image formation in diagnostic ultrasound -...

1997 IEEE International Ultrasonics SymposiumShort Course

Image Formation in Diagnostic

Ultrasound

J. Nelson Wright

Consultant, Medical Imaging and Signal Processing

2672 Bayshore Parkway, Suite 532Mountain View, CA 94043

650/969-1271 FAX 650/[email protected]

Image Formation in Diagnostic Ultrasound

Copyright 1997 by J. Nelson Wright

What is an Image? 4Imaging as a Transformation 4Desirable Characteristics of an Imaging System 5Common Measures of Imaging System Performance 6Canonical Ultrasound Imaging System 7

Apertures and Geometrical Optics 8The Near Field/Far Field Crossover 8Azimuthal Resolution and the Rayleigh Criterion of a Linear Aperture 10Depth of Focus of a Focused Linear Aperture 14Duality 20Summary 22

Linear Shift-Invariant Systems and Transforms 23Representation of Signals in One Dimension 23Some Properties of Fourier Transforms in One Dimension 25The Multidimensional Fourier Transform 26Separability 26Rotational Symmetry 26The Hankel Transform of a Rotationally Symmetric Function 27Radial Symmetry: The Hankel Transform of a Two Dimensional Function 27Spherical Symmetry: The Hankel Transform of a Three Dimensional Function 27Some Properties of Fourier Transforms in Multiple Dimensions 28Marginal Fourier Transforms in Multiple Dimensions 29Multidimensional Fourier Transforms in Space and Time; Plane Waves 30The Multidimensional Dirac Delta Function 31

Acoustic Propagation and Reflection 32The Scalar Wave Equation 32Propagation Velocity and Losses 32Propagation as a Linear Shift Invariant System 33Low-Level Scattering 38Acoustic Sources, Sensors, and Boundary Conditions 40Backpropagation and the Angular Spectrum 42The Pulse Echo Equation for a Fixed Focus Planar Transducer 44



One-Way Beamformation and Spatial Filtering 46Reception Beamformation as Matched Filtering 46Reception Beamformation as Backpropagation 47Reception Beamformation as Inverse Filtering 49Summary 51

Beamformation and the Point Spread Function 53The Three Dimensional Point Spread Function 53A Local Expansion of the Two Dimensional Point Spread Function 55The Fraunhofer Expansion of the Two Dimensional Point Spread Function 62The Transmission/Reception Point Spread Function 64Summary 64

Sampling and Image Processing 65Sampling Requirements in Two Dimensions 65Post-Detection Sampling Requirements 68Shift-Variance in Diagnostic Ultrasound 69Pre-Detection Image Processing 70



What is an Image?

Imaging as a Transformation

Object Field

Image Field

Imaging System

(1.1)

An image is a visual representation of a real object. An image is generally of reduceddimensionality and characterizes some of the object’s properties.

An image is thus a concrete abstraction of the object.



Desirable Characteristics of an Imaging System

• Superposition The characteristic by which an image of two objects is the sum of two images of theindividual objects, i.e., I o o I o I o1 2 1 2+{ } = { } + { } .This is a local and globalcharacteristic.

• Shift Invariance

A characteristic by which small displacements between an object field and animaging system result only in correspondingly small displacements of the resultingimage field, i.e., if I o ix x( ){ } = ( ) then I o ix x+( ){ } = +( )δ δ . This is a localcharacteristic.

• Image Uniformity

The global characteristic by which desirable properties are maintained throughout thefield of view.



Common Measures of Imaging System Performance

• Detail Resolution A measure of the minimum spacing of distinguishable point targets; this is a localcharacteristic.

• Contrast Resolution

A measure of the differentiability of a region of one echogenecity within a region ofdifferent echogenecity. This is also a local characteristic.

• Temporal Resolution

A measure of the number of independent images per unit time that can be acquired byan imaging system; this is a global characteristic.

• Dynamic Range

The ability to simultaneously resolve small targets in the presence of large targets.This is both a local and global measure.

• SensitivityThe ability of the system to detect low level echoes. This is a global property.



Canonical Ultrasound Imaging System

All modern diagnostic ultrasound systems form their images using beamformers andarrays. While there are exceptions to this generalization, we won’t investigate them inthis course.

The block diagram of a canonical imaging system using beamformers is shown below.

TransmissionBeamformer

ReceptionBeamformer

Pre-DetectionProcessing

Detection

Post-DetectionProcessing

Display TransducerArray

(1.2)

While we generally think of the image as that visual entity which appears on the screen,in this course we will take a somewhat narrower and more abstract view. We will focusour attention on beamformation and pre-detection processing, and we will usually thinkof the image as the abstract representation of the object field prior to detection. Thisabstract representation may or may not have a unified physical expression within anyspecific imaging system, but its characteristics are nonetheless central to the system’sperformance.

Linearity and shift-invariance are desirable characteristics of an imaging system. As wewill see, acoustic propagation and diffraction are linear and shift-invariant, and we arenaturally led to linear system theory to analyze, synthesize, and understand the process ofimage formation.

We start with the problem of focusing a linear acoustic aperture using geometrical optics,and deriving some basic geometrical relationships.



Apertures and Geometrical Optics

The Near Field/Far Field Crossover

Consider a uniform linear aperture of extent a excited with a monochromatic signalhaving wavelength λ . Consider a point target at a distance R from the aperture. Theradiation from each incremental part of the aperture arrives at the target according to thepropagation path length associated with the incremental aperture position and the targetlocation. We can analyze this with the following figure.

aR

R x2 2+

(2.1)

The differential path length ∆ associated with a point x on the aperture and a range R canbe evaluated using simple geometry.

∆ = + −

= +

−

≅

R x R

Rx

RR

x

R

2 2

2

2

1

2

(2.2)

This differential error across the aperture is thus essentially quadratic, and can be reducedarbitrarily be increasing R. That is, in the far field the radiation from each point on theaperture arrives (essentially) coherently, adding constructively. As we move the pointtarget closer to the aperture, the delay error increases inversely with R until, at somecrossover range RC between the near field and far field, it becomes non-negligible. Wedefine this range R RC= (rather arbitrarily) as that for which the maximum error is

∆ = λ 8 (2.3)

As the maximum error will always be associated with the ends of the aperture, wesubstitute x a= 2 and use ∆ = λ 8 in eq. (2.2) to obtain



R

a

aC =λ

(2.4)

That is, the crossover range, measured in apertures, is equal to the aperture, measured inwavelengths.

The far field is often called the Fraunhofer region, where we can ignore the differentialphase terms associated with different propagation lengths. The near field is often calledthe Fresnel region, characterized by the (approximately) quadratic phase attributable todifferent propagation lengths from different aperture points.

Let us calculate the near field/far field crossover of a practical ultrasonic apertureoperating at a center frequency of 3.5 MHz. Let

a mm

mm

==

28

44λ .

This aperture might have 128 elements spaced at λ 2 . The transition between theFraunhofer region and the Fresnel region occurs at

R mmC = 1782

or almost two meters! All modern diagnostic ultrasonic imaging occurs in the extremenear field. This distinguishes the discipline from many other imaging technologies andpresents a notable engineering challenge.

We note that there are many alternative definitions of RC possible, and they can be foundin the literature. Outside of a specific engineering context, they are all rather arbitrary.Our choice is consistent with a delay error of one eighth of a wavelength, which may bedevastating in some applications and benign in others. The designer of near field imagingsystems has to be constantly aware of systematic phase errors (and their sources, andtheir effects) across the aperture.



Azimuthal Resolution and the Rayleigh Criterion of a Linear Aperture

Consider again a uniform linear aperture of extent a excited with a monochromatic signalhaving wavelength λ . Consider a point target in the far field, moved around a circular arcof constant range. Because we are in the far field, there is negligible phase error when theangle of the target is zero (i.e., the radiation from each aperture point arrives in phase).But as we traverse the circular arc, the phase error increases until some parts of theaperture start contributing destructively.

R

R

R x Rx2 2 2+ − sinθ

θ a

(2.5)

The differential path length ∆ between an aperture point x and the center, as our pointtarget moves along the circular arc, is

∆ = + − −

= +

−

−

R x Rx R

Rx

R

x

RR

2 2

2

2

1 2

sin

sin

θ

θ(2.6)

Expanding this in terms of x R and θ

∆ ≅ − +

≅ −

xx

Rx

θ

θ

2

2 (2.7)

As in the last section, we ignore the quadratic term on the basis of far field operation.

Destructive interference occurs when the magnitude of any differential path length errorexceeds λ 4 , so we want to calculate the angular extent for which

− ≤ ≤λ λ4 4∆ (2.8)



As the maximum error will always be associated with the ends of the aperture, wesubstitute x a= ± 2 and use ∆ = ±λ 4 in eq. (2.7) to get the angles associated with thismaximum error.

θ λλ∆=± = ±4 2a

(2.9)

We define this angular extent as the angular resolution θR.

θ λR a= (2.10)

That is, the angular resolution, measured in radians, is the inverse of the aperture,measured in wavelengths.

It is also convenient to define the ratio of the range and the aperture, which re-occurs indifferent contexts, as the f-number.

fR

a# = (2.11)

The distance around the circular arc associated with θR is simply R Rθ , yielding theazimuthal resolution uR .

u fR = #λ (2.12)

That is, the azimuthal resolution, measured in wavelengths, is the f-number.

Let us return to eq. (2.7) to consider the diffraction pattern as we traverse an arc in the farfield. Let the arc length u R= θ . A differential path length of ∆ = −xθ will induce adifferential phase of − =2 2π λ π λ∆ xu R (the sign arising because a positive differentialpath length results in a later arrival time). If each incremental part of the aperture isradiating a complex exponential of the form exp j f t2 0π( ), then the ratio of the signalreceived at our point target to the total excitation signal is



dx j f t

dx j f t

dx j xu R

f dx j xu

u

f

a

a

a

aa

a

f

f

exp

exp

exp

exp

sinc

#

#

#

#

2

2

2

2

0

2

2

0

2

22

2

1

2

1

2

π λ

π

π λ

λ π

λ

λ

λ

−[ ]( )

( )= ( )

= ( )

=

−

−

−

−

∫

∫∫

∫

∆

(2.13)

This last integral is simply the (inverse) Fourier transform of f f x# #rectλ λ( ) , yielding thefamiliar sinc function for the far field diffraction pattern associated with a uniform linearaperture. Using eq. (2.13) to scale the abscissa, we see that f#λ is the distance from theorigin to the first zero; it is also a good measure of the width of the main lobe. It iscommonly used as a measure of the resolving power of an optical system, and in thatcontext is referred to as the Rayleigh criterion.

f#λ

u

(2.14)



Let’s consider an example. Suppose we have a reception aperture of 28mm operating atλ = .220mm , corresponding to 7 MHz. Immediately we see that our angular resolution inradians is θ λR a= = 1 128, corresponding to about .45° . If we have to scan through a90° field of view with this aperture, we will need about 200 scan lines.

We hasten to add that this is a first-order approximation. In particular we have ignoredthe obliquity factor (the foreshortening of the aperture by the cosine of the steering angle)and the effects of two-way operation (the extra resolution afforded by the transmissionbeamformer). While these (and other) effects can be accommodated, geometrical optics isultimately limited. Its principal virtue is its simplicity in concept and expression.



Depth of Focus of a Focused Linear Aperture

We can bring the far field diffraction pattern into the near field by focusing the aperture.This is done by applying compensating delays to incremental portions of the aperture. (Inthe figure below, we show the compensating delays as corrections to path length.)

a

R

R x2 2+− +R x2 2

−R

Compensating Delays

Focal

Point

(2.15)

As we move a target along a circular arc through the focal point, we will measure the sincfunction of fig. (2.14); this is because the compensating delays cancel, to first order, thedifferential delays associated with the near field. Our angular resolution is independent ofour focal range R, but our azimuthal resolution improves as we focus further into the nearfield (because our f-number decreases with R). There is, of course, a price to pay for thisimproved resolution. As we move radially away from the focal point, the compensatingdelays no longer match the actual delays, and our azimuthal resolution degrades. We canestimate this depth of focus by considering the delay errors across the aperture as wemove a point target a radial distance r from the focus.

a

R r−

R r x−( ) +2 2− +R x2 2

−R

Compensating Delays

r

Focal

Point

(2.16)

The differential delay between the center of the aperture and any other aperture point x(including the compensating delays) is



∆ = −( ) + − +[ ] − −( ) −[ ]

= −

+

− +

+

R r x R x R r R

Rr

R

x

R

x

R

r

R

2 2 2 2

2 2 2

1 1(2.17)

Expanding this in a two dimensional Taylor’s series in terms of r R and x R andkeeping the lowest order term yields

∆ = x r

R

2

22(2.18)

The delay error across the aperture is thus seen to be essentially linear in r and quadraticin x. As we move closer to the aperture, the sign of the error is positive, and further awayyields a negative error. Similar to our analysis of the near field/far field crossover, wedefine the depth of focus as the extent of the incremental range r for which

− ≤ ≤λ λ8 8∆ (2.19)

over the aperture. Substituting x a= 2 and ∆ = ±λ 8 into eq. (2.18), and using f R a# = ,we see that the depth of focus is bounded by

r f∆=± = ±λ λ82

# (2.20)

so the total depth of focus rD is

r f fD = ± =# #2 22λ λ (2.21)

That is, the one-sided depth of focus, measured in wavelengths, is the square of the f-number. Other definitions for rD can be found in the literature, based on criteria otherthan the λ 8 differential error we have employed here. Let us look at the effects ofmoving our point target to the edge of the depth of focus and beyond.



Let our target traverse an arc of radius R-r in front of an aperture focused at R.

θ

R r x R r x−( ) + − −2 2 2( ) sinθ

Focal

Pointa

R r−

− +R x2 2

−R

Compensating Delays

R r−

r(2.22)

The differential delay is

∆ = −( ) + − −( ) − +[ ] − −( ) −[ ]

= −

+

− −

− +

+

R r x R r x R x R r R

Rr

R

x

R

r

R

x

R

x

R

r

R

2 2 2 2

2 2 2

2

1 2 1 1

sin

sin

θ

θ(2.23)

Expanding this in a three dimensional Taylor’s series in terms of r R, x R and θ ,keeping the lowest order terms, and setting the arc length u R r= −( )θ yields

∆ ≅ − +−

+

≅ − +

xu

R

R r

R r

x r

R

xu

R

x r

R

2

2

2

2

2

2

(2.24)

The diffraction pattern can be determined as in eq. (2.13).

12 2 2 2 22 2

2

22

1

2

1

2

adx j xu R x r R f dx j x r j xu

a

a

f

f

exp exp exp#

#

#

π λ λ λ π λ π

λ

λ

−[ ]( ) = −( ) ( )− −

∫ ∫ (2.25)

This is the (inverse) Fourier transform of f f x j x r# #rect expλ λ π λ( ) −( )2 22 , and of coursewhen r=0 it reduces to eq. (2.13). Below we plot the log magnitude of eq. (2.25) fordifferent values of r.



Figure (2.26) is an idealized diffraction pattern of a focused linear aperture (i.e., a sincfunction), plotted in decibels, when r=0.

0

-10

-20

(2.26)

As the point target is moved to the edge of the depth of focus ( r f= ± #2λ ) the diffraction

pattern degrades due to the quadratic delay error peaking at ±λ 8.

0

-10

-20

(2.27)



When we displace the target radially by ±2 2f# λ (beyond the edge of the depth of focus),the quadratic delay error across our aperture peaks at ±λ 4.

0

-10

-20

(2.28)

If we displace our target to ±4 2f# λ (well out of the depth of focus) the diffraction patternbecomes severely distorted.

0

-10

-20

(2.29)



Depth of focus is a major engineering issue in the design of diagnostic ultrasoundsystems, because all modern systems use dynamic focusing on reception. This technologyrequires the compensating delays to be updated fast enough to focus on the echoes of apropagating pulse; each update has to be done in the round-trip propagation time of thedepth of focus. Let’s look at an example.

Consider a reception aperture working at f# = 2 and a center frequency of 5 MHz,corresponding to λ = .308mm . The depth of focus is thus 2 5. mm , and the round-trippropagation time (assuming the propagation velocity c mm s0 1 54= . µ ) is 3 2. µs . Thusevery 3 2. µs the aperture has to be re-focused in order to maintain the azimuthalresolution available with an f-number of two. If our reception beamformer has 128channels, this requires that a new channel be updated, on average, every 25ns.

More generally, if we are working with a center frequency of f0 , then a dynamic focused

aperture has to be re-focused at a rate of 2 2 420

1

02f c f f# #λ ⋅[ ] =

−.

The choice of 2 2f# λ for our depth of focus is an historically conservative one, andultimately has to be considered in a specific context. In ultrasound, the use of 4 2f# λ andeven 8 2f# λ is not uncommon, and such liberties certainly ease the engineeringdifficulties associated with dynamic focusing. For modern high resolution imaging,however, these measures are inadequate.



Duality

Consider our aperture of extent a focused at range R, yielding azimuthal resolutionu fR = #λ . There is a dual of this system, which has an aperture of extent uR focused atrange R yielding azimuthal resolution a, as can be readily verified. In fact, this dualityextends to other measures, as can be seen in the table below. Perhaps most interesting(and helpful to the memory) is that the depth of focus of our canonical systemcorresponds to the near field/far field crossover of our dual system.

Duality at Range R System DualAperture a uR

Azimuthal Resolution uR af-number f# θR

−1

Angular Resolution θR f#−1

Depth of Focus ± f#2λ ±a2 λ

Near Field/Far Field Crossover a2 λ f#2λ

R

R

Aperture a= Resolution = f#λ

Resolution = aAperture f= #λ

Depth of Focus f= #2λ

Crossover f= #2λ

Crossover a= 2 λ

Depth of Focus a= 2 λ

(2.30)



There is one range that has a special significance in this scheme: when we set R RC=(the near field/far field crossover) then our system becomes the dual of itself.

Duality at Range R=RC System DualAperture a aAzimuthal Resolution a af-number f# f#

Angular Resolution θR θR

Depth of Focus ±RC ±RC

Near Field/Far Field Crossover RC RC



Summary

The geometrical relationships of this section, although approximate, are rules of thumbthat all ultrasound engineers should know. They are applicable to virtually all moderndiagnostic ultrasound systems, and yield insights into visible aspects of theirperformance.

Geometrical Optics Summaryf-number f# R aAngular Resolution θR λ aAzimuthal Resolution uR f#λDepth of Focus rD ± f#

2λNear Field/Far Field Crossover RC a2 λ

However, the geometrical approach is ultimately limited as an analytical tool forunderstanding many issues of critical interest. For example, do our compensating delaysyield azimuthal resolution that is optimal in any sense? And how do we conceptualize thetransformation between object and image? For these sorts of questions we turn to Fouriertheory.



Linear Shift-Invariant Systems and Transforms

Representation of Signals in One Dimension 1,2,3

In one dimension, we can expand a function several ways, but there are two canonicalapproaches.

Expansion of a Signal in Dirac Delta Functions

We can think of an idealized sample of a function g t( ) at time τ as

g tτ δ τ( ) −( ) (3.1)

Integrating these contributions we obtain the original signal

g t d g t( ) = ( ) −( )−∞

∞

∫ τ τ δ τ (3.2)

by virtue of the characteristics of the delta function under integration. For a LSI system,knowing the impulse response h t( ) lets us infer the response for any input. Using thesymbol ⇒ to indicate the operation of a LSI system, with the input on the left and theoutput on the right, it follows that

δ τ ττ δ τ τ τ

τ τ δ τ τ τ τ

τ τ τ

t h t

g t g h t

d g t d g h t

g t d g h t

−( )⇒ −( )( ) −( )⇒ ( ) −( )

( ) −( )⇒ ( ) −( )

( )⇒ ( ) −( )

−∞

∞

−∞

∞

−∞

∞

∫ ∫

∫

(3.3)

In this way, we can think of the familiar convolution integral as the shifted, weightedsuperposition of impulse responses, in which the weights are the expansion coefficientsof the input signal.



Expansion of a Signal in Complex Exponentials. The Fourier Transform

The Fourier transform expresses a signal g t( ) as an expansion of complex exponentials.

g t df G f j ft( ) = ( ) ( )−∞

∞

∫ exp 2π (3.4)

where

G f dt g t j ft( ) = ( ) −( )−∞

∞

∫ exp 2π (3.5)

We refer to eq. (3.4) as the inverse Fourier transform and eq. (3.5) as the (forward)Fourier transform of the signal g t( ) .

Complex exponentials are eigenfunctions of LSI systems; that is, if we excite such asystem with a function of the form exp j ft2π( ) then the output is of the formH f j ft( ) ( )exp 2π , where H f( ) is a complex weight dependent on f but independent of t. Itfollows that

exp exp

exp exp

exp exp

exp

j ft H f j ft

G f j ft G f H f j ft

df G f j ft df G f H f j ft

g t df G f H f j ft

2 2

2 2

2 2

2

π π

π π

π π

π

( )⇒ ( ) ( )( ) ( )⇒ ( ) ( ) ( )

( ) ( )⇒ ( ) ( ) ( )

( )⇒ ( ) ( ) ( )

−∞

∞

−∞

∞

−∞

∞

∫ ∫

∫

(3.6)

These two expansions, in the context of LSI systems, are duals of each other. If we takethe Fourier transform of both sides of the first line of eq. (3.3)

ℑ −( ){ }⇒ ℑ −( ){ }− ⇒ ( ) −( )δ τ τ

π τ π τ

t h t

j f H f j fexp( ) exp2 2(3.7)

we get the first line of eq. (3.6) in its essentials. This is because shifted delta functionsand complex exponentials are Fourier transform pairs.



Some Properties of Fourier Transforms in One Dimension

Let h t( ) and H f( ) be Fourier transform pairs, and their relationship represented with thesymbol ⇔; likewise g t( ) and G f( ). The operator ∗ represents one dimensionalconvolution.

Reversal/Conjugationh t H f

h t H f

h t H f

h t H f

( ) ⇔ ( )−( ) ⇔ −( )( ) ⇔ −( )−( ) ⇔ ( )

∗ ∗

∗ ∗

(3.8)

Convolution/Multiplicationh t g t H f G f

h t g t H f G f

( ) ∗ ( ) ⇔ ( ) ( )( ) ( ) ⇔ ( )∗ ( )

(3.9)

Shift/Modulationh t a H f e

h t e H f a

j af

j at

−( ) ⇔ ( )( ) ⇔ −( )

− 2

2

π

π(3.10)

Power/Rayleigh’s Theorem

dt h t g t df H f G f

dt h t df H f

( ) ( ) = ( ) ( )

( ) = ( )

∗

−∞

∞∗

−∞

∞

−∞

∞

−∞

∞

∫ ∫

∫ ∫2 2

(3.11)

Definite Integral/Central Ordinate

dt h t H

h df H f

( ) = ( )

( ) = ( )

−∞

∞

−∞

∞

∫

∫

0

0

(3.12)

Similarity

ht

aa H af

a h at Hf

a

⇔ ( )

( ) ⇔

(3.13)



The Multidimensional Fourier Transform 1,4,5

Consider a multidimensional function represented in Cartesian coordinates as

h x x x hn1 2, ,K( ) = ( )x . Its Fourier transform and its inverse are

H d h j

dx dx dx h x x x j u x u x u xn n n n

u x x u x( ) = ( ) − ⋅( )= ( ) − + +[ ]( )∫∫

exp

, , exp

2

21 2 1 2 1 1 2 2

π

πK K K(3.14)

h d H j

du du du H u u u j u x u x u xn n n n

x u u u x( ) = ( ) ⋅( )= ( ) + +[ ]( )∫∫

exp

, , exp

2

21 2 1 2 1 1 2 2

π

πK K K(3.15)

where the integrations are taken over all space.

We can view these integral relationships as straightforward extensions of the one-dimensional case. Here, a multidimensional function is expressed as an expansion ofcomplex exponentials of the same dimensionality. Note that the function and itstransform may represent a signal, or the impulse response of a multidimensional system.

Separability

Let h x x xn1 2, ,K( ) be separable, such that

h x x x h x x x h x x xn m m m n1 2 1 1 2 2 1 2, , , , , ,K K K( ) = ( ) ( )+ + (3.16)

Then its Fourier transform is similarly separable.

H u u u H u u u H u u un m m m n1 2 1 1 2 2 1 2, , , , , ,K K K( ) = ( ) ( )+ + (3.17)

Rotational Symmetry

In the special case that h x( ) is rotationally symmetric, then H u( ) is likewise rotationallysymmetric. Both are representations of one independent variable, even though thefunctions are multidimensional.

h h r r

H H s s

x x

u u

( ) = ( ) =

( ) = ( ) =

ˆ ;

ˆ ;(3.18)



The Hankel Transform of a Rotationally Symmetric Function

Under the special condition of rotational symmetry, the one dimensional Hankeltransform is identically an n-dimensional Fourier transform. Using the notation of eq.(3.18), and where Jn ⋅( ) is the Bessel function of the first kind of order n, the transformfor a rotationally symmetric function is

ˆ ˆ JH s

s

dr r h r srn

n

n( ) = ( ) ( )− −

∞

∫22

21

2

21

0

π π (3.19)

ˆ ˆ Jh r

r

ds s H s srn

n

n( ) = ( ) ( )− −

∞

∫22

21

2

21

0

π π (3.20)

Radial Symmetry: The Hankel Transform of a Two Dimensional Function

Applying the above to a two-dimensional radially symmetric function, we get what issometimes called the Fourier-Bessel transform.

h x y h r r x y

H u v H s s u v

, ˆ ;

, ˆ ;

( ) = ( ) = +

( ) = ( ) = +

2 2

2 2(3.21)

ˆ ˆ JH s dr r h r sr( ) = ( ) ( )∞

∫2 20

0

π π (3.22)

ˆ ˆ Jh r ds s H s sr( ) = ( ) ( )∞

∫2 20

0

π π (3.23)

Spherical Symmetry: The Hankel Transform of a Three Dimensional Function

Applying the above to a three-dimensional spherically symmetric function yields

h x y z h r r x y z

H u v w H s s u v w

, , ˆ ;

, , ˆ ;

( ) = ( ) = + +

( ) = ( ) = + +

2 2 2

2 2 2(3.24)

ˆ ˆ sinH ss

dr r h r sr( ) = ( ) ( )∞

∫22

0

π (3.25)

ˆ ˆ sinh rr

ds s H s sr( ) = ( ) ( )∞

∫22

0

π (3.26)



Some Properties of Fourier Transforms in Multiple Dimensions

Several elementary properties are easily derived from the transform definition, and arestraightforward extensions of their one dimensional counterparts. The notation ∗

n denotes

n-dimensional convolution.

Reversal/Conjugationh H

h H

h H

h H

x u

x u

x u

x u

( ) ⇔ ( )−( ) ⇔ −( )( ) ⇔ −( )−( ) ⇔ ( )

∗ ∗

∗ ∗

(3.27)

Convolution/Multiplicationh g H G

h g H Gn

n

x x u u

x x u u

( )∗ ( ) ⇔ ( ) ( )

( ) ( ) ⇔ ( )∗ ( )(3.28)

Shift/Modulationh H e

h e H

j

j

x a u

x u a

a u

a x

−( ) ⇔ ( )( ) ⇔ −( )

− ⋅

⋅

2

2

π

π(3.29)

Power/Rayleigh’s Theorem

d h g d H G

d h d H

x x x u u u

x x u u

( ) ( ) = ( ) ( )

( ) = ( )

∗ ∗∫ ∫∫ ∫2 2

(3.30)

Definite Integral/Central Ordinate

d h H

h d H

x x 0

0 u u

( ) = ( )

( ) = ( )∫

∫(3.31)

Similarity

hx

a

x

a

x

aa a a H a u a u a u

a a a h a x a x a x Hu

a

u

a

u

a

n

nn n n

n n nn

n

1

1

2

21 2 1 1 2 2

1 2 1 1 2 21

1

2

2

, , , ,

, , , ,

K K K

K K K

⇔ ( )

( ) ⇔

(3.32)



Marginal Fourier Transforms in Multiple Dimensions

It is often useful in multidimensional Fourier analysis to transform with respect to someof the independent variables but not all of them. In so doing we obtain a marginaltransform. The properties of marginal transforms extend simply from the properties offull transforms, and are easily derived. We illustrate here with a two-dimensionalexample.

H u y dx h x y j ux

h x y du H u y j ux

x

x

, , exp

, , exp

( ) = ( ) −( )

( ) = ( ) ( )

−∞

∞

−∞

∞

∫

∫

2

2

π

π(3.33)

We augment our ⇔ notation to represent the marginal transform relationship.

h x y H u yx

x, ,( )⇔ ( ) (3.34)

The most important properties include:

Convolutionh x y g x y H u y G u y

x y xx

yx, , , ,

,( ) ∗ ( )⇔ ( )∗ ( ) (3.35)

Power

dx dy h x y g x y du dy H u y G u yx x, , , ,( ) ( ) = ( ) ( )∗

−∞

∞

−∞

∞∗

−∞

∞

−∞

∞

∫∫ ∫∫ (3.36)

Definite Integral

dx h x y H yx, ,( ) = ( )−∞

∞

∫ 0 (3.37)



Multidimensional Fourier Transforms in Space and Time; Plane Waves

In working with four dimensional functions of space and time, we will use x y z, , as thespatial variables, u v w, , as the corresponding spatial frequency variables, t as thetemporal variable, and f as the corresponding temporal frequency variable. Asappropriate, we will also use the vector notation x and u . The transform pairs are thus

h x y z t du dv dw df H u v w f j ux vy wz f t

H u v w f dx dy dz dt h x y z t j ux vy wz f t

, , , , , , exp

, , , , , , exp

( ) = ( ) + + +[ ]( )

( ) = ( ) − + + +[ ]( )−∞

∞

−∞

∞

−∞

∞

−∞

∞

−∞

∞

−∞

∞

−∞

∞

−∞

∞

∫∫∫∫

∫∫∫∫

2

2

π

π(3.38)

or, equivalently

h t d df H f j f t

H f d dt h t j f t

x u u u x

u x x u x

, , exp

, , exp

( ) = ( ) ⋅ +[ ]( )

( ) = ( ) − ⋅ +[ ]( )−∞

∞

−∞

∞

∫∫

∫∫

2

2

π

π(3.39)

The Fourier kernel exp ± + + +[ ]( )j ux vy wz ft2π is a plane wave, so a transformrepresents a function as an expansion of complex plane waves. Of course, not all planewaves will propagate in a medium—only those whose spatial wavelength and temporalfrequency are related by the propagation velocity of the medium.



The Multidimensional Dirac Delta Function

As a simple extension of the one-dimensional case, the multidimensional Dirac deltafunction also has the sifting property. Using vector notation

h d h nx x x x x00( ) = ( ) −( )∫ δ (3.40)

It also has a similar Fourier transform relationship

n jδ πx x u x0 0−( ) ⇔ − ⋅( )exp 2 (3.41)

To help secure some of the notation for the course, these relationships can be expressedin four dimensions of space and time. The sifting property of eq. (3.40) thus becomes

h x y z t dx dy dz dt h x y z t x x y y z z t t0 0 0 04

0 0 0 0, , , , , , , , ,( ) = ( ) − − − −( )−∞

∞

−∞

∞

−∞

∞

−∞

∞

∫∫∫∫ δ (3.42)

and the forward and inverse transforms of eq. (3.41) becomes

40 0 0 0

0 0 0 0

0 0 0 0

40 0

2 2

2

δ

π π

π

δ

x x y y z z t t

du dv dw df j ux vy wz f t j ux vy wz f t

j ux vy wz f t

dx dy dz dt x x y y z

− − − −( ) =

− + + +[ ]( ) + + +[ ]( )

− + + +[ ]( ) =

− −

−∞

∞

−∞

∞

−∞

∞

−∞

∞

∫∫∫∫

, , ,

exp exp

exp

, , −− −( ) − + + +[ ]( )−∞

∞

−∞

∞

−∞

∞

−∞

∞

∫∫∫∫ z t t j ux vy wz f t0 0 2, exp π

(3.43)

Expressed in all their glory, these relationships can appear quite intimidating (which isone good reason for resorting to vector notation). Its important to keep in mind that theserelationships are, in fact, no more complex than their familiar one-dimensionalcounterparts, and that one should always focus on the underlying geometry of theproblem, as it is often simpler than its analytical expression.



Acoustic Propagation and Reflection

The Scalar Wave Equation

The driven scalar wave equation governing the propagation of energy in an idealizedmedium is

∇ ( ) − ( ) = ( )2

02

2

2

1φ ∂∂

φ ψx y z tc t

x y z t x y z t, , , , , , , , , (4.1)

The scalar φ x y z t, , ,( ) is usually taken to be pressure or the velocity potential, andψ x y z t, , ,( ) is the driving function. Taking the marginal Fourier transform with respect totime yields the Helmholtz equation

∇ ( ) + ( ) = ( )22 2

02

4φ π φ ψf f fx y z ff

cx y z f x y z f, , , , , , , , , (4.2)

Propagation Velocity and Losses 6

A commonly accepted idealized velocity of propagation for diagnostic ultrasound is 1540m/s. Adding viscosity to the medium introduces losses that are small but non-negligible.A common (and quite rigorous) technique for introducing this effect analytically is todefine the velocity of propagation as a complex quantity, dependent on the frequency fand conjugate symmetric over f. The simplest model for diagnostic ultrasound is

1

11 0

10

11 0

0

0

0

c

cj f

cf

cj f

=

−( ) >

=

+( ) <

α

α

;

;

;

(4.3)

where c0 is real and α ≥ 0 is the loss tangent of the medium. The conjugate symmetryensures that a real forcing function ψ x y z t, , ,( ) will excite a real propagating functionφ x y z t, , ,( ). The Helmholtz equation thus becomes

∇ ( ) + ( ) = ( )22 2

2

4φ π φ ψf f fx y z ff

cx y z f x y z f, , , , , , , , , (4.4)

For soft biological tissue in the frequency range commonly used in diagnostic ultrasound,a useful approximation is α ≅ × −3 10 3.



Propagation as a Linear Shift Invariant System 7

The scalar wave equation is linear and shift invariant in both time and space. We canthink of acoustic propagation as a LSI system, and use multidimensional Fouriertechniques for analysis, insight, and inspiration.

In taking the approach to follow, we make a number of important simplifyingassumptions:

• The propagation medium is homogeneous • The only non-negligible boundary condition is the forcing function • The only scattering is low-level

It is certainly easy to find specific imaging situations where one or more of theseassumptions are violated, but the success of diagnostic ultrasound is evidence of theirgeneral applicability. More importantly, deep understanding of the simple cases willafford insights into the more complex ones.

Following the development we used for the one-dimensional LSI case, we argue thatknowledge of the impulse response g x y z t, , ,( ) or the transfer function G u v w f, , ,( )(which are Fourier transform pairs) of propagation in the medium of interest will let uscharacterize the response to any spatio-temporal input.

Propagationψ x, y, z,t( )

( )

Ψ u v w f, , ,

φ x, y, z,t( )( )

Φ u v w f, , ,

g x y z t

G u v w f

, , ,

, , ,

( )( )

(4.5)

Let us start by considering the lossy Helmholtz equation. If the forcing function is a four-dimensional delta function in time and space, the marginal transformation to temporalfrequency results in a three-dimensional delta function. By definition, the output in thiscase is the marginal impulse response g x y z ff , , ,( ).

∇ ( ) + ( )= ( )22 2

234

g x y z ff

cg x y z f x y zf f, , , , , , , ,

π δ (4.6)

Taking the three-dimensional Fourier transform of each side yields

− − −( ) ( ) +

( ) =4 4 4 4 12 2 2 2 2 2 2

2

π π π πu v w G u v w ff

cG u v w f, , , , , , (4.7)



Recognizing the spherical symmetry, we let s u v w2 2 2 2= + + , which yields

G s ff

cs

,( ) =

−

1

4 22

2π(4.8)

It follows that the marginal transform g x y z ff , , ,( ) will also exhibit spherical symmetry,so we can use the inverse Hankel transform for three dimensions, eq. (3.26).

g r fr

ds s G s f srf , , sin( ) = ( ) ( )∞

∫22

0

π (4.9)

Formal convergence of this integral occurs only for f c having an imaginary component,thus moving the poles of G s f,( ) away from the real axis in the complex f plane. Thisrepresents a loss mechanism in any realizable system, and we use eq. (4.3) in thisdevelopment to model it. Eq. (4.9) can then be evaluated using contour integration.

g r fj

f

cr

r

jf

cr

rf r c

f ,exp

exp

exp

( ) = −−

= −−

⋅ −( )

2

4

2

420

0

π

π

π

ππ α

(4.10)

This model thus presents a separable term exp −( )2 0π αf r c representing the loss of the

system. This corresponds to about 55α dB/wavelength; when α = × −3 10 3, the loss isabout 1dB/cm MHz, a commonly reported value in diagnostic imaging.

Taking the inverse Fourier transform on eq. (4.10) yields the impulse response of thepropagating medium represented by the scalar wave equation.

g r tt r c

r

r c

r c tt,( ) = −

−( )∗ ( )( ) +

δπ π

αα

0 0

0

2 241

(4.11)

We refer to the second function as an attenuation filter. Using the definite integralproperty of Fourier transforms, we see that the area under the attenuation filter is unity,regardless of α . It similarly follows from Fourier considerations that

limα π

αα

δr

r c

r c tt

→

( )( ) +

= ( )

0

0

0

2 2

1(4.12)



Thus in the limit of negligible loss

g r tt r c

r,( ) = −

−( )δπ

0

4(4.13)

This impulse response is often referred to as the free space Green’s function. Note that itexists only for t ≥ 0, this causality arising from the loss mechanism we introduced.

Having argued that propagation loss can be modeled as a linear filter in space and time,let us set the issue of loss aside and consider the underlying characteristics of losslesspropagation. Note that eq. (4.13) is four dimensional, implying that, in addition to itsFourier transform, there are a total of fourteen marginal transforms. Spherical symmetrymakes all but six of these redundant, and it is instructive to organize them in a cycle.

In the tables below, time and space are the variables in the lower right, and temporalfrequency and spatial frequency occupy the upper left. Each function has as its neighborsits marginal transform taken with respect to one independent variable. The second table isthe same as the first, but with the equations normalized for greater simplicity.

This cycle is not intended to model a particular medium, but rather to abstract out theprocess of propagation so we can consider it in its essentials. Indeed, these relationshipsexist only as mathematical abstractions; there is no way to generate an acoustic impulse,nor is there any medium that can support one. However, when applied as convolutionkernels and filter operators, these functions have all the familiar power of impulseresponses and transfer functions.



The Fourier Transform Cycle of Ideal Propagation

G s ff

cs

j

sf

cf

c

G s tc c st

st

G s z f

j jf

cz

sc

f

t

z

, ,sin

ˆ, ,

expˆ

( ) =

−

+−

( ) = − ( ) ( )

( ) =

− −

1

4 8

2

2

2 1

2

0

2

2

0

0

0 0

0

0

2

π

δ

π

ππ

µ

π

−

( ) = −

−

( )

( ) = −

4 1

2 1

2 2

2

0

0

2

0 00

2

0

0

π

π

µ

π

f

c

sc

f

G s z t

c c stz

c t z

c tt

g r u f

jf

c

z t

u f

ˆˆ, ,

J ˆ

rect

˜, ,

K

,

0

,

0 ˜

˜, ,

cos˜

˜rect

˜

,

exp

ruc

fg r u t

c utr

c t

tr

c t

r

c tt

g r f

j

u

f

1

2

2 1

2 12

2

0

2

00

2

0

20

−

( ) = −

−

−

( )

( ) = −−

π

π

π

µ

ππ

π

δ

π

π π

f

cr

rg r t

tr

c

r

Notes

jf

cz

sc

fj

f

cr

uc

f

0 0

0

0

2

0

0

2

4 4

2 1 2 1

( ) = −

−

− −

−

,

•ˆ

˜

• This is the causal impulse response of the lossless scalar wave equation.

In the evanescent regime, the real part of and of are positive.

• The function The function is the Dirac delta function. The function is the Heaviside step function. J is the Bessel

function of the first kind, order zero. K is the modified Bessel function of the second kind, order zero.

•

0

0

δ µ⋅( ) ⋅( ) ⋅( )⋅( )

= + +

= + +

= +

= +

r x y z

s u v w

r y z

s u v

2 2 2 2

2 2 2 2

2 2 2

2 2 2

•

• ˜

• ˆ



The Normalized Fourier Transform Cycle of Ideal Propagation

G s ff s

js f

fG s t

st

st

G s z f

j j f zs

f

fs

t

z

, ˆˆ

ˆ

ˆˆ , ˆ

sin ˆˆ

ˆ, , ˆ

exp ˆ ˆˆ

ˆ ˆˆ

( ) =−[ ] +

−( ) ( ) = −( ) ( )

( ) =

− −

−

1

4 8

2

2

2 1

4 1

2 2 2

2

π

δ

πππ

µ

π

πff

G s z t

stz

t z

tt

g r u f

j f ru

f

z t

u f

( ) = −

−

( )

( ) = −

−

2

2

2

2 1

2 2

2 1

2

ˆ ˆ, , ˆ

J ˆ ˆˆ

rectˆ

ˆ

˜, , ˆ

K ˆ˜ˆ

ˆ

,

0

,

0

π

µ

π

πgg r u t

utr

t

tr

t

r

tt

g r fj f r

rg r t

t r

u

f

˜, , ˆ

cos ˆ ˜ˆ

ˆ ˜ˆ

rect˜ˆ

ˆ

, ˆexp ˆ

ˆ , ˆˆ

( ) = −

−

−

( )

( ) = −−( ) ( ) = −

−( )

2 1

2 12

2

4

2

2

π

π

µ

π

πδ

44

2 1 2 1

00

2 2

π

π π

δ µ

r

Notes

ff

ct c t

j f zs

fj f r

u

f

• The Fourier cycle is normalized such that and , and the similarity theorem applied.

• In the evanescent regime, the real part of and of are positive.

• The function is the Dirac delta function. The function is the Heaviside step function. J is the Bessel0

ˆ ˆ

ˆ ˆˆ

ˆ ˜ˆ

= =

− −

−

⋅( ) ⋅( ) ⋅( ) function of the first kind, function of the first kind, order zero. K is the modified Bessel function of the second kind, order zero.

•

0 ⋅( )= + +

= + +

= +

= +

r x y z

s u v w

r y z

s u v

2 2 2 2

2 2 2 2

2 2 2

2 2 2

•

• ˜

• ˆ



Low-Level Scattering 8

Consider an illumination wavefield γ x y z t, , ,( ) propagating in a medium containing adiscontinuity of arbitrarily small dimensions at location x y z0 0 0, , . The discontinuity will

induce a scattered wavefield φ x y z t x y z, , , , ,0 0 0( ) that will appear as originating from a

point source of the form

ψ γ δx y z t x y z x y z t o x y z t x x y y z zt

, , , , , , , , , , , , ,0 0 0 0 0 0 0 0 03

0 0 0( ) = ( )∗ ( ) − − −( ) (4.14)

That is, our point discontinuity becomes an induced source, radiating a weighted andtemporally filtered version of its illumination. The resulting scattered wavefield is simplythe convolution of the induced point source ψ ⋅( ) with the impulse response (i.e., the freespace Green’s function) g ⋅( )

φ γx y z t x y z x y z t o x y z t g x x y y z z tt t

, , , , , , , , , , , , , ,0 0 0 0 0 0 0 0 0 0 0 0( ) = ( )∗ ( )∗ − − −( ) (4.15)

Low-level scattering is the assumption that the illumination wavefield is substantiallyunchanged by the presence of scatters and scattering. In that case, we can associate adistinct weight o x y z t0 0 0, , ,( ) to every point in space and superpose the distinct responses.The total scattered wavefield is

φ φ

γ

x y z t dx dy dz x y z t x y z

dx dy dz x y z t o x y z t g x x y y z z tt t

, , , , , , , ,

, , , , , , , , ,

( ) = ( )= ( )∗ ( )∗ − − −( )

∫∫∫∫∫∫

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

(4.16)

whence

φ γx y z t x y z t o x y z t g x y z tt x y z t

, , , , , , , , , , , ,, , ,

( ) = ( )∗ ( )[ ] ∗ ( ) (4.17)

In this formulation, the object field o x y z t, , ,( ) is a four-dimensional entity representingacoustic reflectivity under the assumption of low-level scattering. The challenge ofacoustic imaging is to come up with a representation of the object field, generally in twodimensions, that has utility.



We may use various marginal transforms on eq. (4.17) for different perspectives on theprocess. Taking the marginal transform with respect to time yields

φ γf f fx y z

fx y z f x y z f o x y z f g x y z f, , , , , , , , , , , ,, ,

( ) = ( ) ( )[ ] ∗ ( ) (4.18)

To see how this particular relationship might be used, consider the problem ofdetermining the scattered response at a single point in space (where we might have asmall transducer); without loss of generality let our receiver point be the origin. Then eq.(4.18) becomes

φ γf f f ff dx dy dz x y z f o x y z f g x y z f0 0 0, , , , , , , , , , , ,( ) = ( ) ( ) ( )∫∫∫ (4.19)

Taking a further marginal transform with respect to x and y

Φ Γz zu v

zz

zu v z f u v z f O u v z f G u v z f, , , , , , , , , , , ,,

( ) = ( ) ∗ ( )

∗ ( ) (4.20)

Suppose we have a planar receive aperture in the plane z=0. Eq. (4.20) reduces to

Φ Γz zu v

z zu v f dz u v z f O u v z f G u v z f, , , , , , , , , , , ,,

0( ) = ( ) ∗ ( )

( )∫ (4.21)



Acoustic Sources, Sensors, and Boundary Conditions 7

The transmission (and the reciprocal process, reception) of acoustic signals requirestransducers coupled to the propagation medium. Their transfer function is the subject ofscalar diffraction theory. Recall that Green’s theorem and the Sommerfeld radiationcondition lead to the following integral equation

φφ

φff

f ffx y z f dx dy

n n, , ,( ) = ∂

∂−

∂∂

−∞

∞

−∞

∞

∫∫ ΓΓ

(4.22)

describing radiation from a source in the plane z=0. Here Γf is an artfully chosenGreen’s function and ∂ ∂n is the spatial derivative normal to the transducer plane, i.e., inthe z direction. Two Rayleigh-Sommerfeld canonical equations, derived using twodifferent Green’s functions, are well-known. We write them here as two dimensionalconvolutions, and use pressure as the scalar field function.

p x y z fp x y z f

z

jf

cr

rff

zx y

, , ,, , ,

exp

,( ) = ∂ ( )

∂

∗ −

−

=

2

2

40

0

π

π(4.23)

p x y z f p x y f

jf

cr

r

z

r

j f r c

rf fx y

, , , , , ,

exp

,( ) = ( )[ ] ∗ −

−

⋅ +

2 0

2

41 20 0

π

ππ

(4.24)

(Note that corresponding expressions in the literature often have an inconsequentialreversal of the sign of f.)

The two equations are, of course, equally valid. The choice of one expression or the otherin a specific application is usually governed by convenient assumptions about thecharacteristics of the planar surface from which the signal is transmitted9, or by the needto match empirical data.10

Specifically, consider the case of a rigid plane, such that the normal derivative of pressure(proportional to the temporal derivative of the normal velocity) is zero beyond thephysical limits of the forcing function. Then the natural expression to use is eq. (4.23),and the resulting convolution limits are the physical limits of the forcing function. In thiscase, the forcing function is usually the temporal derivative of the normal velocity at thetransducer face.



Alternatively, consider the plane of interest as a pressure release surface, on which thepressure is zero beyond the physical limits of the forcing function. Then eq. (4.24) is themore natural choice, and the forcing function is the pressure at the transducer face.

For our subsequent development we will always use eq. (4.23), because the convolutionkernel is the free space Green’s function. This unifies our approach to propagation,scattering, transmission, and reception.

In so doing we make no assumptions about the impedance of the transducer plane, butrather recognize that the driving function will not, in general, be bound by the physicaldimensions of the transducer. To be more specific on the last point, imagine an array oftransducer elements in the plane z=0. For any one element driven electrically with a unit-amplitude complex sinusoid with frequency f, we can associate some elemental functionw x y f, ,( ) that will serve in the Rayleigh-Sommerfeld convolution integral of eq. (4.23).This association depends on the details of the transducer design, and will be knowableempirically if not theoretically. Any realizable aperture function will then be comprisedof weighted sums of these elemental functions.



Backpropagation and the Angular Spectrum 7,11

Consider the lossless Helmholtz equation in a region of space which includes no sources.

∇ ( ) + ( ) =22 2

02

40φ π φf fx y z f

f

cx y z f, , , , , , (4.25)

We know that this equation is satisfied by any plane wave of the form

exp ;j ux vy wz ft u v wf

c2 2 2 2

0

2

π + + +[ ]( ) + + =

(4.26)

The terms uc

f

vc

f

wc

f0 0 0, ,

are the direction cosines, specifying the direction of

propagation. When the sign of w and f are the same, propagation is in the negative zdirection; when the same sign of w and f are opposite, propagation is in the positive zdirection.

We want to know the wavefield φ ⋅( ) in an arbitrary plane parallel to z=0 givenknowledge of it in the plane z=0 (perhaps by measuring it with a planar array). If we takethe marginal transform of eq. (4.25) with respect to x and y we get

∂∂

( ) + − − +

( ) =

2

22 2 2 2

2 2

024 4

40

zu v z f u v

f

cu v z fz zΦ Φ, , , , , ,π π π

(4.27)

The general solution to this is straightforward. Recognizing the radial symmetry, wesubstitute s u v2 2 2= + .

Φz u v z f A jf

cz

sc

fB j

f

cz

sc

f, , , exp

ˆexp

ˆ( ) = −

+ − −

2 1 2 1

0

0

0

0π π (4.28)

The first term corresponds to waves propagating in the negative z direction, and thesecond term to the positive z direction. This can be demonstrated by taking the Fouriertransform of both sides; the right side of the equation will consist of two delta functions,the first constraining w to be the same sign as f, and the second constraining w to be theopposite sign as f. Hence our solution is



ΦΦ

z

z

u v z f

u v f

jf

cz

sc

f

jf

cz

sc

f

, , ,

, , ,

expˆ

expˆ

;

( )( ) =

−

− −

0

2 1

2 1

0

0

0

0

π

π

; propagation towards negative z

propagation towards positive z

(4.29)

This a fascinating result. It says that if we have knowledge of a wavefield (and itsdirection of propagation relative to the z axis) at z=0, and we characterize the wavefieldin terms of its two spatial frequencies and one temporal frequency, then we can determinethe wavefield on any parallel plane, thus all (source-free) space. That is, one of thedimensions of the four dimensional representation of the wavefield is redundant by virtueof the wave equation.

The propagation filter expˆ

jf

cz

sc

f2 1

0

0π −

is often used for backpropagation; that

is, we characterize (perhaps measure) at z=0 a wavefield originating in positive halfspace and multiply by this filter to determine the wavefield at an earlier time and agreater distance. The filter is dispersive (non-linear phase) and all-pass up for s f c≤ 0 .

We refer to this regime as the propagation regime. For s f c> 0 the exponent becomesnegative real, and the amplitude rolls off extremely quickly. This regime is theevanescent regime. In most practical applications, the response in the evanescent regimeis neglected.



The Pulse Echo Equation for a Fixed Focus Planar Transducer

Let’s put all this together. Consider an imaging system consisting of a planartransmission aperture excited by a pulse, illuminating an object field consisting of low-level scatterers, and a planar reception aperture followed by a filter.

The pulse echo equation can be formulated with a distinct beamformation perspective, inthat the transmission and reception apertures are specified with weights w x y ft , ,( ) andw x y fr , ,( ) respectively. We consider w x y ft , ,( ) and w x y fr , ,( ) fixed.

Illumination Wavefield

γ f t tx y

fx y z f H f w x y f g x y z f, , , , , , , ,,

( ) = ( ) ( ) ∗ ( ) (4.30)

Reflected Wavefield

φ γf f fx y z

fx y z f x y z f o x y z f g x y z f, , , , , , , , , , , ,, ,

( ) = ( ) ( )[ ] ∗ ( ) (4.31)

Sensed Wavefield

φ γf f fx y z

fz

x y f x y z f o x y z f g x y z f, , , , , , , , , , , ,, ,

00

( ) = ( ) ( )[ ] ∗ ( )

=

(4.32)

Received Signal

R f H f dx dy w x y f x y fr r f( ) = ( ) ( ) ( )∫∫ , , , , ,φ 0 (4.33)

The Pulse Echo Equation

Making the substitutions and denoting H f H f H fr t r t( ) ( ) = ( ) yields

R f H f dx dy dz o x y z f w x y f g x y z f w x y f g x y z fr t f rx y

f tx y

f( ) = ( ) ( ) ( ) ∗ ( )

( ) ∗ ( )

∫∫∫ , , , , , , , , , , , , ,

, ,(4.34)

Eq. (4.34) can be rewritten simply as

R f dx dy dz o x y z f s x y z ff f f( ) = ( ) ( )∫∫∫ , , , , , , (4.35)

where

s x y z f H f w x y f g x y z f w x y f g x y z ff r t rx y

f tx y

f, , , , , , , , , , , , ,, ,

( ) = ( ) ( ) ∗ ( )

( ) ∗ ( )

(4.36)



This last equation is an expression of the response of the system to a spatial impulse atlocation x,y,z. There are three simple components to s x y z ff , , ,( ) .

• H fr t ( ) : The round-trip pulse, in the temporal frequency domain.

• w x y f g x y z ftx y

f, , , , ,,

( ) ∗ ( )

: The wavefield associated with transmission.

• w x y f g x y z frx y

f, , , , ,,

( ) ∗ ( )

: The spatial sensitivity associated with reception.

This expression of the pulse-echo response is quite general, and can be used as the basisfor analyzing a wide variety of acoustic imaging systems, including synthetic aperture,fixed-focus lenses, dynamic-focus beamformers, and others. We will use it as the basisfor our study of beamformer-based imaging systems.

In this context, we see the latter two components of s x y z ff , , ,( ) as identical formulationsof one-way beamformation. In the next section we take a look at one-way beamformationto gain some insight into the appropriate aperture weights to use.



One-Way Beamformation and Spatial Filtering

Ultrasonic beamformation for diagnostic imaging is first a science, and ultimately an art.We will explore some of the former aspects of beamformation with the goal of providingcontext and inspiration for the latter. We start by considering the problem of one-waybeamformation, with the goal of imaging a point source.

Reception Beamformation as Matched Filtering

Consider a point source, real or induced, at x y z0 0 0, , that transmits or scatters a signalh tt ( ). The resulting wavefield will be

φ δx y z t h t x x y y z z g x y z ttx y z t

, , , , , , , ,, , ,

( ) = ( ) − − −( )[ ] ∗ ( )30 0 0 (5.1)

or, taking the marginal Fourier transform with respect to time and using vector notation

φππf tf H f

j f cx

x x

x x0

0

,exp

( ) = ( ) −− −( )

−

2

40

(5.2)

Assume we have an array of sensors, or an aperture that lets us sense the wavefield atsome locations in space. Further assume that each sensor or incremental portion of theaperture introduces additive uncorrelated white noise to the signal representing the sensedwavefield. We can pose the following problem. Given the sensor locations and thewavefield of eq. (5.1), determine the optimum spatio-temporal linear filter to maximizethe signal-to-noise ratio (SNR) at filter output x y z t x y z, , , , , ,( ) = ( )0 0 0 0 .

Denote the impulse response of the candidate filter a x y z tM , , ,( ). The convolution integralcorresponding to the point for which we want to maximize the SNR is

φ φ

φ

x x x x x x

x x x x

x0

0

0

, , , ,

, ,

, , , ,

t a t d dt t a t

d df f a f

x y z t

M M

f fM

( ) ∗ ( )

= ( ) − −( )

= ( ) −( )

∫ ∫

∫ ∫0 (5.3)

where the spatial integration is taken over the location of the sensors. This is simply themultidimensional form of the matched filter problem. Using Schwartz’ inequality and thecharacteristics of the additive noise, it follows that SNR is maximized when, within anarbitrary scale factor,

a f ffM

fx x x0 −( ) = ( )∗, ,φ (5.4)

whence



a f H fj f c

fM

txx

x,

exp( ) = − ( ) ( )∗ 2

40π

π(5.5)

If we interpret eq. (5.3) as the recipe for forming a beam at the point x0 (i.e., delay, filter,and sum), we see that the matched filter beamformation weights are

w f H fj f c

Mtx x

x x

x x00

0

,exp( ) = − ( )

−( )−

∗ 2

40π

π(5.6)

which can be described operationally as follows:

• To each signal acquired at location x introduce a time advance equal to x x0− c0

and an amplitude weight equal to 1 4π x x0−( ) .

• Sum all such time-advanced and weighted signals.• Temporally filter the resulting sum with h tt

∗ −( ), i.e., the filter matched to h tt ( ).• Sample at t=0.

Reception Beamformation as Backpropagation 12

Let us now suppose that our sensing aperture is planar, and is in the plane at z=0, so thesensed signal is φ x y t u v f

x y tz, , , , , ,

, ,0 0( )⇔ ( )Φ . An alternative approach to imaging the point

source at x y z0 0 0, , is to backpropagate the received signal to the plane at z z= 0 and toevaluate the result at x y0 0, . We take the point source to be in positive half space, so thesensed wavefield is propagating towards negative z and we only have to be concernedwith non-negative z.

Because we want to cast backpropagation as a filtering operation, the output of which isultimately sampled at a point in time (chosen arbitrarily to be t=0), we introduce abackpropagation filter a x y z ff

B , , ,( ) such that

φ φx y t a x y z t df x y f a x y z f

df u v f A u v z f

x y t

Bf

x yfB

u v z fB

, , , , , , , , , , , ,

, , , , , ,

, , , ,

,

0 0

0

0

1

( ) ∗ ( )

= ( ) ∗ ( )

= ℑ ( ) ( ){ }[ ]∫

∫ −

x x

x

0 0

0

Φ(5.7)

By inspection we can associate A x y z ffB , , ,( ) with the backpropagation operator of eq.

(4.29). For generality we also introduce a reception filter H fr ( ) . Thus



a x y z f H f jf

cz

sc

f

H fz

G s z f

H fz

g x y z

fB

r u v

r u v z

r f

, , , expˆ

ˆ, ,

, ,

,

,

( ) = ( ) ℑ −

= ( ) ℑ ∂∂

( )

= ( ) ∂∂

− −

−

− ∗

∗

1

0

0

2

1

2 1

2

2

π

,,

exp

f

H f z jf

cx y z

jf

cx y z

x y zr

( )[ ]

= ( ) + + −

+ +

+ +( )2 2 1

2

40

2 2 2 0

2 2 2

2 2 23

2π

π

π

(5.8)

Note that this filter, as defined, operates in the propagation regime; in the evanescentregime where s f c> 0 , it serves to further attenuate the signal. The backpropagationfilter can thus be written

a f H fz j f c j f c

fB

rxx

xx

x

x,

exp( ) = ( ) −

( )2 1 2 2

40 0π π

π(5.9)

where x = + +x y z2 2 2 . Substituting this in the convolution integral of eq. (5.7) andinterpreting it as a beamforming integral, we see the backpropagation beamformationweights are

w f H fz j f c j f c

Brx x

x x

x x

x x

x x

x x00

0

0

0

0

,exp( ) = ( )

−− −

−

−( )−

2 1 2 2

40 0 0π π

π

which can be described operationally as:

• To each signal acquired at location x introduce a time advance equal to x x0− c0 ,

an amplitude weight equal to z0

32π x x0− , and a filter with a frequency response of

H f j f cr ( ) − −( )1 2 0π x x0 .

• Sum all such time-advanced, weighted, and filtered signals.• Sample at t=0.

This is essentially, for most situations of interest, the matched filter case with H ft∗( )

replaced by the combination of a differentiating filter j f2π and a reception filter H fr ( ) .The backpropagation approach offers no guidance as to the desirable characteristics ofthe reception filter.



Reception Beamformation as Inverse Filtering

Let our sensing aperture once again be planar in the plane at z=0. A third approach toimaging the point source at x y z0 0 0, , is to use an inverse filter to reverse the effects ofpropagation. We again take the point source to be in positive half space. Our sensedsignal is φ x y t u v f

x y tz, , , , , ,

, ,0 0( )⇔ ( )Φ where

φ δx y z t h t x x y y z z g x y z ttx y z t

, , , , , , , ,, , ,

( ) = ( ) − − −( )[ ] ∗ ( )30 0 0 (5.10)

from which follows, taking the marginal Fourier transform with respect to time and thetwo spatial coordinates x and y

Φz t z

t

u v f H f G u v z f j ux vy

H f

j jf

cz

sc

f

f

c

sc

f

j ux vy

, , , , , , exp

expˆ

ˆexp

0 2

2 1

4 1

2

0 0 0

00

0

2

0

0

2 0 0

( ) = ( ) ( ) − +[ ]( )

= ( )− −

−

− +[ ]( )

π

π

π

π(5.11)

(We subsequently ignore the distinction between z0 and z0 because of the positive half

space constraint.) Let us first introduce a temporal reception filter H fr ( ) that operatesacross the aperture; we thus replace H ft ( ) with H f H f H fr t rt( ) ( ) = ( ).

Note that eq. (5.11) is phase linear in z, which suggests a Fourier transform if we identifythe correct change of variables. We do this as follows.

df u v f H fj

f

c

sc

ff c s w

H c s w j ux vy wz

z r

rt

Φ , , ,ˆ

sgn

sgn exp

04

1

2

0

0

2

0

0 0 0 0

( ) ( ) −

− ⋅ { }( )

= ⋅ { }( ) − + +[ ]( )∫ π δ

π

(5.12)

Here, s is the radial frequency variable in spherical coordinates, such thats u v w s w2 2 2 2 2 2= + + = +ˆ . Eq. (5.12) can be verified by direct substitution, taking carewith the signs of f and w. It is remarkable in that the right side of the equation (in thelimit as H frt ( ) is broadband and zero phase) is the three dimensional Fourier transform

of 30 0 0δ x x y y z z− − −( ), , , which is our original source.

To identify the associated inversion filter, we take the inverse marginal Fourier transformof the left side of eq. (5.12).



ℑ ( ) ( ) −

− ⋅ { }( )

= ( ) ∗ℑ ( ) −

−

−

∫

∫

u v w z r

fx y

u v w r

df u v f H fj

f

c

sc

ff c s w

df x y f H fj

f

c

sc

f

, ,

,, ,

, , ,ˆ

sgn

, , ,ˆ

1

0

0

2

0

1

0

0

04

1

04

1

Φ π δ

φ π

− ⋅ { }( )

2

0δ f c s wsgn

(5.13)

The inversion filter is thus

a f H fj

f

c

sc

ff c s wf

Iu v w rx,

ˆsgn, ,( ) = ℑ ( ) −

− ⋅ { }( )

−1

0

0

2

0

41

π δ (5.14)

Taking the inverse Fourier transform with respect to w, using the sifting properties of theDirac delta

a f H fj

f

cj

f

cz

sc

ffI

u v rx, expˆ

,( ) = ℑ ( ) −

−1

02

0

0

24

2 1π π (5.15)

But this is just the backpropagation filter multiplied by some new temporal frequencyterms. Using eq. (5.9)

a f H fz j f

c

j f c j f cfI

rxx

xx

x

x,

exp( ) = − ( )

−

( )4 2 1 2 2

402

0 0π π ππ

(5.16)

The inversion beamformation weights are thus

w f H fz j f

c

j f c j f cI

rx xx x

x x

x x

x x

x x00

0

0

0

0

,exp( ) = − ( )

−

− −−

−( )−

4 2 1 2 2

40

02

0 0π π ππ

(5.17)

Operationally this is:

• To each signal acquired at location x introduce a time advance equal to x x0− c0 ,

an amplitude weight equal to z c0 02 3π x x0− , and a filter with a frequency response

of H f j f j f cr ( ) ( ) − −( )2 1 2 0π π x x0 .

• Sum all such time-advanced, weighted, and filtered signals.• Sample at t=0.

Inversion beamformation is essentially the same as backpropagation beamformation withthe addition of a second differentiating filter j f2π . The reception filter H fr ( ) should bechosen to make the total response H frt ( ) broadband and zero phase.



Summary

These three formulations, cast as filtering problems and reduced to beamformationweights, are summarized below.

Matched Filter

df x y z f a x y z ffx y z

fMφ , , , , , ,

, ,( ) ∗ ( )∫

a f H fj f c

fM

txx

x,

exp( ) = − ( ) ( )∗ 2

40π

π

w f H fj f c

Mtx x

x x

x x00

0

,exp( ) = − ( )

−( )−

∗ 2

40π

π

Backpropagation Filter

df x y f a x y z ffx y

fBφ , , , , , ,

,0( ) ∗ ( )∫

a f H fz j f c j f c

fB

rxx

xx

x

x,

exp( ) = ( ) −

( )2 1 2 2

40 0π π

π

w f H fz j f c j f c

Brx x

x x

x x

x x

x x

x x00

0

0

0

0

,exp( ) = ( )

−− −

−

−( )−

2 1 2 2

40 0 0π π

π

Inversion Filter

df x y f a x y z ffx y

fIφ , , , , , ,

,0( ) ∗ ( )∫

a f H fz j f

c

j f c j f cfI

rxx

xx

x

x,

exp( ) = − ( )

−

( )4 2 1 2 2

402

0 0π π ππ

w f H fz j f

c

j f c j f cI

rx xx x

x x

x x

x x

x x00

0

0

0

0

,exp( ) = − ( )

−

− −−

−( )−

4 2 1 2 2

40

02

0 0π π ππ



These approaches are all motivated by the Fourier relationships of propagation and LSIfiltering, rather than geometrical arguments. The latter two use planar reception apertures.Even though we solved three different problems to get three different beamformationweights, the three results have features in common.

• They all introduce identical compensating delays (i.e. focusing).• They all introduce similar amplitude weighting (i.e., apodization).

In addition, the latter two introduce filtering, in the form of temporal derivatives, that isweakly aperture-dependent.

All three techniques work in the propagation regime, and, by design, cut off sharply inthe evanescent regime. As a consequence, there are effectively no spatial frequenciespreserved above f c0 . This is not so much a matter of choice as it is a physical limitdictated by the characteristics of propagation.

We have ignored rudimentary practical issues, such as finite aperture effects, temporalbandwidth, and the like. As a consequence, we’ve not yet developed guidance forsidelobe control, range resolution, and so on. That comes next.



Beamformation and the Point Spread Function

The Three Dimensional Point Spread Function

Consider an acoustic imaging system with a planar aperture with which we want to imagethe point x0 = x y z0 0 0, , . Recall the pulse-echo equation (4.34), rewritten here using vectornotation

R f H f d o f w f g f w f g fr t f rx y

f tx y

f( ) = ( ) ( ) ( ) ∗ ( )

( ) ∗ ( )

∫ x x x x x x, , , , ,

, ,(6.1)

To set up the one-way imaging equation, we restrict candidate weighting functions forreception to the form

w f a f j f cr rx x x x x0 0, , exp( ) = −( ) −( )2 0π (6.2)

Note that all three of the beamformation weights developed in the prior section are of thisform. This expresses the idealized requirement that shifting our point of focus merelyshifts our aperture weighting function. Three points are important.

• This is a requirement for a shift-invariant imaging system.• This requirement cannot be met in general with a finite-aperture system, as aperture

end effects will ultimately come into play. With care, however, this requirement canbe substantially met over localized regions of the field of view.

• This is the basis for dynamic focusing and scanning.

In order to characterize the one-way beamformation performance, we use anomnidirectional illumination function, i.e., a point source transmitter at the origin. Thetransmission aperture thus becomes a sampled version of eq. (6.2).

w f x y a f j f c

x y a f j f c

t t

t

x x x x x

x x

0 0

0 0

, , , exp

, , exp

( )= ( ) −( ) −( )= ( ) −( ) ( )

20

20

2

2

δ π

δ π(6.3)

We subsequently ignore the transmission weight at −( )x0 , as it can be incorporated in the

transmission pulse H ft ( ) . Following our development of the prior section, we integrateout temporal frequency to obtain the desired point of our image.



i df H f d o f

a f j f cj f c

j f cj f c

r t f

rx y

x x x

x x x xx

x

xx

x

0

0 0

0

( ) = ( ) ( )

−( ) −( ) ∗−( )

( ) −( )

∫∫ ,

, expexp

expexp

,2

2

4

22

4

00

00

πππ

πππ

(6.4)

For simplicity of expression this can be cast as a convolution.

i df H f d o f s fr t f fx x x x x0 0( ) = ( ) ( ) −( )∫∫ , , (6.5)

The impulse response of the imaging system is thus

p df H f s fr t fx x( ) = ( ) ( )∫ , (6.6)

where the convolution kernel is given by

s f a f j f cj f c

j f cj f c

f rx y

x x x x x xx

x

xx

x

0 0 0

0

−( ) = −( ) −( ) ∗−( )

( ) −( )

, , expexp

expexp

,2

2

4

22

4

00

00

πππ

πππ

(6.7)

The one-way impulse response of the imaging system focused at x0 can thus beexpressed as

p df H f d a fj f c

rt rδδ δ

δ δx x x x

x x x x x x

x x x0 0

0 0 0 0

0 0

( ) = ( ) ( )− − − −( ) + − −[ ]( )( ) − −( ) −∫ ∫ ,

exp 2

4

0

2

π

π(6.8)

This function is usually referred to as the point spread function of the system, in thetradition of optics. Its Fourier transform is the transfer function of the system.

The distance terms in eq. (4.29) have very simple geometrical interpretations.

• x x0− The distance from aperture point x to focal point x0

• x x0− −( )δ The distance from aperture point x to field point x0 − δ• x0 The distance from the origin to focal point x0

• x0 − δ The distance from the origin to field point x0 − δ



A Local Expansion of the Two Dimensional Point Spread Function

Because the point spread function in a well designed imaging system is compact, we willexpand eq. (6.8) assuming δ is small. For simplicity, we will consider just twodimensions, range and azimuth. The geometry is shown below.

R0 = x0

x x0−

x0 − δ x x0− −( )δ

x

Focal Point

Field Point

θ0θ φ0 +

(6.9)

In polar coordinates, we locate the focal point at R0 0,θ (where, obviously, R0 = x0 ) andthe field point at R r0 0+ +,θ φ , where we will be interested in small r and φ . The

associated arc length is u R= =0φ φx0 as can be seen in the figure below, and we willuse r and u as our incremental parameters for the expansion.



arc length u R= =0φ φx0

Focal Point x0

Field Point x0 + δ

radial length r

(6.10)

We expand the exponent of eq. (6.8) in a two-dimensional Taylor’s series around x0 andkeep the lowest order terms; the denominator is similarly expanded. This yields

p r u df H f dxa x f

jx f

cr j

x f

cu

rtr,

,

expsin

expcos

( ) ≅ ( ) ( )( ) −

− + −−

−

−

∫ ∫ 4

2 2

2

0

0

0

0

π

πθ

π θ

x x x

x x x

x x x x

0 0

0 0

0 0

(6.11)

This can be simplified with the following construction.



x x0−

x

Focal Pointψ x( )

θ0

R0 = x0

(6.12)

We define the angle ψ x( ) as the observation angle; it is the angle seen at the focal pointbetween the origin and the point x on the aperture. As usual, counterclockwise rotationcorresponds to increasing angle. It is readily shown using the law of sines that

−−

= ( )xx

cossin

θ ψ0

x x0

(6.13)

and, using the law of cosines

x

x x0

0

−−

= ( )xx

sincos

θψ0 (6.14)

Hence, substituting into eq. (6.11)

p r uR

df dxx

H f a x f

j xf

cr j x

f

cu

rt r,cos

cos,

exp cos exp sin

( ) ≅ ( )( ) +( ) ( ) ( )

+ ( )[ ]

( )

∫ ∫1

4

2 1 2

0

20

0

0 0

πψ θ

θ

π ψ π ψ

(6.15)

As the aperture is finite in extent, we can associate some full observation angle anddenote it Ψ ; by construction, the origin can be chosen such that it bisects the fullobservation angle. In this way the full observation angle is bound by Ψ < π and the angleto any aperture point by ψ πx( ) < 2. We shall see that the azimuthal resolution of thissystem is ultimately limited by Ψ .

We see that eq. (6.15) can be written as a two dimensional Fourier transform with thefollowing change of variables.



ρ ψ

υ ψ

= + ( )[ ]

= ( )

f

cx

f

cx

0

0

1 cos

sin(6.16)

By construction, there is a one-to-one correspondence between ρ υ, space and f x,space, and a coordinate transformation may be performed. The resulting Fourier integralfrom eq. (6.15) is

p r u d d P j r u, , exp( ) = ( ) +[ ]( )∫∫ ρ υ ρ υ π ρ υ2 (6.17)

We refer to P ρ υ,( ) as the transfer function of the imaging system under consideration, asit is the Fourier transform of the impulse response. The interested reader may wish toexpand P ρ υ,( ) in terms of H frt ( ) and a x fr ,( ) . We will confine our investigation hereto the geometry of this coordinate transformation with the goal of obtaining some insightsinto point spread functions.

First note that

υ ρ ψψ

ρ ψ

= ( )+ ( )

= ( )

sincos

tan

x

x

x

1

2

(6.18)

We infer from this the following.

• Lines of constant location x (on the aperture) in f x, space map to diagonal lines inρ υ, space, passing through the origin at an angle of ψ x( ) 2 with the ρ axis.

Second, it is easily shown that

ρ υ−

+ =

f

c

f

c0

2

2

0

2

(6.19)

from which we infer

• Lines of constant frequency f in f x, space map to circles in ρ υ, space, centered at

ρ υ, ,( ) =

f

c0

0 with radii of f c0 .



To visualize this, consider an imaging system for which the product H f a x frt r( ) ( ), has abounded region of support (i.e., finite aperture and bandwidth) in f x, space as shownbelow.

f

x

(6.20)

After the coordinate transformation, the transfer function P ρ υ,( ) also has a finite regionof support, but of a different geometry.

ρ

υ

angle = Ψ 2

(6.21)

There are two obvious consequences.

• The effects of frequency weighting and amplitude weighting are not orthogonal.• In the limit of infinite bandwidth and aperture, the ρ υ, plane is only half filled (as Ψ

is limited to π ), implying a natural limit to one-way resolution.



Let’s examine the azimuthal resolution of this system for a broadside, narrowbandexcitation. Set

H f a x f f frt r( ) ( ) = −( )=

, δθ

0

0 0(6.22)

We also want to evaluate the azimuthal resolution only, so we set r = 0 . The fullobservation angle is Ψ , which corresponds to the range of integration from some xmin toxmax. In this case, eq. (6.15) simplifies to

p uR

df dx x f f j xf

cu

Rdx x j x

f

cu

Rd j

u

x

x

x

x

01

42

1

42

1

42

0

2 00

0

20

0

0

20

, cos exp sin

cos exp sin

exp

min

max

min

max

( ) = ( ) ( ) −( ) ( )

=( ) ( ) ( )

=( )

−∞

∞

∫ ∫

∫

πψ δ π ψ

πψ π ψ

πξ π ξ

λ−( )

( )

∫sin

sin

Ψ

Ψ

2

2

(6.23)

Here we have made the customary substitution f c0 0 01= λ . The integral is simply the

(inverse) Fourier transform of rect sinξ 2 2Ψ( )[ ], hence

p uR

u02 2

4

2 2

0

20

,sin

sincsin( ) = ( )

( )

( )

Ψ Ψπ λ

(6.24)

The Rayleigh criterion for the azimuthal resolution of this system can be seen byinspection; as Ψ is bounded, the resolution also has an upper limit.

resolution = ( ) <λ λ0 0

2 2 2sin Ψ(6.25)

This limit is the Rayleigh limit for a one-way linear aperture.

In the broadside illumination case we have used here, the resolution can be expressed interms of the f-number. By construction



a 2

R0

Ψ 2

(6.26)

Using f R a# = 0 we see that the resolution can be expressed as

λ λ

λ

0 0

02 2

0 2

2 2 2

11

4

sin

##

Ψ( ) =+ ( )

= +

a R a

ff

(6.27)

Thus our rule of thumb that azimuthal resolution equals f#λ0 holds for f-numbers greaterthan about one; below that, adding more aperture is increasingly ineffectual. The graphbelow plots azimuthal resolution (in wavelengths) versus f-number; both eq. (6.27) andour rule of thumb are plotted.

0.5 1 1.5 2 2.5 3

0.5

1

1.5

2

2.5

3

(6.28)



The Fraunhofer Expansion of the Two Dimensional Point Spread Function

The Fraunhofer expansion of the point spread function is a further approximation to thefull integral expression, eq. (6.8). Its advantage is its simplicity.

The local expansion in the prior section approximated

p df H f d a fj f c

rt rδδ δ

δ δx x x x

x x x x x x

x x x0 0

0 0 0 0

0 0

( ) = ( ) ( )− − − −( ) + − −[ ]( )( ) − −( ) −∫ ∫ ,

exp 2

4

0

2

π

π(6.29)

by expanding the exponent and the denominator separately in a two dimensional Taylor’sseries, in r and u. The Fraunhofer expansion approximates the integral by similarlyexpanding terms in a three dimensional Taylor’s series, in r and u and x.

p r uR

df dx H f a x f jf

cr j

x

R

f

curt r, , exp exp

cos( ) ≅ ( ) ( ) ( )

−

∫ ∫1

42

22

0

20

0

0 0ππ π θ

(6.30)

The Fourier relationship

p r u d d P j r u, , exp( ) = ( ) +[ ]( )∫∫ ρ υ ρ υ π ρ υ2 (6.31)

follows directly from the substitutions

ρ

υ θ

=

= −

2

0

0

0 0

f

c

x

R

f

c

cos(6.32)

The interested reader is invited to explore the geometry of this coordinate transformation.Even with this simplification, the effects of amplitude weighting and frequency weightingare still not orthogonal.

Orthogonality can be achieved with the narrowband Fraunhofer expansion. In this, eq.(6.32) is written assuming some center frequency f0 . This has the advantage that, whenar ⋅( ) is amplitude weighting only, the resulting approximation is separable:

p r uR

df H f jf

cr dx a x j

x

R

f

cu

d P j r d P j u

p

rt r, exp expcos

exp exp

( ) ≅ ( ) ( )

( ) −

= ( ) ( )[ ] ( ) ( )[ ]=

∫ ∫

∫ ∫

1

42

22

2 2

0

20

0

0

0

0ππ π θ

ρ ρ πρ υ υ πυ

rr p u( ) ( )

(6.33)



where we can see by inspection (neglecting scale factors, and using the standard notationfor Fourier pairs a x A ur r( ) ⇔ ( ))

p r hc

r P Hc

p u AR

u P aR

rt rt

r r

( ) =

⇔ ( ) =

( ) =

⇔ ( ) =

220

0

0

0 0

0 0

0

ρ ρ

θλ

υ λθ

υcoscos

(6.34)

This is the approximation that is used most often in rough characterization of acousticimaging systems, and it’s basically the result we saw earlier using simple geometricalarguments. It says:

• The range resolution is the imaging pulse, scaled by two for the round trip.• The azimuthal resolution is the Fourier transform of the aperture, appropriately scaled

with the obliquity factor, range, and wavelength.

One should use this approximation with care. In modern ultrasound systems, both thenarrowband assumption and the small aperture (i.e., large f-number) assumption are oftenviolated.



The Transmission/Reception Point Spread Function

Thus far we have confined ourselves to the discussion of the one-way, or reception, PSFby modeling the transmitter as a point source which provides no azimuthal resolution. Inactual systems, of course, designers use transmission beamformers to focus acousticenergy.

The resulting PSFs are complex to model accurately. Using the techniques of the priorsections for two-way analysis, and assuming a narrowband Fraunhofer expansion, we cancome to a rough description of the resolution characteristics of a fixed-focustransmit/dynamic-focus receive system.

• The range resolution is unchanged from our prior description, being dominated by theround-trip imaging pulse h trt ( ) .

• The azimuthal resolution is equivalent to a one-way system whose aperture is theconvolution of the transmission and reception apertures.

The last point means that a system will exhibit essentially one-way azimuthal resolutionaway from the transmission focus, and improved two-way azimuthal resolution within thedepth-of-focus of the transmission beam.

The complexity of analysis for two-way operation should not obscure a central fact: theimage field is the convolution of the object field and the two-way PSF. The PSF is thus afilter through which the object is seen, and image quality depends in the mostfundamental way on this filter.

Summary

Images are formed through the convolution of the object field with the system pointspread function (or, equivalently, weighting the Fourier transform of the object field withthe system transfer function). This is a direct consequence of: (1) the LSI character ofpropagation and low-level scattering; (2) our choice of LSI processing for formation ofthe image; and (3) the design of PSFs that are compact and relatively invariant as thesystem scans through a neighborhood.

The design of PSFs for ultrasound can be complex because of the interaction of range andazimuthal characteristics, and there is no substitute for simulation tools and carefulempirical evaluation in a diagnostic context.



Sampling and Image Processing

Sampling Requirements in Two Dimensions

We have seen how the point spread function of an imaging system is a convolutionkernel—equivalently, how its transfer function is a filter—through which the systemmaps an object field to an image field

Imaging

System

o

O u v

x, y( )( )

, ,

i

I u v

x, y( )( )

,

p x y

P u v

,

,

( )( )

(7.1)

We need to be mindful that this model applies only in localized regions in even the bestdesigned systems, as the PSF will vary in a gradual way over space. Within any suchlocalized region, however, the PSF can be viewed as a filter, suppressing some spatialfrequencies and passing others. When our aperture is finite (as it always is) and ourimaging pulse is band-limited (which is a function of system design), the transferfunction of the imaging system has a finite region of support, i.e., it is two-dimensionallyband-limited, as we saw earlier. As a direct consequence, the image is likewise confinedto the same region of support.

ρ

υ

angle = Ψ 2

(7.2)

Because we are dealing with real signals and systems, the Fourier transform of the imagefield (confined to the regions shown above) will exhibit hermetian symmetry. As aconsequence, only one of the regions is necessary to fully characterize the image. Using



the narrowband Fraunhofer expansion we can estimate the extent of either region;neglecting scale factors:

p r hc

r P Hc

p u AR

u P aR

rt rt

r r

( ) =

⇔ ( ) =

( ) =

⇔ ( ) =

220

0

0

0 0

0 0

0

ρ ρ

θλ

υ λθ

υcoscos

(7.3)

It is convenient to define the extent of P υ( ) using the two way f-number, and the extentof P ρ( ) using the inverse full fractional bandwidth of the imaging pulse (in this context,full fractional bandwidth refers to the region of support of the imaging pulse frequencyresponse, not the usual 3 dB bandwidth). In the figure to follow, let

f

ff

B

# =

= =

System two way f - number

full bandwidth of imaging pulseInverse full fractional bandwidth0

We can think of fB as the imaging pulse analog of the f-number. Applying these limits toeq. (7.3):

ρ

υ

2

0fBλ

cos

#

θλ

0

0f

(7.4)

So far, we have considered the imaging process to be a continuous one, and of course it isnot. Consider an image sampled in both range and azimuth. As is well known, adequatespatial sampling (permitting perfect reconstruction in principle) of the bandlimited signalrepresented above requires:



azimuth sampling period

range sampling period

=f

=f

#

B

λθ

λ

0

0

0

2

cos(7.5)

For example, consider a system with a receive f# = 2 and a transmit f# = 2 , steeredstraight ahead. At the transmit focus, the system has an effective two-way f# = 1, so thespacing for scan lines should be λ0. (Note that the Rayleigh criterion for resolution is thesame as the Nyquist criteria for sampling.) If the full fractional bandwidth is 50%, thenfB = 2 and the range sampling period should also be λ0.

There are some practical points to consider.

• Any real system will not exhibit a sharp, idealized cutoff. In azimuth, any apodizationapplied to the transmit or receive apertures will taper the response smoothly to zero;in range, the envelope of the imaging pulse will usually be smoothly tapered as well.This means that there is latitude to adjust the sample grid to optimize the system forspecific situations.

• Sampling in azimuth is almost always more expensive than sampling in range. As aconsequence, compromises in sample rates are usually most egregious there.

• The two most compelling reasons to compromise sampling density (especially inazimuth) are: (1) to meet frame rate requirements, and (2) to satisfy some systemlimitations (such as memory limits, data rate limits, etc.).



Post-Detection Sampling Requirements

The image we have been considering thus far is the result of LSI processing. Theexception to this is the sampling process (and any other process to shift the spectrum inthe frequency domain), but these are reversible as long as certain criteria are met. Theprocess of detection—necessary for the display of the image—fundamentally changes thesampling requirements of the image.

Consider a squared-magnitude detection process. (While diagnostic ultrasound generallyuses log-magnitude detection, the conclusion to follow is not qualitatively affected.) Asmultiplication in one domain is convolution in the other, the region of support of the post-detection image spectrum is approximately four times the area of the pre-detection imagespectrum.

ρ

υ

2

0fBλ

cos

#

θλ

0

0f

(7.6)

The conclusion is that our post-detection sampling requirements are approximately fourtimes—twice in azimuth and twice in range—as our pre-detection samplingrequirements13. These theoretical results are often violated in practical diagnosticultrasound, with consequential corruption of image information due to aliasing.

The necessity of an adequately fine sample grid to preserve all the information in thepost-detection image is easily seen using the frequency domain analysis above.Interestingly, there are theoretical grounds14 for the position that adequate sampling ofthe post-detection image is sufficient to preserve all the pre-detection information also, asthe detection is (almost always) reversible in principle.



Shift-Variance in Diagnostic Ultrasound

There are endless opportunities to degrade an acoustic image through the introduction ofshift-variance. We have examined a key requirement for shift-invariance earlier,specifically that the PSF be substantially invariant throughout all localized regions of theimage field. Most modern beamformer-based systems meet this criterion to a greater orlesser degree (although in this regard many fall short of older mechanically-scannedsystems, where PSF invariance was ensured by physical constraints). We will identifyhere two other sources of shift-variance, which can be substantially eliminated withcareful system design.

Shift Variance due to Undersampling

Historically, it has been common in diagnostic ultrasound to use a density of scan linesmore or less adequate to sample an image in azimuth prior to detection. As aconsequence ultrasound images are commonly undersampled post-detection by a factorof two.

There are two consequences of the resulting aliasing. First, the image becomes shift-variant. As is known15, when a signal is undersampled, the resulting “reconstructed”signal will vary with the alignment of the sample grid and the original signal. Inultrasound, this is demonstrable by shifting, in azimuth, the object field relative to thegrid of scan lines by distances that are small relative to the scan line spacing. A shiftinvariant system will exhibit no changes in the image as this translation is performed.

Second, image information is lost. When a signal is undersampled by a factor of two, thefrequency spectrum is folded over in such a way that higher frequencies are overlaid onlower frequencies. This process is in general irreversible; low frequency information iscorrupted and high frequency information is irretrievable.

Shift Variance due to Geometric Distortion

When an imaging system introduces systematic errors in its mapping of points in theobject field to points in the image field, the visible consequence is geometric distortion.This can usually be seen, when present, by suitable translation of the sampling grid overthe object field.

Geometric distortion is often associated technically with the use of multiple simultaneousbeams. The promise of multi-beam technology for imaging, of course, is integer-multipleincrease in temporal resolution. In conventional application, however, the deliberatemisalignment of transmission and reception beams introduces an asymmetry in which theresulting “scan lines” are no longer lines, but curves—indeed, it becomes difficult tosatisfactorily define what loci they follow.



Pre-Detection Image Processing

Manipulation of the image prior to detection is theoretically attractive, as the systemdesigner can synthesize a class of signals, or samples, as an alternative to acquiring themdirectly from the aperture. This can be used to increase performance and to reduce costs.

The linearity of the acquisition process can be extended to image processing, with theadvantage of predictable system performance. For example, if we average two samples ofthe pre-detected image, we can think of the resulting third sample in three different ways:

• as an output sample of a filter on the underlying band-limited image.• as an image sample that would have been acquired by the average of the two point

spread functions associated with the two original samples.• as an image sample that would have been acquired by the average of the two

apertures associated with the two original samples.

It is important to remember that these perspectives are incorrect when applied to similaroperations post-detection, for the detection process breaks the chain of linear operationsused to form the image to this point.

The specific applications for pre-detection image processing in diagnostic ultrasound aremanifold. Some interesting ones include the following.

• Reconstruction, or partial reconstruction, of the image prior to detection16. In anotherwise properly sampled image, this technique can eliminate post-detectionundersampling as a source of shift-variance and information loss without increasingthe time to acquire the image.

• Elimination of geometric distortion in multi-beam operation16. The asymmetryintroduced by multiple beams can be removed by careful filtering of the resulting pre-detection image, thus permitting substantially higher temporal resolution.

• Synthetic aperture. Large, high resolution apertures can be synthesized from smallerones.

• Deconvolution of the image to increase the transmit depth of focus17. In thistechnique, the quadratic phase error associated with the transmission aperture can bepartially corrected.

As the hardware for real-time image processing becomes ever more cost-effective, weexpect growing practical interest and innovation in pre-detection image processing.



1 Ronald N. Bracewell, "The Fourier Transform and its Applications," Second Edition, McGraw-Hill, 1978

2 Athanasios Papoulis, "The Fourier Integral and its Applications," McGraw-Hill, 1962

3 Norbert Wiener, "The Fourier Integral and Certain of its Applications," Dover, 1958; originallyCambridge University Press, 1933

4 Ronald N. Bracewell, "Two-Dimensional Imaging," Prentice-Hall, 1995

5 Athanasios Papoulis, "Systems and Transforms with Applications in Optics," McGraw-Hill, 1968

6 Eugen Skudrzyk, "The Foundations of Acoustics," Springer-Verlag, 1971. See especially section 2.8,"Basic Theory of Internal Friction," and section 13.12, "The Effect of Viscocity."

7 Joseph W. Goodman, "Introduction to Fourier Optics," McGraw-Hill, 1968

8 Philip M. Morse and K. Uno Ingard, "Theoretical Acoustics," Princeton University Press, 1968. Seeespecially chapter 8, "The Scattering of Sound."

9 P. Stepanishen, M. Forbes and S. Letcher, "The Relationship between the Impulse Response and AngularSpectrum Methods to Evaluate Acoustic Transient Fields," J. Acoust. Soc. Am., vol. 90, no. 5, Nov. 1991

10 A. R. Selfridge, G. S. Kino and B. T. Khuri-Yakub, "A Theory for the Radiation Pattern of a Narrow-Strip Acoustic Transducer," Appl. Phys. Lett., vol. 37, no. 1, Jul. 1980

11 Robert C. Waag et. al., "Cross-Sectional Measurements and Extrapolation of Ultrasonic Fields," IEEETrans. Sonics and Ultrasonics, vol. SU-32, no. 1, Jan. 1985

12 Dong-Lai Liu and Robert C. Waag, "Propagation and Backpropagation for Ultrasonic WavefrontDesign," IEEE Trans. UFFC, vol. 44, no. 1, Jan. 1997

13 Dan E. Dudgeon and Russell M. Mersereau, “Multidimensional Digital Signal Processing,” PrenticeHall, 1984. See especially Chapter 1.4, “Sampling Continuous 2-D Signals.”

14 Monson H. Hayes, “The Reconstruction of a Multidimensional Sequence from the Phase or Magnitudeof its Fourier Transform,” IEEE Trans. ASSP, vol. 30, no. 2, Apr. 1992

15 Athanasios Papoulis, “Signal Analysis,” McGraw-Hill, 1977. See especially Chapter 5-1, “Sampling andInterpolation.”

16 J. Nelson Wright et. al., “Method and Apparatus for Coherent Image Formation,” US Patent 5,623,928,Apr. 1997

17 S. Freeman, Pai-Chi Li, and M. O’Donnell, “Retrospective Dynamic Transmit Focusing,” UltrasonicImaging, vol. 17, no. 3, Jul.1995

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