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Chapter 1. Use of Computer-Assisted Tomography in Studying Water Movement around Plant Roots

Chapter 1. Use of Computer-Assisted Tomography in Studying Water Movement around Plant Roots

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2



L. A. G. AYLMORE



I. INTRODUCTION

An appreciation of the physical, chemical, and biological factors determining the supply, availability, and movement of water in soil/plant ecosystems, together with suitable techniques for the measurement of the

forces involved, is essential to the development of an understanding of the

mechanisms and dynamics of water movement in soils and their biological

implications. The importance of this field of study cannot be overemphasized, particularly in semiarid and saline environments, where the availability of scarce water resources for agriculture makes it imperative that the

most efficient water utilization by plants is achieved, and where limits to

growth and production are most commonly set by limitations on our

knowledge of such factors.

A serious difficulty encountered in attempts to relate soil water to plant

response is the fact that the water content in a plant root zone varies

markedly in both time and space. Slatyer (1967) emphasized the importance of the soil water potential at the root/soil interface as the main soil

characteristic controlling the availability of soil water for plant growth,

with its value depending on both the soil water potential of the bulk soil

and the potential gradient from the bulk soil to the root surface, which

develops as a result of water removal by the root. Philip (1966) also

suggested that the value of the water potential at the root surfaces was

critical to the distribution of water potential (and to the possibility of

wilting) throughout much of the plant domain. Furthermore, although the

soil water tension, or matric suction, at a given depth in the root zone may

correlate well with plant response in some circumstances and provide a

useful basis for imgation, a clearer understanding of water availability to

plants requires some means of resolving changes in soil suction or water

content over the entire root zone.

Unfortunately, progress in this area has been severely limited because of

the difficulties associated with direct experimental measurement of soil

water content or potential at the root/soil interface and in the soil immediately around the root. Until recently, techniques for the direct measurement of soil water content or potential have either been destructive (and

hence lacked continuity), have perturbed the sensitive balance being examined, were too slow in their response time, or simply lacked the dimensional resolution necessary for meaningful definition of water content

distributions. Although Dunham and Nye (1973) were able to measure

one-off drawdowns in proximity to curtains of roots by destructive sectioning and So et al. (1976, 1978) were able to determine water potentials

at the root surface by extrapolation using a collar tensiometer - potometer

system, these techniques provided only very limited insights into the dy-



CAT STUDIES OF WATER MOVEMENT



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namics of the availability of soil water for plant growth. Consequently,

questions concerning the relative magnitudes of soil and plant resistances

to water movement under different conditions of soil water potential and

transpirational demand (Newman, 1969a,b), concerning the nature of the

water driving forces (Nobel, 1974), and concerning the extent to which

root/soil contact resistance (Herkelrath et al., 1977), accumulation of soil

solute concentrations (osmotic potentials) (Passioura and Frere, 1967),

etc., influence water availability have remained largely unresolved. Furthermore, conflicting results obtained predominantly by the indirect measuring procedures that previously were the only available techniques

(Dunham and Nye, 1973; So et af., 1976, 1978) raise questions as to the

validity of the physical concepts on which theoretical treatments (Molz,

1981) have been based.

Similar problems had long existed in medical diagnostic radiology in

seeking a method by which the interior of a section of the human body

could be viewed in a nondestructive manner without interference from

other regions. With advances in X-ray physics, detector technology, and

mathematical reconstruction theory, a solution to the problem was essentially achieved in the early 1970s by Hounsfield (1972), who developed the

technique known as computer-assisted tomography (CAT), or more simply, computed tomography (CT). (The word tomography is derived from

two Greek words: tomo, meaning slice or section, and graphy, meaning to

write or display.) CAT enables the three-dimensional, nondestructive

imaging of the internal structure of the object under examination using

measurements of the attenuation of a beam of radiation. The application

of the technique to the attentuation of X-rays (colloquially referred to as

CAT scanning) allowed dramatic advances in medical diagnostic capability

and benefited the medical profession greatly by reducing the need for

exploratory surgery to examine the internal structures of the human body.

For this work Godfrey Hounsfield shared the 1979 Nobel Prize for Medicine with A. M. Cormack, who had earlier, in 1963, developed and applied

a mathematical model that allowed the determination of absorption coefficients at specific points in scanned sections from the measured attenuation of collimated beams of 6oCoy-radiation.

Tomographic imaging in various forms is applicable to a number of

different types of energy beams, including electrons, protons, a particles,

lasers, radar, ultrasound, and nuclear magnetic resonance. However, because of its convenience and versatility in medical, industrial, and scientific applications, most attention has been directed to X-ray CT. In recent

years, the opportunity to use CAT scanning for nonmedical applications

has blossomed, particularly in the United States, Canada, Europe, Australia, and Japan. Hopkins et al. (1981) and Davis et af.(1986) demonstrated



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L. A. G . AYLMORE



its application to industrial problems, particularly for nondestructive testing of timber poles, plastics, concrete pillars, steel-belted automobile tires,

and electronic components. Onoe e? al. (1983) described the use of a

portable X-ray CAT scanner for measuring annual growth rings of live

trees.

The potential applications of CAT scanning in the soil and plant

sciences have also attracted increasing interest over the past decade. Numerous workers (Petrovic er al., 1982; Hainsworth and Aylmore, 1983,

1986; Crestana er al., 1985; Anderson e? al., 1988, 1990; Tollner e? al.,

1987; Tollner and Verma, 1989) have demonstrated that commercially

available X-ray medical scanners can provide excellent resolution for some

studies of the spatial distributions of bulk density and water content in soil

columns, including in particular those near plant roots (Hainsworth and

Aylmore, 1983, 1986; Aylmore and Hamza, 1990; Hamza and Aylmore,

1991, 1992a,b). The quantitative usefulness of such systems in soil studies

has, however, been limited by the polychromatic nature of the X-ray beam

and its inability to distinguish between changes in water content and bulk

density in swelling soils. Furthermore; these instruments are prohibitively

expensive (about $2 million) and hence have not been generally accessible

to soil and plant scientists. Consequently, work in several laboratories has

sought to provide experimentally more suitable systems and to reduce

vastly the cost of the equipment, by the modification of “conventional” y

scanning systems (Gurr, 1962; Groenevelt et al., 1969; Ryhiner and Pankow, 1969) to utilize the CAT approach (Hainsworth and Aylmore, 1983,

1988; Crestana e? al., 1986). pRays are essentially monochromatic, and

the ready availability of sources providing large differentials in energy level

offers the potential to distinguish quantitatively between simultaneous

changes in water content and bulk density. However, the relatively low

photon emission from pray sources compared with X-ray tubes requires

much longer scanning times and has as yet limited measurements by this

means to slow or steady-state processes.

Despite these current limitations there is no doubt that the application of

this exciting new technique will, with further developments, provide a

major tool for soil and plant scientists and has the potential to resolve the

major controversies with respect to the physics of water uptake by plant

roots.



11. COMPUTER-ASSISTED TOMOGRAPHY

The theory and use of the CAT technique for medical purposes has been

reviewed in some detail by Budinger and Gullberg (1974), Brooks and Di

Chiro (1975, 1976), and Panton (198 l), and complete reviews of various



CAT STUDIES OF WATER MOVEMENT



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aspects of CAT scanning have been presented by Newton and Potts (198 1)

and Kak and Slaney (1988). Brief reviews of CAT scanning theory as it

relates to the determination of soil water content have been presented by

Hainsworth and Aylmore (1983), Crestana er al. (1983, and Anderson et

al. (1988). However, as the technique has only recently been introduced in

soil science, an outline of the theory of CAT is given here to familiarize

readers with the technique.



A. THEORY

OF ATTENUATION

In conventional radiography the transmission of radiation through a

three-dimensional object is used to produce a two-dimensional image of

the internal features of the object on a radiation-sensitivefilm. Attenuation

occurs because the photons in the incident beam may be absorbed by the

material and disappear, or may be deflected out of the path of the beam,

leading to a decrease in the detected radiation intensity (Fig. 1). The image

formation relies on the spatial variation of radiation attenuation in the

object, which gives rise to a contrast in the transmitted radiation recorded

on the film. The physical quantity that characterizes the attenuation of

radiation by matter is called the linear attenuation coefficient (p).

The three principal mechanisms of radiation attenuation in matter are

photoelectric absorption, Compton scattering, and electron- positron pair

production (Cullity, 1978). In photoelectric absorption, the photon collides

directly with an atom of the absorber and transfers all of the energy to one

of the orbital electrons, which is ejected from the atom. This is the most

important process for low-energy photons (<500 keV). Because photons

with energy in excess of that required to eject an electron are unlikely to be

absorbed, the photoelectric absorption coefficient decreases rapidly with



Incident

beam



Transmitted

beam



absorbed

scattered

Figure 1. Attenuation of a narrow beam of radiation by absorption and scattering.



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L. A. G. AYLMORE



increasing photon energy. Compton scattering is the predominant scattering process in which a photon collides with an atom and is deflected from

its original direction with the loss of only a portion of its energy. This

energy is transferred to an atomic electron, which recoils out of the atom.

The absorption of photons by Compton scattering is most probable for

intermediate-energy photons (500- 1000 keV). The photon continues on

at a reduced energy to undergo additional Compton scattering or to be

absorbed by photoelectric interaction with a second electron. Of secondary

importance may be Rayleigh scattering, in which a photon may be deflected with no loss of energy and the whole atom recoils under the impact.

This can occur for photons of low energy, i.e., in the region where the

photoelectric effect is dominant. At very high photon energy, > 1000 keV,

a photon may be absorbed in the neighborhood of an atomic nucleus or

atomic electron and produce an electron-positron pair. In soil water

studies, the highest photon energy used is 662 keV from a y-radiation

source of I3’Cs, thus electron- positron production is not important in the

attenuation process.

The attenuation of a collimated beam of monoenergetic photons

of intensity I,, as a result of passing through a sample of material of thickness D, yields a transmitted intensity I behind the sample, as illustrated in

Fig. 2a (Anderson et al., 1988); this can be described by Beer’s Law,



I = I, exp(-pD)



(1)



where p, the linear attenuation coefficient (often referred to as attenuation coefficient),represents the fractional attenuation per unit length of the

material traversed by the radiation. The value of p depends primarily on

the energy of the radiation, the electron density, and the packing density of

the material. Equation (1) assumes that the material is homogeneous in

composition and density over the distance D. For heterogeneous materials,

one can subdivide the length D into n subdivisions (of length d), each

having a different linear attenuation coefficient, and can describe the

attenuation over the length D as the sum of the attenuation of these small

subdivisions (Fig. 2b). The transmitted radiation intensity is then



For real objects, such as soil, the attenuating material is continuously

rather than discretely distributed, so EQ. (2) takes the form of an integral



C A T STUDIES OF WATER MOVEMENT



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I

:

I\



EDEDI



Y



Figure 2. Schematic representation of the attenuation of monochromatic radiation of

initial intensity I,, by (a) homogeneous material, (b) nonhomogeneous material consisting of

discrete units with different attenuation coefficients,and (c) nonhomogeneous material consisting of a variable attenuation coefficient,p x , over the distance x from the source. (Afler

Anderson et aL, 1988.)



(Fig. 2c),



where x is the distance from the radiation source and varies between 0 and

D, the thickness of the sample.

Equation (3) can be rearranged to yield



The logarithm of Zo/Z is effectively a sum of the attenuation coefficients

along the ray path and is called a ray sum or ray projection. Obviously, a

single ray sum cannot give any information about the distribution of

attenuation coefficientsat discrete points within the material along the ray

path. Thus interpretative difficulties can arise because the image obtained

is really a two-dimensional projection of a three-dimensional object. The



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L. A. G. AYLMORE



Figure 3. Coordinate system for calculation of the photon attenuation coefficient at a

given point. Points within the object are described by fixed (x, y ) coordinates. Rays (dashed

lines) are specified by their angle (4)with the y axis and their distance (r)from the origin. The

S coordinate denotes distance along the way.



aim of the CAT technique is to overcome this difficulty and to reveal the

spatial distribution of attenuation coefficients unambiguously.



B. PRINCIPLES

OF COMPUTER-ASSISTED

TOMOGRAPHY

In CAT, multiple scans from different angles in a given plane provide a

large number of ray sums or projections. Using these projections, a two-dimensional image of the slice is reconstructed numerically to give the

distribution-of attenuation coefficients at discrete points within the slice.

For image reconstruction, an (x, y ) coordinate system (Fig. 3) is used to

describe points in the slice. As the slice is scanned, ray paths through the

slice can be defined by 4, the angle of the ray with respect to the y axis, and

r, its distance from the origin. The distance of a point from the source on

any ray path is given by the coordinate S, which varies from 0 to S.

The contribution of each point to the attenuation of a ray ( I , 4) with

initial intensity Z, and transmitted intensity I is denoted by



Equation ( 5 ) can be rearranged to obtain the projection value, p, of the



CAT STUDIES OF WATER MOVEMENT



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Figure 4. Schematic illustration of a parallel-beam CAT scanning procedure showing a

single projection consisting of a set of parallel ray sums and the linear and rotational

movementsinvolved in the collection of data prior to image reconstruction for a cross-section

of the object.



ray ,(I



4)

P(r, 4) = W o / ~ , +=)



I,



AX,V ) ds



(6)



In Eq. (6),p(x, y) is determined using many independent views or projections through the object. In the simplest scanning systems these are obtained using a scanning procedure involving both linear and rotational

movements of the source detector system (Fig. 4). A complete set of

parallel ray sums represents a single projection for that view of the crosssectional layer of the object.

Ideally, p(x, y) is a continuous two-dimensional function and an infinite

number of projections are required for reconstruction. Because in practice

it is physically impossible to obtain an infinite number of projections,

p(x, y ) is calculated at a finite number of points from a finite number of

projections. If the object is confined to a circular domain of diameter d,

and the image is reconstructed at points arranged rectangularly with spacing w, then there are n = d / w points along a principal diameter. Each

square cell of width w is called a pixel (an acronym for picture element). It

is also assumed that there are rn projections spaced equally from 0 to 180",

each consisting of n ray sums at intervals w. The minimum number of



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L. A. G. AYLMORE



rotations required for accurate image reconstruction is given by an/4

(Panton, 1981).

1. Numerical Reconstruction



In theory, if p(r, 4) is known for every line of width w passing through a

pixel of dimension w, then p(x, y) can be determined if Eq. (6) can be

inverted (Panton, 198I). Bracewell (1956) was the first to devise a numerical reconstruction technique for determining,u(x, y ) from Eq. (6), and with

subsequent advances there are now more than a dozen different approaches available for reconstructing p(x, y ) (Budinger and Gullberg,

1974). However, these approaches can be classified broadly into three

methods: ( 1) back-projection, (2) iterative reconstruction, and (3) filtered

back-projection (Brooks and Di Chiro, 1975, 1976).

a. Back-Projection Reconstruction

In the back-projection method, reconstruction is performed by applying

the magnitude of each projection to all points that make up the ray, or, in

other words, the back-projected p(x, y ) value is obtained by superimposing

projections together. The process can be described by Eq. (7),



+



where rj = x cos 4j y sin 4j,the distance of the ray from the origin; $ j is

the jth projection angle and A 4 is the angular distance between projections

(i.e., A 4 = a/m)and the summation extends over all rn projections.

Back-projection, however, does not produce a good reconstruction because each ray sum or projection is applied not only to points of high

density but to all points along the ray. This defect shows up most strikingly

with discrete areas of high density, producing a star artifact. The star

artifact causes the density function to vary from the true density function

by an intolerable margin, hence this method is rarely used these days.

b. Iterative Reconstruction

In the iterative methods of reconstruction, the basic strategy is to apply

corrections to arbitrary initial p(x, y) values in an attempt to match the

measured ray projections. Because former matchings are lost as new

corrections are made, the procedure is repeated until the calculated projections agree with the measured ones within the desired accuracy. Iterative

methods are primarily classified according to the sequence in which

corrections are made and incorporated during an iteration, as this choice



CAT STUDIES OF WATER MOVEMENT



11



has a significant effect on the performance of the method. Three such

variations have been proposed:

1. Simultaneous correction: all projections are calculated at the beginning of the iteration and corrections are applied simultaneously to all

points (x, y). This method has sometimes been referred to as the iterative

least-squares technique (ILST) (Goitein, 1972).

2. Point-by-point correction: each point is corrected simultaneously for

all rays passing through it and corrections are incorporated before moving

to other points. This technique was introduced in electron microscopy by

Gilbert ( 1972), who named it the simultaneous iterative reconstruction

technique (SIRT).

3. Ray-by-ray correction: a given set of ray projections is calculated and

corresponding corrections are applied to all points. The updated p ( x , y )

values are then used for calculating the next projection. Ray-by-ray correction was used in the original version of the EM1 scanner (Hounsfield, 1972)

and was independently discovered in electron microscopy by Gordon et al.

( I 970), who named it the algebraic reconstruction technique (ART). The

iterative reconstruction methods are slow and hence are not very popular.



c. Filtered Back-Projection

Filtered back-projection methods are the most commonly used and are

considered to be the most accurate. The basis of the analytical methods

involves a fundamental relationship between the Fourier transform of the

linear attenuation function p(x, y ) and the projection function p(r, 4).

Such a relationship filters the projection, accounting for the portions of the

projection that may pass outside a given pixel, thereby eliminating the star

artifact mentioned earlier in the back-projection reconstruction method.

The filtered projections are then back-projected, i.e., the reconstructed

p ( x , y ) is analogous to Eq. (7) except that the p is replaced by the filtered

version p *:



A number of filtering techniques (Fourier, Radon, convolution filtering,

etc.) have been developed, but the performance of filtered back-projection

is not greatly affected by the choice of filtering technique. Derivation of the

formula for the different filters can be found in works by Brooks and Di

Chiro (1976), Herman (1980), and Kak and Slaney (1988).

A schematic illustration of a reconstructed image with direct back-projection and back-projection after filtration of projections (profiles) measured for a homogeneous cylinder is shown in Fig. 5. After direct back-



L. A. G . AYLMORE



12



a



b



Figure 5. Schematic illustration of a reconstructed image with (a) direct back-projection

and (b) back-projectionafter filtration of profiles measured for a homogeneous cylinder.



projection of the profiles measured, image details would be smeared if no

further corrections were carried out. For this reason, back-projection is

preceded by a filtration or convolution process.

Filtered back-projection has the advantage that the data can be processed as they are collected and the image can be built up and displayed

projection by projection, thus allowing a useful saving in measurement

time, particularly when data rates are low. An added advantage is that the

final image is ready virtually immediately after the scans are completed

and image quality can be progressively assessed. Iterative techniques need a

larger data set before image reconstruction can commence, but are capable

of giving better results from fewer projections than is normally required for

the convolution method (Gilboy, 1984).

2. Aliasing Artifacts



Reconstruction procedures are only as good as the data on which they

are based and a number of errors or artifacts can arise through the availability of insufficient data or by the presence of random noise in the

measurements. An insufficiency of data may occur either through undersampling of projection data or because not enough projections are

recorded. The distortions that arise as a result of inadequate data are called

aliasing artifacts and generally appear as streaks, blumng, rings, or interference patterns in the reconstructed image. Aliasing distortions may also

be caused by using an undersampled grid for displaying the reconstructed

image (Brooks and Di Chiro, 1976; Kak and Slaney, 1988). Regions of

overestimated or underestimated attenuation associated with sharp

changes in attenuation and appearing as rings or oscillations are called the

Gibbs effect. Moire patterns are interference patterns that can dominate

the entire image. The occurrence of these effects depends largely on the

relative number of projections and rays in each projection, and it has been

shown (Kak and Slaney, 1988) that a balanced image reconstruction re-



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