# A FRACTAL MULTI-DIMENSIONAL ULTRASOUND SCATTERER DISTRIBUTION MODEL

**ABSTRACT** This paper presents a multi-dimensional point scatterer distribution model for the context of ultrasound image simulation. The model has a simple parameterisation, has low computational requirements and is flexible enough to model spatial organisation of scatterers ranging from highly clustered to nearly regular. The model extends an existing 1D model by mapping 1D scatterer positions to a Hubert space-filling curve. The flexibility of the heuristic model is illustrated through experiments where common statistical models of ultrasonic speckle are fitted to simulated data. The results agree with theoretical predictions.

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**ABSTRACT:**This paper describes a simple, yet powerful ultrasound scatterer distribution model. The model extends a 1-D generalized Poisson process to multiple dimensions using a Hilbert curve. The model is intuitively tuned by spatial density and regularity parameters which reliably predict the first and second-order statistics of varied synthetic imagery.IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control 03/2009; · 1.82 Impact Factor - SourceAvailable from: Jean-Louis Dillenseger[Show abstract] [Hide abstract]

**ABSTRACT:**The goal of our work is to propose a fast ultrasound image simulation from CT volumes. This method is based on a model elaborated by Bamber and Dickinson that predict the appearance and properties of a B-Scan ultrasound image from the distribution of point scatterers. We propose to extend this model for the standard medical ultrasound image simulation by taking into account the acoustical tissue properties (scatterer distribution) and the geometry and the specifications of the ultrasound probe (circular probe, number and size of transducers, US pulse frequency and bandwidth, etc.). Simulations have been obtained in a fairly fast computation speed and qualitatively they present most of the real ultrasound image characteristics.Computers in biology and medicine 02/2009; 39(2):180-6. · 1.27 Impact Factor

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A FRACTAL MULTI-DIMENSIONAL ULTRASOUND SCATTERER DISTRIBUTION MODEL

Catherine Laporte, James J. Clark and Tal Arbel

Centre for Intelligent Machines, McGill University

3480 University Street, Montreal, Qc, H3A 2A7 CANADA

ABSTRACT

This paper presents a multi-dimensional point scatterer dis-

tribution model for the context of ultrasound image simula-

tion. The model has a simple parameterisation, has low com-

putational requirements and is flexible enough to model spa-

tial organisation of scatterers ranging from highly clustered

to nearly regular. The model extends an existing 1D model

by mapping 1D scatterer positions to a Hilbert space-filling

curve. The flexibility of the heuristic model is illustrated

through experiments where common statistical models of ul-

trasonic speckle are fitted to simulated data. The results agree

with theoretical predictions.

Index terms: acoustic imaging, simulation, fractals, statistics,

speckle

1. BACKGROUND

An ultrasound (US) image simulator is a useful validation

platformforimageprocessingapplicationswheregroundtruth

pertainingtoimagecontentoracquisitionconditionsisneeded

but not available from US data without additional equipment.

It also serves as a test platform during application software

development when dataacquisition is impossible orawkward.

Typical simulators take a list of point scatterers with their

strength and position as input along with US transducer spec-

ifications and simulate the resulting backscattered signal.

The density and spatial organisation of scatterers are two

of many factors which determine the appearance and statistics

of US speckle. In order to validate image processing applica-

tions or study image properties across different tissue types,

it is useful to be able to generate random lists of point scat-

terers with varying density and spatial organisation, ranging

from highly clustered to random to nearly regular. There is

currently a lack of models displaying such flexibility along

with computational efficiency and simple parameterisation.

The Neyman-Scott point process model has been used for

studying the effect of blood cell aggregation on ultrasonic

signals [1]. The model creates random cluster centers and

spawns daughter points surrounding them. It is not appropri-

ate for generating quasi-periodic patterns. Such patterns can

Catherine Laporte is funded by a doctoral scholarship from the Natural

Science and Engineering Research Council of Canada.

be generated with varying regularity by perturbing a regular

point lattice [2], but this approach cannot generate clustered

point patterns. An alternative is a Gibbs-Markov area interac-

tion process which imposes pairwise repulsive and attractive

constraints between points. This model was used to study

US backscattering from aggregates of non-overlapping blood

cells [3]. This type of model is computationally demanding

and fails to produce a broad enough variety of clustered pat-

terns when no repulsive constraints are used to maintain a

minimal distance between points [4]. In short, previous at-

tempts to parameterise variation in the spatial organisation

of scatterers have been geared towards specific applications

and have not modeled the full continuum of spatial organisa-

tionsrangingfromclusteredtoregular. Onenotableexception

is the 1D marked regularity model of Cramblitt and Parker

[5]. This model is a generalisation of the Poisson point pro-

cess with an additional parameter which tunes the variance of

inter-scatterer spacing. Unfortunately, the model is one di-

mensional, and not suitable for 2D or 3D simulations.

This paper proposes a heuristic extension to Cramblitt and

Parker’sscattererdistributionmodelforthemulti-dimensional

case. The proposed approach is to map the results from the

original 1D model to a Hilbert space-filling curve. It is shown

that this mapping preserves some of the statistical properties

of the 1D model. The statistics of resulting simulated images

are analysed, showing that the multi-dimensional extension

can produce a broad variety of image patterns corresponding

to commonly used parametric models of speckle statistics, in

agreement with theory.

The rest of this paper is structured as follows. Cramblitt

and Parker’s 1D model is described in section 2. The pro-

posed multi-dimensional extension is described in section 3.

The validation methods used for experiments are given in sec-

tion 4, with the results presented in section 5.

2. THE 1D MODEL

The 1D marked regularity model of Cramblitt and Parker [5]

generates a random scatterer function of the form

s(x) =

?

i

aiδ(x − Xi),

(1)

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wherexdenotespositiononthereallineandaiandXidenote

the strength and position of scatterer i. In this model, the

distance di= |Xi− Xi+1| is gamma distributed with shape

and scale parameters α and β:

f(d) =dα−1exp(−d/β)

Γ(α)βα

α,β,d > 0.

(2)

This distribution can be usefully reparameterised in terms of

the mean and variance in scatterer spacing,¯d = αβ and

σ2

in terms of scatterer density (1/¯d) and regularity (controlled

by α). With α = 1, the model reduces to a Poisson point

process with exponentially distributed inter-scatterer spacing,

and scatterers are placed at random. For α < 1 (high vari-

ance), scatterers tend to group in clusters, whereas for α > 1

(low variance), scatterers become more evenly spaced.

Incidentally, anidenticalrenewalpointprocessmodelwas

independently studied in the field of economic statistics [6].

In this context, it was shown that statistics on the lengths of

inter-event (inter-scatterer) intervals have a simple translation

in terms of asymptotic statistics on event counts within inter-

vals of fixed length. Specifically, as the length of the fixed

interval tends towards infinity,

d= αβ2=¯d2/α. This allows the model to be tuned

E{N}

V AR{N}→ α,

(3)

where N is the scatterer count and E{.} and V AR{.} are the

expectation and variance operators, respectively.

3. EXTENSION TO MULTIPLE DIMENSIONS

While the α = 1 (Poisson) case is easily generalised to multi-

ple dimensions, there is no obvious extension of the model for

other α. A simple heuristic to generate a 2D scatterer function

would be to rearrange short segments of a 1D scatterer func-

tion on a 2D grid in a raster scan fashion. Such a mapping is

notdesirablebecauseitonlypreservesthespatialorganisation

of the points along one direction. The raster scan mapping is

but one of many possible mappings of a line segment onto a

finite multi-dimensional space known as discrete space-filling

curves, which have the property of traversing every single

point of a discrete m-dimensional grid of a given precision.

Better results can be obtained by choosing a different type of

space-filling curve with better preservation of locality, such

that distances measured along the original line correlate well

with distances measured in the multi-dimensional space. A

good choice is the Hilbert curve, a fractal curve shown to

nearly achieve theoretical bounds in terms of preservation of

locality [7]. The Hilbert curve of precision k traverses (2k)m

points on an m-dimensional grid. Discrete approximations

of the 2D Hilbert curve for different levels of precision are

shown in figure 1.

The algorithm introduced in [9] can be used to determine

the Hilbert curve mapping of a point in 1D space to m-D

Fig. 1. Discrete approximations of the 2D Hilbert curve [8].

The level of precision of the curves increases from left to right

from 1 to 5.

space. This algorithm takes as input a fixed point mk bit rep-

resentation of the point’s position along the 1D line segment,

recursively removes m bits from the left of this input and uses

them as an index to one of 2mequally sized sub-cells within

the current cell of the m-dimensional space. The precision of

the resulting m-dimensional coordinates is equal to k. In 2D

or 3D, the process is equivalent to decoding the indexing of

cells in a quadtree or octree, respectively.

The Hilbert curve mapping is applied to each point scat-

terer in the 1D function sampled from Cramblitt and Parker’s

model, yielding a new set of scatterers in 2D or 3D whose

spatial arrangement exhibits similar characteristics to the 1D

version. This was empirically verified by generating scatter-

ing functions with different parameters 1/¯d and α over large

spaces in 1D, 2D and 3D, breaking these spaces into equal

numbers of equally sized non-overlapping bins and calculat-

ing the ratio of mean to variance in scatterer count over these

bins. As predicted by Eq (3), this ratio is close to α for large

bin sizes in the 1D case. In 2D and 3D, it was found that the

agreement with theory is also good provided that the statis-

tics are measured on full sub-cells of the Hilbert curve (which

themselves are Hilbert curves of lower precision). This means

that when using the (heuristic) multi-dimensional extension

of the model, one should use the data generated for an integer

number of full Hilbert curve sub-cells in order to guarantee

preservation of the 1D statistics. The agreement between the-

ory and the samples for a 262,144 unit sized space divided

into 64 bins is illustrated in figure 2. The agreement was

found to be of good quality for all values of 1/¯d.

Fig. 2. Ratio of mean to variance in scatterer count versus α.

Solid line: theory, squares: 1D model samples, circles: 2D

model samples, crosses: 3D model samples.

Page 3

4. APPLICATION TO US IMAGE SIMULATION

The proposed scatterer distribution model was validated on

the basis of its usefulness in the context of 2D US image sim-

ulation. The model should permit simulation of a broad vari-

ety of speckle images. The statistics of these images should

be reliably predictable from the model parameters (as with the

original 1D model) and in agreement with theoretical consid-

erations.

4.1. US speckle amplitude statistics

Theory proposes several parametric models for the statisti-

cal distribution of the US echo amplitude A. The three main

models studied in [5] were retained for this study. The first is

the Rayleigh distribution model:

f(A) =A

σ2exp(−A2/2σ2),

(4)

whereσ2reflectsthevarianceinscattererstrength. Themodel

assumes a large number of scatterers distributed uniformly in

space (α close to 1), resulting in random interference between

backscattered waves.

The second model is the K distribution model:

?bA

where b = 2?

an accurate statistical description of US speckle for low scat-

terer density or high scatterer clustering (α < 1), reflected in

the additional parameter µ. The model reduces to a Rayleigh

distribution with parameter σ for µ → ∞ .

The third model is the Rician distribution model:

f(A) =

2b

Γ(µ)2

?µKµ−1(bA)

µ > 0,

(5)

µ

2σ2

?1/2and Kµ−1(.) is the modified Bessel

function of the second kind and order µ − 1. The model is

f(A) =

A

σ2exp(−(A2+ s2)/2σ2)I0

where I0(.) is the modified Bessel function of the first kind

of order 0, and the parameter s describes the amount of co-

herent scattering associated with highly organised scatterer

structures causing specular reflection and other forms of con-

structiveinterference. Thesemayoccurforα > 1. Themodel

reduces to the Rayleigh distribution model for s = 0.

?As

σ2

?

s ≥ 0,

(6)

4.2. Simulation and model selection

Forquantitativevalidation, USimagesweresimulatedbycon-

volving 1D and 2D scattering functions with a gated cosine

pulse, yielding a radio-frequency (RF) signal. The square

pulse envelope was chosen over a Gaussian envelope because

the former allows for better defined scatterer/resolution cell

count. The resolution cell size T (the same in 1D and in 2D)

was chosen such that the scatterer density, in terms of scat-

terers per resolution cell, T/¯d, varied from 5 to 50, with¯d

set to either 0.5 or 1.0 cycle of the cosine pulse (in 1D, these

Fig. 3. 2D simulated images obtained with high scatterer den-

sity and increasing values of α.

correspond to destructive and constructive interference cases,

respectively). The regularity parameter α was increased from

0.1 to 30. The scatterer strengths were independently sam-

pled from a lognormal distribution with mean 1 and variance

0.1, as suggested in [5]. The amplitude signal was obtained

through envelope detection of the RF signal via the Hilbert

transform. Ten simulated images of size 1024T were gener-

ated for each combination of T/¯d, α and¯d.

For each of the three speckle models described earlier, a

maximum likelihood fit to the 1024 data samples at the cen-

ters of sequential image patches of size T was calculated. A

closed form expression for the maximum likelihood value of

σ exists for the Rayleigh model. The Fletcher-Reeves con-

jugate gradient algorithm was used to optimise parameters of

the Rician distribution model. For the K distribution model,

whose log-likelihood function has complicated derivatives,

the derivative free Nelder-Mead simplex algorithm was used.

The“best”modelwaschosentominimiseSchwarz’sBayes

Information Criterion (BIC) [10]:

BIC = −2L + plogn,

(7)

where L is the logarithm of the likelihood function at its max-

imum, p is the number of parameters in the model analysed

and n is the number of data samples. The BIC was chosen

over a simple goodness of fit criterion because it measures

the complexity of the model in addition to its goodness of fit,

choosing simpler models over more complex ones for similar

goodness of fit levels. The BIC is particularly appropriate for

this study, where the compared models are all generalisations

of the simple Rayleigh model.

5. RESULTS

For illustration, examples of simulated ultrasound images ob-

tained with high scatterer density and varying α are shown in

figure 3. A Gaussian pulse envelope was used for realism, and

the images were logarithmically compressed for better visual-

isation. As α increases and the scatterers are more uniformly

placed, it can be seen that the images become smoother.

Figures 4 and 5 show how often each speckle model was

chosen out of then ten trials for each simulation case. For

small scatterer densities and high clustering (α < 1), the K

distributionmodelispreferred, inagreementwiththeory, both

in 1D and 2D. For random scatterer placement (α = 1), small

Page 4

α = 0.1

α = 0.3

α = 1

T

¯d= 5

T

¯d= 10

T

¯d= 20

T

¯d= 50

Fig. 4. Number of times each speckle model was chosen out

of 10 trials for simulation cases with α ≤ 1. Black: 1D

model,¯d = 1.0 pulse cycle; dark grey: 1D model,¯d = 0.5

pulse cycle; light grey: 2D model,¯d = 1.0 pulse cycle; white:

2D model,¯d = 0.5 pulse cycle.

scatterer densities yield K distributed amplitudes, which be-

come Rayleigh distributed as the scatterer density increases.

This transition seems to occur for smaller densities in the 2D

case than in the 1D case.

The statistics of the simulated images when regular struc-

ture is present (α > 1) depend on¯d. In 1D, constructive inter-

ference occurs for large values of α, irrespective of scatterer

density, when¯d = 1 pulse cycle, yielding Rician amplitude

statistics. Destructive interference occurs for¯d = 0.5 pulse

cycle, yielding K (for small densities), or Rayleigh (for large

densities) distributed statistics. In 2D, the average scatterer

spacing¯d is mapped onto a non-linear segment of the Hilbert

curve, making it difficult to generate scatterers at exact multi-

ples of half-wavelengths in the direction of wave propagation.

However, Ricianamplitudestatisticscanstillbeobtainedwith

the 2D model for large values of α, high scatterer densities

and small inter-scatterer spacing. This is illustrated by the 2D

case with¯d = 0.5 pulse cycle. This case effectively simulates

a compact, solid structure yielding specular reflection.

While the 1D and 2D results may differ for some pa-

rameter settings, both make theoretical sense in their context.

Moreover, the type of speckle obtained by simulation can be

reliably predicted from the parameters of the 2D model, as it

is with the 1D model.

6. CONCLUSIONS

This paper presented a new multi-dimensional scatterer dis-

tribution model which can be used in the context of US im-

age simulation. The model builds on a flexible 1D model

whose desirable characteristics it preserves well. Simulation

results show that textures corresponding to commonly used

parametric models of speckle statistics can be obtained from

this model and well predicted from the model parameters.

α = 3

α = 10

α = 30

T

¯d= 5

T

¯d= 10

T

¯d= 20

T

¯d= 50

Fig. 5. Number of time each speckle model was chosen out

of 10 trials for simulation cases with α > 1.

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