low spatial frequen-

cies of the object, which will subsequently result in a “blurred” recon-

struction. Therefore, to remove the blurring in real space and to restore

the correct “frequency balance” in Fourier space, it is necessary to

apply weighting schemes. The weighted backprojection consists of two

steps: First the aligned projections are weighted in Fourier space by a

function that characterizes the different sampling density in Fourier

space. Assuming an inﬁ nitely small tilt increment the weighting has

to be proportional to the amount of the spatial frequency orthogonal

to the tilt axis (“analytical” weighting); more precise weighting (“exact”

weighting) has to consider the shape function of the object to approxi-

mate the sampling density in Fourier space, normally using a sine

function, which is retained only up to its ﬁ rst zero crossing (Harauz

and van Heel, 1986; Hoppe and Hegerl, 1980). The second step is the

backprojection of the weighted micrographs into the reconstruction

body, most frequently by trilinear interpolation.

Chapter 7 Cryoelectron Tomography (CET) 581

ART formulates the Radon transform as a system of algebraic equa-

tions. The inversion of this algebraic system is performed iteratively

until self-consistency is achieved. The solutions of both reconstruction

methods should be similar; however, ART determines the correct

weighting parameters automatically whereas weighted backprojection

requires a priori knowledge about the specimen to provide a correct

weighting parameter for exact weighting. Despite this fact weighted

backprojection is used in most software packages for 3D reconstruction

due to its computational speed. However, the development of parallel-

ized implementations as described in Fernandez et al. (2002) might

establish ART as an alternative.

Several methods have been introduced to increase the quality of

tomographic reconstructions, i.e., improving the SNR of the data and

ﬁ lling unsampled regions of the object’s Fourier space with consistent

data. All of these methods incorporate additional constraints to restrict

the solution of the reconstruction. Application of solvent ﬂ attening in

single particle analysis (van Heel, 2000), projection onto convex sets

(Carazo and Carrascosa, 1987), or the Gerchberg–Fienup algorithm

(Spence et al., 2003) in electron crystallography can improve the

Figure 7–21. Illustration of the sampling ‘density’ problem in Fourier space.

The large number of sampling points at low frequencies (darker area) are in

contrast to the periphery (high frequencies), very only few sampling points

are included in the single projections (brighter area). This sampling imbalance

will result in ‘blurred’ reconstructions, which can be overcome by weighting

schemes.

582 J.M. Plitzko and W. Baumeister

resulting reconstructions considerably. Despite this fact, in most cases

collection of sufﬁ cient data is preferred since conﬁ ning the basis set of

the reconstruction can cause artifacts. In cryo-ET there is typically no

a priori information available; furthermore the data are not fully con-

sistent because some features occurring at high tilt angles are not

present in low tilt angles, i.e., the “unit cell is only partially deﬁ ned”

(Hoppe and Hegerl, 1980). Therefore, reﬁ nement techniques are com-

monly not used in cryo-ET, despite unproven claims of resolution

improvement close to the subnanometer regime (Sandin et al., 2004).

To extend the interpretability of tomograms beyond the ﬁ rst zero of

the CTF it would be necessary to correct for the effects of the CTF as

is done in single-particle analysis. Corrections based on exit-wave

reconstruction, which rely on very few projections, have been pro-

posed to extend the achievable resolution of tomograms (Han et al.,

1996). For practical implementation of a CTF correction for tomography

the lateral focus gradient, particularly for high tilt angles, needs to be

incorporated, which make exit-front reconstructions considerably more

complicated. A simpler restoration method, which requires only one

micrograph per tilt and incorporates the lateral focus gradient, has

been realized recently (Winkler and Taylor, 2003). However, this res-

toration method is designed primarily for thin specimens. In cryo-ET,

CTF corrections have not been established yet. This is primarily due

to the fact that the SNR of the individual micrographs is still too low,

particularly because of the poor MTF of the CCD cameras, which pro-

hibits a precise determination of the CTF.

3.5.3 Visualization and Image Analysis

The interpretation of tomograms at the ultrastructural level requires

decomposition of a tomogram into its structural components, e.g., the

segmentation of intracellular membranes or the assignment of organ-

elles. Currently, a manual assignment of features is commonly used

because human anticipation is still superior in most cases to available

segmentation algorithms, although machine-based segmentation is in

principle more objective (Frangakis and Hegerl, 2002; Volkmann, 2002).

Instead of addressing the ultrastructure (Ladinsky et al., 1999), cryo-ET

provides the basis for interpreting tomograms even at the molecular

level. However, the analysis and 3D visualization are hampered by a

very low SNR. To increase the SNR, so-called denoising algorithms

have been developed (reviewed in Frangakis and Foerster, 2004). These

algorithms aim to identify noise and remove it from the tomogram, but

in practice, they also remove a certain fraction of the signal, resulting

in data with reduced information but higher SNR. The simplest denois-

ing techniques used diverse linear ﬁ ltering operations, such as a simple

low-pass ﬁ ltering in Fourier space. Better signal preservation can be

achieved by nonlinear ﬁ ltering algorithms, such as nonlinear anisotro-

pic diffusion (NAD) (Frangakis and Hegerl, 2001; Fernandez and Li,

2003) or bilateral ﬁ ltering (Jiang et al., 2003b). NAD is particularly

useful for the visualization of the ultrastructural features because it

can enhance features such as membranes. In addition, these ﬁ lters

preserve the signal, without major alterations, which is especially suit-

Chapter 7 Cryoelectron Tomography (CET) 583

able for visualization at the molecular resolution level. Denoising based

on the wavelet transformations (Stoschek and Hegerl, 1997) maintains

the high-resolution content, but requires immense computational

efforts, thus currently favoring the former approaches.

CET enables us to resolve large macromolecular complexes such

as the 26S proteasome inside intact cells (Medalia et al., 2002). High-

resolution structures of numerous macromolecules are available from

X-ray crystallography or nuclear magnetic resonance (NMR). The

purpose of many ongoing structural genomics projects is to generate

a comprehensive library of protein structures. Therefore, molecular

recognition is the task of locating a priori known structures in the

context of cells or other complex biological samples rather than clarify-

ing structurally novel features, e.g., as intended with denoising

methods. Mapping of macromolecules on the basis of their structural

signature requires the quantitative comparison of tomogram data with

a library of macromolecular structures. Ideally, the search of tomo-

grams should be exhaustive and would reveal a cellular atlas of resolv-

able macromolecules. Simulations and experiments with “phantom

cells,” i.e., liposomes encapsulating macromolecules, indicated that

such an approach is feasible (Boehm et al., 2000; Frangakis et al., 2002).

The information addressing the spatial relationship of different com-

plexes fosters and complements other proteomic methods and will be

indispensable for structural proteomic approaches (Sali et al., 2003).

The most common molecular detection algorithm is a locally normal-

ized, matched ﬁ lter, introduced in a different context (Roseman, 2000,

2003). It was modiﬁ ed to account for the missing-wedge effect (Fran-

gakis et al., 2002) and applied to cryotomograms (Rath et al., 2003).

These applications demonstrated that it is feasible to identify large

macromolecular complexes (>500 kDa) within cryotomograms with

high ﬁ delity. Furthermore, the high computational demand of template

matching is signiﬁ cantly reduced by parallelization. The information

an individually resolved macromolecule contains is limited because of

the low electron dose and the missing-wedge effect. Averaging tech-

niques aim to overcome the dose limitation of resolution by explicit

averaging of different reconstructions from the same particle. Iterative

algorithms are used to align subtomograms of arbitrarily oriented

copies of a particle and obtain a consistent average. Medium resolution

(2–3 nm) structures of the thermosome and tricorn could be obtained

from cryotomograms of puriﬁ ed complexes (Walz et al., 1997b; Nitsch

et al., 1998). Generally, the resolution from cryo-ET is inferior to

conventional single-particle reconstructions, which use 2D projections

of particles recorded on ﬁ lm to obtain 3D structures. Considering

the aforementioned dose fractionation theorem (see Section 3.2), the

higher information content of tomograms should favor averaging of

tomograms over the averaging from projections. However, the low

resolution of CCD cameras compared with that of ﬁ lm, the alignment

error of the micrographs prior to 3D reconstruction, and the current

inability to correct reliably for the CTF argue against averaging of

tomograms. Nevertheless, cryo-ET can provide medium-resolution

structures of protein complexes without using extensive puriﬁ cation

584 J.M. Plitzko and W. Baumeister

procedures (Foerster et al., 2005). Furthermore, structures obtained by

cryo-ET can be used as starting models that are reﬁ ned by single-

particle techniques or can be combined with other higher resolution

structural techniques to provide comprehensive descriptions of molec-

ular complexes.

3.5.4 Identiﬁ cation Strategies of Macromolecular Complexes

A very common approach to the task of locating features of known

structure in an input scene is “matched ﬁ ltering.” Matched ﬁ ltering is

basically the computation of a suitable cross-correlation function (CCF)

of the template, a macromolecule of known structure, and the input

scene, the cryo-electron tomogram (Figure 7–22). Cross-correlation

functions are straightforward to compute as a function of spatial

parameters r, since they can be calculated using use the fast Fourier

transform (FFT) (F):

CCF I T F F I F T

rr r

=⋅=

()

⋅

()

[]

′′

−

′

∑

The gray values of the input scene are denoted I

and the template’s

gray values are T

. The maxima of this cross-correlation function should

Figure 7–22. Detection and identiﬁ cation of individual macromolecules in cellular tomograms is

based on their structural signature. Because of the crowded nature of the cytoplasm and ‘contamina-

tion’ with noise, an interactive segmentation and feature extraction is not feasible. It requires sophis-

ticated pattern recognition techniques to exploit the information contained in the tomograms. A

volume rendered presentation of a 3D image is presented on the left. Even though some high-density

features may be visible, an unambiguous identiﬁ cation of individual structures would be difﬁ cult if

not impossible given the residual noise. An approach, which has proven to work, is based on template

matching. Templates of the macromolecules under scrutiny are obtained by a high- or medium resolu-

tion technique (X-ray crystallography, NMR, electron crystallography or single particle analysis).

These templates (4 times magniﬁ ed in this ﬁ gure; 20S proteasome and thermosome) are then used

to search the entire volume of the tomogram systematically for matching structures by three-

dimensional cross-correlation and the result is reﬁ ned by multivariate statistical analysis. In principle

the 3D image has to be scanned for all possible Eulerian angles ϕ,ψ and θ around three different axes,

with templates of all the different protein structures one is interested in e.g. the thermosome (blue)

and proteasome (yellow). The search procedure is computationally very demanding but can be paral-

lelized with respect to the different angular combinations in a highly efﬁ cient manner. Finally, the

position and orientation of the different complexes can be mapped directly in the 3D image. (See color

plate.)