introduccion matem atica a la tomograf a...verano de las matem aticas, del 28 de junio al 24 de...

Introducción matemáticaa la tomograf́ıa

J. Héctor Morales Bá[email protected]

Grupo de Matemáticas Aplicadas

Verano de las Matemáticas, del 28 de junio al 24 de julioCIMAT - Guanajuato

Tomograf́ıa [1/116]

Objetivo

Mostrar de forma simple un procedimiento

matemático que se emplea en la reconstrucción

de imágenes tomográficas en diversos sistemas.

Verano de las Matemáticas, del 28 de junio al 24 de julio. CIMAT


Temario

I. Introducción: Los problemas inversos y los métodos no invasivos.

II. La tomograf́ıa proyectiva y el teorema de la rebanada.

III. La transformada de Fourier y la transformada de Radon.

IV. La transformada inversa de Radon.

V. Ejemplos numéricos.



I. IntroducciónLos problemas inversos

y los métodos no invasivos



Sismoloǵıa (ondas mecánicas)80 M.P. Lamoureux and G.F. Margrave

Fig. 1 Seismic wave experiment

∂2ϕ

∂x2+

∂2ϕ

∂y2+

∂2ϕ

∂z2=

1c2

∂2ϕ

∂t2,

where ϕ(x, y, z, t) is the wave function and c = c(x, y, z) is the (non-constant)speed of propagation of the seismic wave. Numerical calculations based onthis equation are key to recovering an image of the subsurface.

In practice, real seismic experiments are performed by exploding dynamiteon the surface (or near surface) of the earth, and recording the vibrationsproduced by the explosive energy using sensitive geophones. The signals ofinterest are those acoustic waves that have propagated down to some inter-esting geological formation and been reflected back to the surface. Figure 1shows a simplified seismic setup, with instruments placed on the surface ofthe earth, and seismic energy traveling along raypaths within the earth. Thegeophones are typically placed at the surface of the earth,1 and are sensitiveenough to record vibrations that have traveled from the dynamite source,down five kilometers or more through rock, and return the same distanceback to the surface. Hundreds of geophones are monitored and the signaldata collected from them are recorded in a computer; dozens of dynamiteblasts are recorded, exploded at different locations, and independently at dif-ferent times. This recorded data is then processed to create an image of thesubsurface, and is often used in the search for hydrocarbons (oil and gas).

In marine seismic imaging, the experiments take place at sea rather thanon land. Typically, thousands of hydrophones attached to floating cables aretowed behind a ship traveling back and forth across a target area. A signal

1 In Vertical Seismic Profiling, or VSP, the geophones may be placed deep within the earth,usually down the borehole of an oilwell.

An Introduction to Numerical Methods of Pseudodifferential Operators 81

Fig. 2 Data collected from a seismic experiment. Reflectors are the images of interest

is initiated by setting off an air gun at the ocean surface, which starts anacoustic wave traveling down through the water and into the ocean floor,which then propagates through the rock in the form of a seismic wave. En-ergy that is reflected back travels through rock, then water, and then to thehydrophones where the data is recorded. It is also possible to use an OceanBottom Cable (OBC), where a string of geophones is actually placed on theocean floor and data recorded directly from the ocean bottom. This is a muchmore expensive setup. In either case, a given marine survey may collect dataover several days, covering dozens of square kilometers of territory. A hugeamount of data can be collected in one ocean survey, often amounting toterabytes of computer files.

The raw data contains much information, and much noise. In Fig. 2, wehave organized the time series data from a sequence of geophones placed ina line, so that a useful raw image appears. The first breaks and ground rollare due to surface propagation of seismic energy, and are considered noise.The hyperbolas represent reflections from interesting geological structures,and contain the information of a true image of the subsurface. Many of thedevelopments in seismic data processing are techniques in signal processingand include steps to remove noise due to first breaks and ground roll (muting,f–k filtering), to straighten the hyperbolas into proper reflectors (migration),and to sharpen the image (deconvolution), among others.

Thus, this problem combines aspects of physical modeling via PDEs, andsignal processing. Time–frequency analysis, and in particular pseudodifferen-tial operators, are well-suited for combining these approaches.



Radioloǵıa (rayos X)Chapter 10. Tomographic Evaluation ofc Materials of254

serts (Permision Commissariat à l’Energie Atomique, CEA/DAMRI, Clefs Cea, N° 34 Hiver 1996-1997. La radioactivité)

In different techniques used in applications, the mathematical method of solution for data inversion is the same, namely Radon s transform and its inverse.

10.2 X-rays tomography

The principle of X-ray tomography is based on Radon’s transform R.When a ray L passes through the body, its global attenuation m along the ray L can be measured experimentally. It is equal to the sum of unknown local attenuation density µ(x) along the ray L

Figure 10.2: The Aphrodite statue of the Louvre Museum revealing metallic in-

,

However, less than one year later, Wilhelm Roentgen succeeded increating the first image of an X-rayed hand and demonstrated thathuman tissues behave differently depending on their density. Thisfirst radiography of Mrs Roentgen’s hand, dating from 1896,became famous and opened the way for a new discipline of medi-cine, radiology. The recognition of the value and advantages of thistechnology was such that special radiology services were createdwithin five years, and military medicine services in particular devel-oped mobile imaging services.

The atom model, with its nucleus made of protons and neutronsaround which electrons are gravitating, was proposed by ErnestRutherford (1871-1937; 1908 Chemistry Nobel Prize winner) in1911. This model was improved by Niels Bohr (1885-1962; 1922Physics Nobel Prize winner) in 1913. The concept of isotope wasintroduced by Frederick Soddy (1877-1956; 1921 ChemistryNobel Prize winner) only in 1913.

26 NUCLEAR MEDICINE

Figure 2. Radiography of Mrs Roentgen’s hand, taken in 1896.



Estructura molecular de los ácidos nucléicos: 1953



Radar (ondas electromagnéticas)Radar inverse scattering R3

Figure 1. A generic radar detection system.

and locating radar targets1. Active radar systems accomplish these tasks by transmitting anelectromagnetic waveform and measuring the reflected field as a time-varying voltage in theradar receiver. If R denotes the distance from the transmitter to the target, then the radiationcondition guarantees that the energy in the transmitted field will decay as R−2 when R issufficiently large (this is the usual case). The energy reflected by the target will also obey thisradiation condition and so, in monostatic radar configurations (in which the radar transmitterand receiver are co-located), the energy collected by the receiver will be reduced from thattransmitted by a factor of R−4. In practice, this received energy is quite small, and when thetarget is far away the measured signal competes for attention with the thermal noise voltage ofthe radar system itself. Accurately separating the wanted signal from the unwanted noise is animpressive engineering feat and it is not surprising that a fair amount of device-specific jargonhas evolved over the years. This section will briefly explain the methods employed.

Figure 1 shows the main elements of a generic (and greatly simplified) radar measurementsystem (cf [11, 82, 96, 127], and references cited therein). An incident time-varying signalsinc(t) is mixed with a carrier signal with (angular) frequency ω0 chosen to conform withfavourable properties of the transmission medium (typically, atmospheric windows are soughtfor which free space parameters are good approximations). This mixed signal is used to excitecurrents on an antenna that cause an electromagnetic waveform to be launched toward thetarget. (This antenna is rarely an isotropic radiator: it is usually designed to have some degreeof directionality (gain), and this directionality can also be used to estimate target bearing.) Thewaveform is reflected from the target and the reciprocal situation occurs in the receiver: thereflected field induces antenna currents; the signal voltage is mixed with a conjugate carrier toremove (beat down) the carrier frequency dependence; and the time-varying voltage srec(t) isoutput to a signal processor. Note that, while high frequencies dominate the target scatteringphysics (ω0 is typically 1–35 GHz), the signal processing block usually sees much lowerfrequencies (ω ! 0.1ω0).

2.1. I and Q signals

The system signals are real-valued voltages and are often expressed in terms of circularfunctions p(t) = a(t) cos φ(t) so that the concepts of amplitude and phase may beintroduced. It is convenient, however, to consider the signals to be complex-valued, withs(t) = 12 (p(t) + iq(t)) for some q(t). Of course, the choice of q(t) is not unique and,1 A radar target is any object of interest illuminated by the radar. In contrast, radar clutter is composed of collateraland undesired illuminated objects.

R14 B Borden

Figure 4. Example ISAR image. The image is the |ρ(t, ν)|2 of a Boeing 727 jetliner withorientation as in the inset. The radar centre frequency and bandwidth were 9.25 GHz and 500 MHz,respectively. Compare this figure with the range profile of figure 3.

of the library are called templates: τn ∈ LN where n indexes the N known targets in the library.In the case of range profile images the templates are functions of the range variable t and areparametrized by θ .

Let ρ̂(t) denote a range profile measured using an HRR radar. There are many ways tocompare ρ̂(t) with τn(t; θ), but one of the most common is based on the maximum correlationcoefficient

C(n, θ) = maxt∈R

∫

Rρ̂(t ′) τn(t ′ + t; θ) dt ′. (28)

Template correlation methods compute C(n, θ) for each n = 1, . . . , N and each θ =θ1, . . . , θM . Target type and orientation are then estimated as

n, θ = arg maxn′=1,...,N

θ ′=θ1,...,θM

{

C(n′, θ ′)}

. (29)

The search over the M target orientations is required because it is impossible to divinetarget aspect from ρ̂(t) (and it must be implicitly understood that θ generally refers to anordered pair of spherical angles). Typical libraries are set up with N ≈ 25 ‘targets of interest’.But, as we shall see below, there is little freedom in the choice of M .



Radioastronoḿıa (ondas electromagnéticas)

Figura 1: (a) Radio, (b) Infra-rojo, (c) Visible y (d) Rayos X



Radioastronoḿıa



Acústica de inversión temporal (sonido)

medical imaging, where one wishes to send the ultrasoundthrough fat, bone and muscle to targets such as tumors orkidney stones. Pulse-echo detection with a TRM can cir-cumvent this problem.

Pulse-Echo Detection

First, one part of the array sends a brief pulse out throughthe distorting medium to illuminate the region of inter-est. Next, the wave that is reflected back to the array by atarget is recorded, time-reversed and reemitted. The time-re-versal process ensures that this reversed wave focuses on thetarget despite all the distortions of the medium.

When the region contains only one target, this self-focus-ing technique is highly effective. If there are several targets,

the problem is more complicated, but a single target can beselected by repeating the procedure. Consider the simplestmultitarget case, in which the medium contains two targets,one more reflective than the other. The echoes produced bythe initial pulse will have a somewhat stronger componentfrom the brighter target than from the weaker one. There-fore, the first time-reversed signal will focus a wave on eachtarget but with a more powerful wave on the brighter tar-get. The echoes from these waves will have an even greaterbias toward the brighter target, and after a few more itera-tions one will have a signal that focuses primarily on thattarget. More complex techniques let one select the weakerreflectors.

Among the medical applications of pulse-echo TRM, theclosest to fruition is the destruction of stones in kidneys and

96 Scientific American November 1999 Time-Reversed Acoustics

KIDNEY STONES can be targeted and broken up with ultra-sound by using the self-focusing property of a time-reversal mir-ror. An ultrasonic pulse emitted by one part of the array (a) pro-duces a distorted echo from the stone (b). A powerful time-reverse

of this echo passes through intervening tissues and organs, fo-cuses back on the stone (c) and breaks it up. Iterating the proce-dure improves the focus and allows real-time tracking as thestone moves because of the patient’s breathing.

ULTRASONIC PULSE ECHO FROM STONE TIME-REVERSED WAVE

CONTROL SYSTEM

RUBBER MEMBRANE

TRANSDUCER ARRAY

TIME-REVERSAL MIRROR

CYLINDRICAL TUB OF WATER

KIDNEY STONE

TRANSDUCERARRAY

PULSE OF ULTRASOUND

SOUND REFLECTEDFROM KIDNEY

STONE

HIGH-POWERTIME-REVERSEDPULSE

ALF

RED

T.K

AM

AJI

AN

a b c

Copyright 1999 Scientific American, Inc.



Emisión tomográfica (de positrones)72 NUCLEAR MEDICINE

PET imaging process

Production of the radioactive isotope

Injection of the tracer and data acquisition

Data processing Image/Interpretation

Incorporation in a molecule

Figure 8. As a consequence of the short half-life of Fluorine 18, PET imaging requires afour-step process that has to be performed in the shortest possible time: manufacturing ofFluorine 18 in a cyclotron (1); synthesis of the FDG molecule in a radiopharmacy unit (2);transportation of the doses to the imaging centre; injection to the patient and acquisitionof the image with a PET camera (3). The last step, which consists in processing data andinterpreting the final images (4), is obviously independent from the previous steps.



Resonancia magnética (emisión en ondas de radio)



Premio Nobel de f́ısica de 1979

EARLY TWO-DIMENSIONALRECONSTRUCTIONANDRECENT TOPICS STEMMING FROM IT

Nobel Lecture, 8 December, 1979

byALLAN M . CORMACKPhysics Department, Tufts University, Medford, Mass., U.S.A.

In 1955 I was a Lecturer in Physics at the University of Cape Town whenthe Hospital Physicist at the Groote Schuur Hospital resigned. SouthAfrican law required that a properly qualified physicist supervise the useof any radioactive isotopes, and since I was the only nuclear physicist in

Cape Town, I was asked to spend 1 1 / 2 days a week at the hospitalattending to the use of isotopes, and I did so for the first half of 1956. I wasplaced in the Radiology Department under Dr. J. Muir Grieve, and in thecourse of my work I observed the planning of radiotherapy treatments. Agirl would superpose isodose charts and come up with isodose contourswhich the physician would then examine and adjust, and the processwould be repeated until a satisfactory dose-distribution was found. Theisodose charts were for homogeneous materials, and it occurred to me thatsince the human body is quite inhomogeneous these results would be quitedistorted by the inhomogeneities - a fact that physicians were, of course,well aware of. It occurred to me that in order to improve treatmentplanning one had to know the distribution of the attenuation coefficient oftissues in the body, and that this distribution had to be found by measure-ments made external to the body. It soon occurred to me that this informa-tion would be useful for diagnostic purposes and would constitute atomogram or ser ies of tomograms , though I did not learn the word“tomogram” for many years.

At that time the exponential attenuation of X- and gamma-rays had beenknown and used for over sixty years with parallel sided homogeneous slabsof material. I assumed that the generalization to inhomogeneous materialshad been made in those sixty years, but a search of the pertinent literaturedid not reveal that it had been done, so I was forced to look at the problemab initio. It was immediately evident that the problem was a mathematicalone which can be seen from Fig. 1. If a fine beam of gamma-rays ofintensity I, is incident on the body and the emerging intensity is I, then themeasurable quantity g = In(I0/ I) = SLfds, where f is the variable absorp-

tion coefficient along the line L. Hence if f is a function in two dimensions,and g is known for all lines intersecting the body, the question is: “Can f bedetermined if g is known ?“. Again this seemed like a problem which would

55 I

COMPUTED MEDICAL IMAGING

Nobel Lecture, 8 December, 1979

BYGODFREY N . HOUNSFIELDThe Medical Systems Department of Central Research Laboratories EMI,London, England

In preparing this paper I realised that I would be speaking to a generalaudience and have therefore included a description of computed tomo-graphy (CT) and some of my early experiments that led up to the develop-ment of the new technique. I have concluded with an overall picture of theCT scene and of projected developments in both CT and other types ofsystems, such as Nuclear Magnetic Resonance (NMR).

Although it is barely 8 years since the first brain scanner was construct-ed, computed tomography is now relatively widely used and has beenextensively demonstrated. At the present time this new system is operatingin some 1000 hospitals throughout the world. The technique has succesful-ly overcome many of the limitations which are inherent in conventional X-ray technology.

When we consider the capabilities of conventional X-ray methods, threemain limitations become obvious. Firstly, it is impossible to display withinthe framework of a two-dimensional X-ray picture all the informationcontained in the three-dimensional scene under view. Objects situated indepth, i. e. in the third dimension, superimpose, causing confusion to theviewer.

Secondly, conventional X-rays cannot distinguish between soft tissues. Ingeneral, a radiogram differentiates only between bone and air, as in thelungs. Variations in soft tissues such as the liver and pancreas are notdiscernible at all and certain other organs may be rendered visible onlythrough the use of radio-opaque dyes.

Thirdly, when conventional X-ray methods are used, it is not possible tomeasure in a quantitative way the separate densities of the individualsubstances through which the X-ray has passed. The radiogram recordsthe mean absorption by all the various tissues which the X-ray has penetrat-ed. This is of little use for quantitative measurement.

Computed tomography, on the other hand, measures the attenuation ofX-ray beams passing through sections of the body from hundreds ofdifferent angles, and then , from the evidence of these measurements, a

computer is able to reconstruct pictures of the body’s interior.Pictures are based on the separate examination of a series of contiguous

cross sections, as though we looked at the body separated into a series ofthin “slices”. By doing so, we virtually obtain total three-dimensional infor-

mation about the body.

568

HOLOGRAPHY, 1948-1971

Nobel Lecture, December 11, 1971

by

D E NN I S G A B O R

Imperial Colleges of Science and Technology, London

I have the advantage in this lecture, over many of my predecessors, that Ineed not write down a single equation or show an abstract graph. One can ofcourse introduce almost any amount of mathematics into holography, but theessentials can be explained and understood from physical arguments.

Holography is based on the wave nature of light, and this was demonstratedconvincingly for the first time in 1801 by Thomas Young, by a wonderfullysimple experiment. He let a ray of sunlight into a dark room, placed a darkscreen in front of it, pierced with two small pinholes, and beyond this, at somedistance, a white screen. He then saw two darkish lines at both sides of abright line, which gave him sufficient encouragement to repeat the experi-ment, this time with a spirit flame as light source, with a little salt in it, toproduce the bright yellow sodium light. This time he saw a number of darklines, regularly spaced; the first clear proof that light added to light canproduce darkness. This phenomenon is called interference. Thomas Young hadexpected it because he believed in the wave theory of light. His great contri-bution to Christian Huygens’s original idea was the intuition that mono-

DARK

EXPLANATION

Fig. 1.Thomas Young’s Interference Experiments, 1801



Métodos y sistemas mini o no invasivos:

• Absorción: Tomograf́ıa computada de Rayos X.

• Emisión: Tomograf́ıa de emisión de positrones (PET), imágenes de reso-nancia magnética (MRI).

• Difusión: Tomograf́ıa óptica.

• Difracción: Microondas, ultrasonido, radares, sonares.



Problemas inversos

en matemáticas



Definición 1. (Problema inverso). El problema de obtener información oinferir propiedades de cantidades desconocidas por medio de observacionesindirectas es lo que se llama un Problema inverso.



Exploración mediante ondas

Por medio de radiación (ondas exploradoras), que viaja a través de un medio

material o que proviene de una fuente desconocida, “resolvemos” problemas

inversos; es decir, determinamos o estimamos parámetros de interés f́ısico,

geométrico o biológico.



Formulación matemática de un problema inverso

• Problema directo

modelo {parámetros del modelo m, fuentes s} → datos d

d = As(m)

• Problema inverso

{datos d, fuentes d} → modelo {parámetros del modelo m}

m = A−1s (d)

{datos d} → modelo y fuentes {parámetros m, fuentes s}

(m, s) = A−1

(d)



Ejemplo: Regresión lineal de observaciones baĺısticas

0 2 4 6 8 10 12100

150

200

250

300

350

400

450

500

550

600

Tiempo (s)

Ele

vaci

on

(m)

datos



Regresión lineal de observaciones baĺısticas

• Modelo de regresión parabólica (Galileo)

d(t) = m1 +m2t−1

2m3t

2,

donde la función d(·) mide la posición del proyectil y t es el tiempo.

• Consideremos datos sintéticos con i = 10 observaciones y errores inde-pendientes distribuidos normalmente (σ = 8,0 m):

mverdadero =

24 10 m100 m s−19,8 m s−2

35




• Sitema de ecuaciones:

Am =

26641 t1 t

21

1 t2 t22

... ...

1 t10 t210

377524m1m2m3

35 =2664d1d2...

d10

3775 = d

• Solución de ḿınimos cuadrados: ecuaciones normales

mL2 = (ATA)−1ATd,

que minimizan el residuo r = d− Am en la norma L2.




0 2 4 6 8 10 120

100

200

300

400

500

600

Tiempo (s)

Ele

vaci

on

(m)

ajustedatos



¿Se puede escuchar la forma de un tambor?



Determinación de la circunferencia de los planetas



Descubrimiento de planetas



Tomograf́ıa por proyecciones

��

��

receptor

Ix

R(x, y)

emisor

I0

y

x

Θ2

Θ1

• Teorema de retroproyección:

m(x, y) =

Z π0

Z +∞−∞

dfdΘPΘ(f)|f |ei2πf(x cos(Θ)+y sin(Θ))



Tomograf́ıa por proyecciones

Figura 2: Johann Radon (1887-1956)

Matemático austriaco. Nació el 16 de

diciembre de 1887 en Tetschen, Bohe-

mia (ahora Decin, República Checa).

Murió el 25 de mayo de 1956 en Viena,

Austria.



II. La tomograf́ıa proyectivay la transformada de Fourier



Tomograf́ıa

Muchas clases de datos (mediciones) se pueden considerar en tomograf́ıa:

• ABSORCIÓN: atenuación de una señal, rayos X, mientras pasan a travésde un objeto. El logaritmo negativo de la atenuación, en una región infini-

tesimal en x ∈ R3, se denota por s(x)dx, con x ∈ (x1, x2, x3)T .



Tomograf́ıa




• EMISIÓN: generación de una señal, como un haz de part́ıculas radiactivaso radiación en radio frecuencia, desde dentro de un objeto. La emisión,

desde una región infinitesimal en la posición x ∈ R3 se denota por s(x)dx.



Tomograf́ıa




• EMISIÓN: generación de una señal, como un haz de part́ıculas radiactivaso radiación en radio frecuencia, desde dentro de un objeto. La emisión,

desde una región infinitesimal en la posición x ∈ R3 se denota por s(x)dx.

• DISPERSIÓN: desviación de una señal desde el interior de un objeto endirecciones espećıficas. La dispersión, desde una región infinitesimal en la

posición x ∈ R3 se denota, nuevamente, por s(x)dx.



Tomograf́ıa proyectiva

Definición 2. PROYECCIÓN: observación basada en atenuación de rayosque provienen de la aproximación de la ÓPTICA GEOMÉTRICA.



Espectro de frecuencias



¿Qué es la óptica geométrica?

• En el ĺımite conceptual cuando λ → 0 se obtiene propagación rectiĺıneade la luz en medios homogéneos y tenemos el dominio idealizado de la ópticageométrica.



Ley de Beer-Lambert (1760 y 1852)

Figura 3: Descubierta por Pierre Bouguer en 1729. Dibujo de Allan M. Cormack

en su disertación Nobel de 1979.



Objetivo de la tomograf́ıa proyectiva

En cada una de estas situaciones, deseamos DETERMINAR a s(x) por medio

de los datos (observaciones o mediciones) proyectados.

Definición 3. La reconstrucción de imágenes mediante proyecciones se lla-ma TOMOGRAFÍA PROYECTIVA (o tomograf́ıa de proyección).

• (Primera aproximación). En muchas aplicaciones es adecuado formar unasimple sección bidimensional de s(x) en la dirección x3.

sk(x1, x2) = s(x1, x2, k∆x3)

• En 3D, el tratamiento de s(x) es computacionalmente muy costoso.



Proyecciones

• Enunciaremos el teorema de la rebanada, que relaciona a una transforma-da de Fourier bidimensional de una señal, con transformadas de Fourier

unidimensionales de ciertas funciones llamadas PROYECCIONES.

• Pero, ¿qué es una transformada de Fourier?



¿Qué es el análisis de Fourier (1807–1822)?

Figura 4: Llamado de la ballena azul: amplitud vs tiempo (sec)



¿Qué es el análisis de Fourier?

• Una función del tiempo t, real o compleja, cuya enerǵıa

Ep =

Z +∞−∞

dt|s(t)|2


¿Qué es el análisis de Fourier?

Figura 5: Harmónicos del llamado de la ballena: potencia vs frecuencia (Hz)



Proyecciones

• Sea s(x1, x2) una señal bidimensional, posiblemente compleja, de enerǵıafinita. La proyección de s(x1, x2) sobre el eje x1 es:

p(x1) =

ZRs(x1, x2)dx2.

• Observe que no hay unicidad en la proyección: dos objetos distintos pueden“proyectar” la misma sombra.



Proyecciones

��

��

��

��

��

��

pθ(t)

x1

s(x1, x2)

x1Intensidad de la sombra



Proyecciones

• En general, la PROYECCIÓN de s(x1, x2) sobre un eje (o ĺınea) a ánguloθ, se define como:

p(t, θ) =

ZRdr s(t cos θ − r sen θ, t sen θ + r cos θ).



Proyecciones

r x2

t

x1

θ

Proyecciones



Ecuación normal

• El ángulo θ especifica una rotación que relaciona a las variables (t, r)con las variables (x1, x2).

θ

t

x1

x2

θ

r

x1

x2(x1, x2)

x1 = t cos θ − r sen θ

x2 = t sen θ + r cos θ

)t = x1 cos θ + x2 sen θ.



Transformada de Fourier bidimensional

• Dada una función compleja s(x), con x ∈ R2, cuya enerǵıa

Ep =

ZZR|s(x)|2dx


Transformada de Fourier bidimensional

• La transformada INVERSA de Fourier (bidimensional) está dada por

s(x) =

ZZRS(f)e

i2πf ·xdf (Teorema)

• Sea S(f) una señal bidimensional, posiblemente compleja, de enerǵıafinita. La rebanada (o corte) de S(f) a lo largo de fx1 es S(f, 0); la

rebanada de S(fx1, fx2) a lo largo de un eje a un ángulo θ es

S(f cos θ, f sen θ).



Transformada de Fourier bidimensional: ejemplos

Ejemplo 1.

rect(t) =

(1, si |t| ≤ 120, si |t| > 12.

Ejemplo 2.

rect(x1, x2) =

(1, si |x1| ≤ 12, |x2| ≤

12

0, de otra forma.



III. La transformadade Radon



Teorema de la rebanada

El siguiente teorema nos dice que la transformada de Fourier unidimensional,

de una proyección de s(x1, x2) es una rebanada de S(fx1, fx2).

Teorema 1 (De la rebanada de proyección). Si pθ(t) es la proyección des(x1, x2) a un ángulo θ, y S(fx1, fx2) es la transformada de Fourier des(x1, x2), entonces

Pθ(f) = S(f cos θ, f sen θ)

es la transformada de Fourier de pθ(t).



Teorema de la rebanada

Demostración.

Pθ(f) =

ZRpθ(t)e

−i2πftdt

=

ZZRs(t cos θ − r sen θ, t sen θ + r cos θ)e−i2πftdrdt.

Empleando un cambio de variables

t = x1 cos θ + x2 sen θ y drdt = Jdx1dx2,

donde J ≡ 1 es el jacobiano de la transformación (rotación unitaria), setiene que

Pθ(f) =

ZZRs(x1, x2)e

−i2π(f cos θx1+f sen θx2)dx1dx2

= S(f cos θ, f sen θ).



• La proyección

pθ(t) =

ZRs(t cos θ − r sen θ, t sen θ + r cos θ)dr

se introdujo considerando a θ como un ángulo fijo.

• Si se define la proyección de esta forma, pero para todo valor del ánguloθ, entonces la proyección es una función de dos variables, a saber, t y θ,

y se llama TRANSFORMADA DE RADON.

Definición 4. Dada una señal bidimensional, s(x1, x2), de enerǵıa finita,la transformada de Radon de s es la función bivariada

p(t, θ) =

ZRs(t cos θ − r sen θ, t sen θ + r cos θ)dr.



La transformada de Radon también se puede definir en un espacio

multidimensional.

El teorema de la rebanada de proyección implica que la transformada de

Radon se puede invertir.

• Si P (f, θ) denota la transformada de Fourier unidimensional dep(t, θ), en la variable t, entonces

P (f, θ) = S(f cos θ, f sen θ).

• Esto es, P (f, θ) es la transformada de Fourier bidimensional des(x1, x2) expresada en coordenadas polares.



Ejemplos

Ejemplo 3. Sea s(x1, x2) = circ(x1, x2), tal que

circ(x1, x2) =

8


de donde se sigue

p(t, θ) =

ZR

circ(t, r)dr,

pero en

x21 + x

22 = t

2+ r

2 ≤1

4se tiene que

circ(t, r) = 1;

es decir,

r ≤r

1

4− t2 =

1

2

p1− 4t2.

Por lo anterior,

p(t, θ) =

Z +12√1−4t2−12√

1−4t2dr =

p1− 4t2, t ≤

1

2.

∴

p(t) =

(√1− 4t2, t ≤ 12

0, t > 12,

que es independiente de θ.



Ejemplo 4. El pulso gaussiano bidimensional

s(x1, x2) = e−π(x21+x

22)

tiene la siguiente transformada de Radon:

p(t, θ) =


=

Z +∞−∞

e−π(t2+r2)

dr

= e−πt2

Z +∞−∞

e−πr2

dr

= e−πt2

,

que es independiente de θ.



IV. La transformadainversa de Radon



En resumen

• Cuando pθ(t) se conoce ∀ θ de 0 a π, el conjunto de proyeccionesconstituye la transformada de Radon de s(x1, x2), y se denota por

p(t, θ) =


• La tarea de reconstruir la imagen s(x1, x2) a partir de sus proyecciones,es la inversión de la transformada de radon.

• El problema de inversión se puede resolver exactamente cuando no hay“ruido” y se conocen las proyecciones ∀ θ.



En resumen

• El teorema del corte proyectado dice que Pθ(f), la transformada deFourier de pθ(t), está dada por

Pθ(f) = S(f cos θ, f sen θ).

∴ una tarea equivalente a la inversión de la transformada de Radon eshallar S(fx1, fx2) cuando esté dada por pθ(t) para todos los valores del

parámetro θ.



Inversión del haz paralelo

Teorema 2. (Teorema de retroproyección, o proyecciónde retroceso o transformada inversa de Radon.)

s(x1, x2) =

Z π0

ZRPθ(f)e

i2πf(x1 cos θ+x2 sen θ)|f |dfdθ.




Demostración. La transformada inversa de Fourier es

s(x1, x2) =

ZZRS(fx1, fx2)e

i2π(fx1x1+fx2x2)dfx1dfx2.

Considerando que 0 ≤ θ ≤ π y que f : R2 → R, cambiamos la integracióna coordenadas polares:

fx1 = f cos θ,

fx2 = f sen θ,

dfx1dfx2 =∂(fx1, fx2)

∂(f, θ)dfdθ,




donde

∂(fx1, fx2)

∂(f, θ)=

˛̨̨̨cos θ sen θ

−f sen θ f cos θ

˛̨̨̨= |f | cos2 θ + |f | sen2 θ = |f |.

Entonces,

s(x1, x2) =

Z π0

ZRS(f cos θ, f sen θ)e

i2πf(x1 cos θ+x2 sen θ)|f |dfdθ.



Si escribimos x1 = r cos θ y x2 = r sen θ, el teorema de retroproyección

también se puede escribir en coordenadas polares como sigue:

Corolario 1.

s(r cos θ, r sen θ) =

Z π0

dθ

ZRdfPθ(t)|f |ei2πfr cos(θ−φ).



• Los métodos de reconstrucción de imágenes, basados en el teorema deretroproyecciones, se refieren como RETROPROYECCIÓN o proyección

de RETROCESO.

• Terminoloǵıa:

• “MATCHED FILTER”→ Procesamiento de señales.

• RETROPOYECCIÓN→ Tomograf́ıa de rayos X.

• MIGRACIÓN→ Geof́ısica o sismoloǵıa.



La estructura de la operación de retroproyección se puede simplificar si es

descompuesta en dos pasos como sigue:

1. gθ(t) =R

R df |f |Pθ(f)ei2πft

2. s(x1, x2) =R π

0dθgθ(x1 cos θ + x2 sen θ)

como se ilustra en el siguiente diagrama:



• La primera integral está dada en la forma de una transformada inversa deFourier, del corte modificado de la transformada de Fourier |f |Pθ(f).

∴ dada la estructura del producto |f |Pθ(f), uno puede suponer queel resultado de la transformada inversa de Fourier se puede describir,

formalmente, como una convolución de la proyección pθ(t) con la función

h(t), cuya transformada inversa de Fourier es la función |f |.

• Aunque la transformada inversa de Fourier de |f | NO EXISTE como unafunción propiamente dicha, EXISTE com una función generalizada.



• La segunda derivada de 12|f | es la función impulso δ(t), tal que |f |,formalmente, posee la transformada inversa de Fourier − 2√

2πt.

• Aśı, conh(t) = −

1

2

1√πt,

la operación de retroproyección se puede escribir como

gθ(t) = h(t) ∗ pθ(t) :=Z

Rdτ h(t− τ)pθ(τ)



• El término “retroproyección” se refiere al hecho de que pθ(t) está, dealguna forma, esparcido por la función h(t) a lo largo de la ĺınea.

Entonces,

s(x1, x2) =

Z π0

dθ gθ(x1 cos θ + x2 sen θ).

• Esta formulación de la retroproyección tiene la “desventaja” de que h(t)es una función generalizada con una sigularidad en t = 0.

• El empleo de una función generalizada se puede evitar siempre que

S(fx1, fx2) ≡ 0 cuandoqf2x1 + f

2x2> B,

para alguna constante B, sea que esta condición se cumple porque

está dada o porque las altas frecuencias son suprimidas mediante un

“filtro”.



• En todo caso, seŕıa suficiente reemplazar a |f | por un filtro de tal formaque H(f) = |f | para |f | ≤ B, y de otra forma H(·) sea una funciónarbitraria pero de enerǵıa finita.

• Nos referiremos a cualquier h(t), con tal H(f), como una función o“filtro rampa”.

• El empleo de un filtro rampa asegura que la transformada de Fourierbidimensional decae, en frecuencia, suficientemente rápida como para

tener definida una transformada inversa de Fourier h(t).



• En resumen, podemos reemplazar a |f | en la fórmula de retroproyecciónpor H(f), ya que

|f |Pθ(f) = H(f)Pθ(f)y

H(f)Pθ(f)↔ h(t) ∗ pθ(t).La función h(t) es llamada el NÚCLEO de la reconstrucción.

• Podemos escoger filtrar a la señal s(·, ·) o a sus proyeciones pθ(·).

• Sea h(t) un filtro unidimensional, con transformada de Fourier H(f),y sea h(x1, x2) el filtro circularmente simétrico bidimensional, cuya

transformada de Fourier

H(fx1, fx2) = H“q

f2x1 + f2x2

”está definida en términos de H(f).



Teorema 3. Sea pθ(t) la proyección a ángulo θ de s(x1, x2). Si gθ(t) =h(t) ∗ pθ(t) y g(x1, x2) = h(x1, x2) ∗ ∗s(x1, x2), entonces gθ(t) es laproyección a ángulo θ de g(x1, x2).

Donde definimos:

h(x1, x2) ∗ ∗s(x1, x2) :=ZZ

Rdξdη h(x1 − ξ, x2 − η)s(ξ, η)

y se cumple que

h ∗ ∗s = s ∗ ∗h,en donde a la función h(·, ·) se conoce como “point spread function” eninglés.

Demostración. La proyección de g(x1, x2) a ángulo θ tendrá transformadade Fourier

G(f cos θ, f sen θ) = H(f cos θ, f sen θ)S(f cos θ, f sen θ)

= H(f)Pθ(f).



Con ayuda del Corolario 4.1, tenemos que

g(r cosφ, r senφ) =

Z π0

dθ

»ZRdf H(f)Pθ(f)e

i2πfr cos(θ−φ)–,

como una forma alternativa a la retroproyección. Esta forma alternativa, que

se muestra en la figura, se conoce como retroproyección filtrada.

• El filtro H(f) se puede escoger de tal forma que las frecuencias de interésen s(x1, x2) sean permitidas.

• En este caso, seah(t) =

ZRdf H(f)e

i2πft.



• Definamosgθ(t) = h(t) ∗ pθ(t).

Entonces,

s(x1, x2) =

Z π0

dθ gθ(x1 cos θ + x2 sen θ).

• Ahora tenemos el cáculo de retroproyecciones separado en un proceso dedos pasos, dado por estas ecuaciones.

• Primero, toda proyección es filtrada por h(t) (respuesta impulsiva),entonces es evaluada una integral.



V. Dispersión de ondas y

formulas de inversión

en sismoloǵıa



Sismoloǵıa

80 M.P. Lamoureux and G.F. Margrave

Fig. 1 Seismic wave experiment

∂2ϕ

∂x2+

∂2ϕ

∂y2+

∂2ϕ

∂z2=

1c2

∂2ϕ

∂t2,

where ϕ(x, y, z, t) is the wave function and c = c(x, y, z) is the (non-constant)speed of propagation of the seismic wave. Numerical calculations based onthis equation are key to recovering an image of the subsurface.

In practice, real seismic experiments are performed by exploding dynamiteon the surface (or near surface) of the earth, and recording the vibrationsproduced by the explosive energy using sensitive geophones. The signals ofinterest are those acoustic waves that have propagated down to some inter-esting geological formation and been reflected back to the surface. Figure 1shows a simplified seismic setup, with instruments placed on the surface ofthe earth, and seismic energy traveling along raypaths within the earth. Thegeophones are typically placed at the surface of the earth,1 and are sensitiveenough to record vibrations that have traveled from the dynamite source,down five kilometers or more through rock, and return the same distanceback to the surface. Hundreds of geophones are monitored and the signaldata collected from them are recorded in a computer; dozens of dynamiteblasts are recorded, exploded at different locations, and independently at dif-ferent times. This recorded data is then processed to create an image of thesubsurface, and is often used in the search for hydrocarbons (oil and gas).

In marine seismic imaging, the experiments take place at sea rather thanon land. Typically, thousands of hydrophones attached to floating cables aretowed behind a ship traveling back and forth across a target area. A signal

1 In Vertical Seismic Profiling, or VSP, the geophones may be placed deep within the earth,usually down the borehole of an oilwell.

An Introduction to Numerical Methods of Pseudodifferential Operators 81

Fig. 2 Data collected from a seismic experiment. Reflectors are the images of interest

is initiated by setting off an air gun at the ocean surface, which starts anacoustic wave traveling down through the water and into the ocean floor,which then propagates through the rock in the form of a seismic wave. En-ergy that is reflected back travels through rock, then water, and then to thehydrophones where the data is recorded. It is also possible to use an OceanBottom Cable (OBC), where a string of geophones is actually placed on theocean floor and data recorded directly from the ocean bottom. This is a muchmore expensive setup. In either case, a given marine survey may collect dataover several days, covering dozens of square kilometers of territory. A hugeamount of data can be collected in one ocean survey, often amounting toterabytes of computer files.

The raw data contains much information, and much noise. In Fig. 2, wehave organized the time series data from a sequence of geophones placed ina line, so that a useful raw image appears. The first breaks and ground rollare due to surface propagation of seismic energy, and are considered noise.The hyperbolas represent reflections from interesting geological structures,and contain the information of a true image of the subsurface. Many of thedevelopments in seismic data processing are techniques in signal processingand include steps to remove noise due to first breaks and ground roll (muting,f–k filtering), to straighten the hyperbolas into proper reflectors (migration),and to sharpen the image (deconvolution), among others.

Thus, this problem combines aspects of physical modeling via PDEs, andsignal processing. Time–frequency analysis, and in particular pseudodifferen-tial operators, are well-suited for combining these approaches.

Objetivo: Obtener velocidad de pro-pagación v(x) de observaciones de

ondas dispersadas.

Hipótesis: Las ondas satisfacen Helm-holtz (frecuencia):

Lu(x,xs, ω) = −F (ω)δ(x− xs)

L := ∆ +ω

v2(x)

x ∈ R3 y ω ∈ R

Condición de radiación de Sommer-

feld:

|x|„∂|x|u−

iω

vu

«→ 0, |x| → ∞



x3 < 0

n̂x1

x2

x3Distancias ≈ kmsTiempos ≈ segundos

densidad constante

Ω

xs, t = 0

F (ω)

velocidad de propagación conocida



Perturbación a la velocidad de propagación

1

v2(x)=

1

c2(x)(1 + α(x))

• α perturbación “pequeña”.

• v = velocidad de propagación de las ondas.

• c = velocidad de propagación de referencia conocida.

L0u(x,xs, ω) = −F (ω)δ(x− xs)−ω2

c2(x)α(x)u(x,xs, ω)

L0 := ∆ +ω2

c2(x)Operador de Helmholtz



• Se descompone al campo de velocidades: u = uI + uS

uI respuesta impulsiva en ausencia de perturbaciones.

uS campo dispersado (respuesta a la perturbación α).

• Campo incidente uI satisface:

L0uI(x,xs, ω) = −F (ω)δ(x− xs)

sujeto a la condición de radiación de Sommerfeld

• Campo dispersado uS satisface

L0uS(x,xs, ω) = −ω2

c2(x)α(x)[uI(x,xs, ω) + uS(x,xs, ω)]



Representación integral del campo dispersado

Teorema y función de Green (densidad constante)

L∗0g∗(x,xg, ω) = −δ(x− xg)Z

Ω

dx[g∗L0u− uL∗0g

∗] =

Z∂Ω

ds[g∗∂nu− u∂ng∗]

xs

fuente

xg1 xg2

receptoresF (ω)

n̂

Ω



Ecuación integral para uS

• Solución al campo dispersado:

uS(xg,xs, ω) = ω2Z

Ω

dxα(x)

c2(x)[uI(x,xs, ω)+uS(x,xs, ω)]g(xg,x, ω)

Ω contiene el soporte de α (subdominio finito de z > 0).

• Aproximación de Born: Si |αuS| � |αuI| entonces

uS(xg,xs, ω) = ω2Z

Ω

dxα(x)

c2(x)uI(x,xs, ω)g(xg,x, ω)



Configuración de fuentes y receptores

• Fuentes y receptores están localizados en una superficie Γ:

Superficiefuentereceptor

xg(ξ1, ξ2)Γ(ξ)

xs

ξ1ξ2



Solución aproximada de uS

• Campo incidente:

uI(x,xs, ω) = F (ω)g(x,xs, ω)

• Función de Green WKBJ (altas frecuencias – óptica geométrica):

g(x,x0, ω) ∼ A(x,x0)eiωτ(x,x0)

donde τ tiempo de viaje de x a x0 y A amplitud (rayo).

uS(xg,xs, ω) ≈ ω2F (ω)Z

Ω

dxα(x)

c2(x)a(x, ξ)e

iωΦ(x,ξ)

Φ(x, ξ) = τ(x,xs(ξ)) + τ(xg(ξ),x)

a(x, ξ) = A(x,xs(ξ))A(xg(ξ),x)



Fórmula de inversión

• Se postula el operador:

α̃(y) =

Zdω

ZdξB(y, ξ)e

−iωΦ(y,ξ)uS(xg,xs, ω)

• B determina la estabilidad de la fórmula.

• Se obtiene substituyendo uS en la expresión para α̃:

α̃(y) =

Zdωω

2F (ω)

ZdξB(y, ξ)

Zdxe

iω[Φ(x,ξ)−Φ(y,ξ)]α(x)C(x, ξ)

donde

C(x, ξ) :=a(x, ξ)

c2(x).



Observación clave

• El kernel de la expresión debe tener (asintóticamente) la propiedad“coladera” de la delta de Dirac

α̃(y) ∼Z

dxδ(x− y)α(x)

es decir

δ(x− y) =Z

dωω2F (ω)

ZdξB(y, ξ)e

iω[Φ(x,ξ)−Φ(y,ξ)]C(x, ξ)

• x = y punto cŕıtico dominante⇒ aproximación de Taylor:

iω[Φ(x, ξ)− Φ(y, ξ)] ≈ ik · (x− y)

k := ω∇xΦ(x, ξ)|x=y

• Cambio de variables (Stolt):

(ω, ξ)→ k.



• En términos de k

δ(x− y) ∼Zd

3kB(y, ξ)ω

2(k)F (ω(k))

a(y, ξ)

c2(y)J(ω, ξ,k)e

ik·(x−y)

donde

ω(k) =k · ∇yΦ(y, ξ)|∇yΦ(y, ξ)|2

,

J(ω, ξ,k) =

˛̨̨̨∂(ω, ξ)

∂(k)

˛̨̨̨. (Determinante de Beylkin)

• El lado derecho de la expresión NO es una tranformada de Fourier.

• Si la expresión fuera una igualdad, la amplitud de la integral seŕıa igual a1/8π3, si además F (ω) ≈ 1 entonces

B(y, ξ) =1

8π3h(y, ξ)c2(y)

a(y, ξ)



Fórmula de inversión aproximada

• En conclusión obtenemos:

α̃(y) =1

8π3

Zdξ

Zdωe

−iωΦ(y,ξ)P (y, ξ)uS(xg,xs, ω)

donde

P (y, ξ) =h(y, ξ)c2(y)

a(y, ξ)



VI. Dispersión de ondas y

formación de imágenes en

radares de apertura

sintética



Los radares (“RAdio Detection And Ranging, 1940”)

1887 - H. R. Hertz. Demostró que se pueden propagar ondas electro-

magnéticas cierta distancia. Radioondas se pueden transmitir a través de

ciertos materiales y otros las reflejan.

1904 - Hülsmeyer (Alemania) patenta el primer radar.

◦ “Hertzian-wave projecting and receiving apparatus adapted to indicateor give warning of the presence of a metallic body, such as a ship or a

train, in the line of projection of such waves”.

1922 - Taylor and Young: Detección sistemática de nav́ıos en Naval

Research Laboratory, Washington, D. C. 1930 - Hyland: Primer detección

de aeronaves.

¡En 1941, un radar del ejército estadounidense detectó aviones japoneses

aproximándose a Pearl Harbor, pero el supervisor a cargo decidió que las

señales eran espurias!



La Segunda Guerra Mundial impulsa el desarrollo de nuevos modelos de

radares.Radar inverse scattering R3

Figure 1. A generic radar detection system.

and locating radar targets1. Active radar systems accomplish these tasks by transmitting anelectromagnetic waveform and measuring the reflected field as a time-varying voltage in theradar receiver. If R denotes the distance from the transmitter to the target, then the radiationcondition guarantees that the energy in the transmitted field will decay as R−2 when R issufficiently large (this is the usual case). The energy reflected by the target will also obey thisradiation condition and so, in monostatic radar configurations (in which the radar transmitterand receiver are co-located), the energy collected by the receiver will be reduced from thattransmitted by a factor of R−4. In practice, this received energy is quite small, and when thetarget is far away the measured signal competes for attention with the thermal noise voltage ofthe radar system itself. Accurately separating the wanted signal from the unwanted noise is animpressive engineering feat and it is not surprising that a fair amount of device-specific jargonhas evolved over the years. This section will briefly explain the methods employed.

Figure 1 shows the main elements of a generic (and greatly simplified) radar measurementsystem (cf [11, 82, 96, 127], and references cited therein). An incident time-varying signalsinc(t) is mixed with a carrier signal with (angular) frequency ω0 chosen to conform withfavourable properties of the transmission medium (typically, atmospheric windows are soughtfor which free space parameters are good approximations). This mixed signal is used to excitecurrents on an antenna that cause an electromagnetic waveform to be launched toward thetarget. (This antenna is rarely an isotropic radiator: it is usually designed to have some degreeof directionality (gain), and this directionality can also be used to estimate target bearing.) Thewaveform is reflected from the target and the reciprocal situation occurs in the receiver: thereflected field induces antenna currents; the signal voltage is mixed with a conjugate carrier toremove (beat down) the carrier frequency dependence; and the time-varying voltage srec(t) isoutput to a signal processor. Note that, while high frequencies dominate the target scatteringphysics (ω0 is typically 1–35 GHz), the signal processing block usually sees much lowerfrequencies (ω ! 0.1ω0).

2.1. I and Q signals

The system signals are real-valued voltages and are often expressed in terms of circularfunctions p(t) = a(t) cos φ(t) so that the concepts of amplitude and phase may beintroduced. It is convenient, however, to consider the signals to be complex-valued, withs(t) = 12 (p(t) + iq(t)) for some q(t). Of course, the choice of q(t) is not unique and,1 A radar target is any object of interest illuminated by the radar. In contrast, radar clutter is composed of collateraland undesired illuminated objects.

R14 B Borden

Figure 4. Example ISAR image. The image is the |ρ(t, ν)|2 of a Boeing 727 jetliner withorientation as in the inset. The radar centre frequency and bandwidth were 9.25 GHz and 500 MHz,respectively. Compare this figure with the range profile of figure 3.

of the library are called templates: τn ∈ LN where n indexes the N known targets in the library.In the case of range profile images the templates are functions of the range variable t and areparametrized by θ .

Let ρ̂(t) denote a range profile measured using an HRR radar. There are many ways tocompare ρ̂(t) with τn(t; θ), but one of the most common is based on the maximum correlationcoefficient

C(n, θ) = maxt∈R

∫

Rρ̂(t ′) τn(t ′ + t; θ) dt ′. (28)

Template correlation methods compute C(n, θ) for each n = 1, . . . , N and each θ =θ1, . . . , θM . Target type and orientation are then estimated as

n, θ = arg maxn′=1,...,N

θ ′=θ1,...,θM

{

C(n′, θ ′)}

. (29)

The search over the M target orientations is required because it is impossible to divinetarget aspect from ρ̂(t) (and it must be implicitly understood that θ generally refers to anordered pair of spherical angles). Typical libraries are set up with N ≈ 25 ‘targets of interest’.But, as we shall see below, there is little freedom in the choice of M .



La ecuación del radar

zy

φ

θx Antena receptora

φ′θ′

Antena transmisora

Reflector

Rt Rr

Ecuación de alcance del radar:

Er = σEt

R4

La propagación de las ondas está caracterizada por la permitividad

eléctrica ε y permeabilidad magnética µ del medio.



Radar de Apertura Sintética (SAR), 60s

~γ(s)

x1

x2

h

x3

~xv

εT

Sistema de monitereo y reconocimiento remoto: Ondas oceánicas,humedad de suelos, ecoloǵıa forestal, cartograf́ıa de planetas, etc.

Antena en movimiento transmite y recolecta pulsos electromagnéticos.

Śıntesis de la apertura efectiva de la antena.

Procesamiento de las señales recibidas para formar imágenes.



Modelo de dispersión electromagnética

Ecuaciones de Maxwell (1861-1873)

(∇×E =− ∂tB, ∇ ·D = ρb,

∇×H =∂tD + J, ∇ ·B = 0.

Relaciones constitutivas

B(t,x) = µ0H(t,x) y D(t,x) = (ε ∗t E)(t,x)

D(t,x) =

Z ∞0

ε(s,x)E(t− s,x)ds, (E(t,x)|t


Permitividad eléctrica del medio

γ(s)

ψ

T

�r

dispersor (inhomogeneidad del medio)→ perturbación a la permitividad

ε(s,x) = ε0εr(s,x) +

término perturbativoz }| {ε0εT (x1, x2)δ0(x3 − x03)δ(s)

∇2E − ∂2t (µ0ε0εr| {z }medio

∗tE) = (µ0ε0εTδ0)| {z }dispersor

∂2tE − (µ0∂tJ)xi| {z }

fuente



Campo total

Campo total E = Ein + Esc

∇2E − ∂2t (µ0ε0εr ∗t E) = (Tδ0)∂2tE − (µ0∂tJ)xi

T ≡ µ0ε0εT (blanco)

Función de Green↔ Transformada de Fourier (1822)

∇2g − ∂2t (c−20 εr ∗t g) = −δ(t)δ(x− y), (c

−20 = µ0ε0)

g(t,x− y) =1

2π

Ze−iω(n(ω)|x−y|/c0−t)

4π|x− y|dω

n(ω) = nR(ω) + i nI(ω) ≡q�r(ω) ∈ C (́ındice de refracción)



Campo incidente

Campo fuente generado en la antena

∇2Ein − ∂2t (c−20 εr ∗t E

in) = (µ0∂tJ)xi = −js

js → δ(y − yc)p(t) = δ(y − yc)Z ωmaxωmin

e−iωt

P (ω)dω (TF pulso)

Ein = (g ∗ js)(t,x)

Ein

(t,x) =

Ze−iω(t−n(ω)|x−yc|/c0)

4π|x− yc|P (ω)dω

Dispersor x está lejos de la antena (isotrópica y puntual) yc = γ(s)

∴ Ein(t,x, s) =Ze−iω(t−n(ω)|x−γ(s)|/c0)

4π|x− γ(s)|P (ω)dω



Campo dispersado y approximación de Born

Campo reflejado por los dispersores

∇2Esc − ∂2t (c−20 εr ∗t E

sc) = (Tδ0)∂

2tE, (T = c

−20 εT (x))

∴ Esc(t,x) = −Zg(t− τ,x− z)T (z)δ0(z3 − z03)∂

2τE(τ, z)dτdz

‖Esc‖ � ‖Ein‖ ⇒ E → Ein (enerǵıa)

EscB (t,x, s) ≈ −

Zg(t− τ,x− z)T (z)δ(z3 − z03)∂

2τE

in(τ, z, s)dτdz

EscB (t,x, s) =

Ze−iω(t−n(ω)(|x−z|+|z−γ(s)|)/c0)

(4π)2|x− z||z − γ(s)|ω

2P (ω)T (z)dωdz

El campo dispersado se colecta en la antena x = γ(s)⇒



Problema directo �

Señal recibida por la antena

EscB (t, s) =

Ze−iω(t−2n(ω)|z−γ(s)|/c0)

(4π)2|z − γ(s)|2ω

2P (ω)T (z)dωdz

P (ω)→ TF del pulso incidente p(t)

Ruido aditivo (ruido térmico) η(t)

d(t, s) = EscB (t, s) + η(t)

Problema inverso→ Determinar T (z) a partir de d(t, s)

F [T ] 7→ d ⇐⇒ M [d] 7→ T



FORMACIÓN

DE IMÁGENES



Imágenes

Información de una región espacial que

se obtiene midiendo el campo dispersado

desde diferentes ángulos y frecuencias



Problema inverso: Formación de imágenes �

Retroproyección (Matched Filter) M [d] 7→ T̃ (z).

T̃ (z) =

Zeiω′(t−2nR(ω

′)|z−γ(s)|/c0)Q(ω′, z, s)d(t, s)dω

′dsdt

◦ Propiedad clave de M : misma fase que CC de F (F [T ] 7→ d).◦ Filtro Q por determinar.

Substituyendo d = F [T ] en T̃ = M [d]:

◦ Cambio de variables (ω, s)→ ξ ∈ Ω,“a ≡ ω

2e−2ωnI(ω)|γ(s)−y|/c0(4π|γ(s)−y|)2

”T̃ (z) =

Zei(z−y)·ξ

Q(ξ, z)a(ξ,y)P (ξ)T (y)J(ξ, z,y)dξdy+ T̃η(z)



Filtro óptimo: Ḿınimo error cuadrático medio �

Reflectividad ideal

T̃Ω(z) =

Zξ∈Ω

ei(z−y)·ξ

T (y)dydξ

Error cuadrático medio (Varianza)

∆(Q,P ) =

Z〈|T̃ (z)− T̃Ω(z)|2〉dz ∈ R

Cálculo de variaciones

0 =d

dρ

˛̨̨̨ρ=0

∆(Q+ ρQρ, P )



Filtro de Wiener

Qopt

=P (ω, s)a(ω, s, z)ST (ω, s)

|P (ω, s)|2J(ω, s, z)a2(ω, s, z)ST (ω, s) + Sη(ω, s)

Densidades espectrales Sη y ST (Transformada de Fourier de la

correlación)

Sη(ω, s) =

Zdτe

−iωτ〈η(τ + t, s)η(t, s)〉

ST (k,k′) =

Zdydy

′e−iy·k

eiy′·k′〈T (y)T (y′)〉

Procesos estocásticos estacionarios: media y varianza no cambian



RESULTADOS

NUMÉRICOS



Implementación numérica: Problema directo d(t, s) �

(a) d(t, s) =ReiωtD(ω, s).

(b) Discretización D(ω, s)→ D(ωi, sj)def= Dij

Dij ≈P (ωj)

(4π)2

Xdispersorm

Tmei2ωjn(ωj)|γ(si)−ym|/c0

|γ(si)− ym|2

(c) ◦ Dispersores localizados en ym1, ym2, etc.◦ Trayectoria de vuelo circular γ(s) con tiempo lento s ∈ [0, 2π].◦ Sinusoide de modulado cuadrado p(t).

(d) Se obtienen los datos directos empleando FFT (60’s): dij = F [Dij].



Experimento numérico

−100−80

−60−40

−200

2040

6080

100

−100−80

−60−40

−200

2040

6080

1000

5

10

T2T1

T2T1

T2T1

T2T1

T2T1

T2T1

T2T1

T2

T1

T2

T1

T2

T1

T2

T1

T2T1

T2T1

h=

10

m

m m

r=100 m

γ(s)

107 108 109−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

[Hz]

Fung-Ulaby model dielectric

nRnI

−10 0 10 20 30 40 50 60 70 80 90 100−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

[ns]

[V/m

]

0 0.05 0.1 0.15 0.2 0.25

−40

−30

−20

−10

0

[ G H z ]

[dB

]



Problema directo: follaje ralo

500 600 700 800 900 1000 1100 1200−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5x 10

−11

[ns]

[V/m

]

[ns]

[rad

]

600 700 800 900 1000 1100

0

1

2

3

4

5

6

500 600 700 800 900 1000 1100 1200

−5

−4

−3

−2

−1

0

1

2

3

4

5

x 10−11

[ns]

[V/m

]

[ns]

[rad

]

600 700 800 900 1000 1100

0

1

2

3

4

5

6



Implementación numérica para formar imágenes I(z)

(a) Discretización I(z)→ I(zi, zj)def= Iij

Iij =Xm

∆sXn

∆ωne(−iωn2nR(ωn)|γ(sm)−zij|/c0)Q

optmnDmn

(b) Se discretiza Qoptmn:

Qoptmn =

Pnanm(zij)ST (zij)

|Pn|2Jnm(zij)|anm(zij)|2ST (zij) + Sn

(c) Sn y ST distribuidos normalmente.

(d) La reconstrucción es sobre una superficie.



Imagen reconstruida sin ruido

Figura 6: Representación en 3D de la imágen reconstruida en un medio con

ı́ndice de refracción constante.



Imagen reconstruida sin ruido

Figura 7: Representación en 3D de la imágen reconstruida en un medio con

ı́ndice de refracción que depende de la frecuencia (medio dispersivo).



Imágenes reconstruidas

[m]

[m]

!

!

!" !# !$ % $ # "

!"

!#

!$

%

$

#

"%&'

%&$

%&(

%

%&)

%&"

%&*

%&+

%&,

'

[m]

[m]

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6 0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

[m]

[m]

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

[m]

[m]

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

60.1

0.2

0.3

0.4

0.5

0.6

0.7




[ m ]

[m]

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

[m]

[m]

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

0.2

0.4

0.6

0.8

1

1.2

[ m ]

[m]

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

[ m ]

[m]

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8



Problema directo: follaje “denso”

107 108 109−0.5

0

0.5

1

1.5

2

2.5

[Hz]

Fung-Ulaby model dielectric

nRnI

800 850 900 950 1000 1050 1100 1150 1200

−6

−4

−2

0

2

4

6

x 10−12

[ns]

[V/m

]

[ns]

[rad

]

700 800 900 1000 1100 1200 1300

0

1

2

3

4

5

6




[ m ]

[m]

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

60.02

0.04

0.06

0.08

0.1

0.12

0.14

[ m ]

[m]

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

[ m ]

[m]

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

6 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

[ m ]

[m]

−6 −4 −2 0 2 4 6

−6

−4

−2

0

2

4

60.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8



Trabajo en colaboración con

• Margaret CheneyInverse Problems Center & Department of Mathematical Sciences.

Rensselaer Rensselaer Polytechnic Institute (RPI), Troy NY.

• Trond VarslotDepartment of Applied Mathematics. The Australian National University,

Canberra.

T. Varslot, J. H. Morales, and M. Cheney. Synthetic-aperture radarimaging through dispersive media. Inverse Problems, 26 (2010), pp. 1–27(025008).



Conclusiones

Cómputo

Cient́ıfico

Métodos

Estad́ısticos

Funcional

ProblemasInversos

Ecuaciones enDerivadas Parciales

Análisis



FINReferencias:

Charles L. Epstein, 2008. Introduction to the Mathematics of Medical

Imaging. 2nd. Edition, SIAM. Philadelphia.

Habib Ammari, 2000. An Introduction to Mathematics of Emerging

Biomedical Imaging. Springer-Verlag.

Norman Bleistein, et al, 2001. Mathematics of Multidimensional SeismicImaging, Migration, and Inversion. Springer-Verlag.

Margaret Cheney, 2001. A Mathematical Tutorial on Synthetic Aperture

Radar. SIAM Review Vol. 43, No. 2, pp. 301-312.

Wade Allison, 2006. Fundamental Physics for Probing and Imaging.

Oxford University Press. New York, USA.



Ejemplo introductorio

Operador de onda en medio homogéneo

∂2tu(x, t)− c

2∆u(x, t) = 0, (x, t) ∈ Rd × R, d > 1 (1)

Condiciones iniciales

u(x, 0) = u0(x) y ∂tu(x, 0) = 0 (2)



Velocidad de propagación constante⇒• T. Fourier desacopla componentes espectrales (frecuencias) de u.• Cada componente una EDO que se resuelve expĺıcitamente.• Solución u(x, t) superposición de modos de Fourier:

u(x, t) =1

2

„Zdye

2πi(x·y+c|y|t)û0(y) +

Zdye

2πi(x·y−c|y|t)û0(y)

«Fase Φ±(x, y) = x · y ± c|y|t



Linealización del problema inverso

Operador diferencial y su perturbación

Lv = g, L = L0 + L1

Campo total

v = vin

+ vds, v

in:= L−10 g

vds

= −L−10 L1vin − L−10 L1v

ds

Solución approximada al campo dispersado

vds ≈ −L−10 L1v

in



Fluido de densidad constante

Índice de refracción. Por determinar: f(x)

n2(x) = 1 + f(x), x ∈ Ω ⊂ Rd

Operador de Helmholtz en un medio de velocidad constante

L0 = ∇2x + k2

Perturbación al operador

L1 = k2f(x)Cuando d = 3 se obtiene el campo dispersado

vds

(k, ξ, η) = −k2

16π2

ZΩ

eik|x−ξ|

|x− ξ|eik|x−η|

|x− η|f(x)dx



Ecuación integral del problema inverso linealizado

v(k, ξ, η) = (−ik)d−1Z

Ω

f(x)a(x, ξ, η)eikφ(x,ξ,η)

dx

Fase y amplitud

φ(x, ξ, η) = φ̂(x, ξ) + φ̃(x, η)

a(x, ξ, η) = Aout

(x, ξ)Ain

(x, η)

a(x, ξ, η) ∈ C∞(Ω̄× ∂Ω× ∂Ω)



Transformada de Radon generalizada

(Rf)(t, ξ, η) =(R

Ωf(x)a(x, ξ, η)δ(t− φ(x, ξ, η))dx, t ≥ 0

0, t < 0

Transformada de Fourier F

F(RF ) = ̂(Rf)(k, ξ, η)

v(k, ξ, η) = (−ik)d−1 ̂(Rf)(k, ξ, η)



Pseudodifferental operators

Operador diferencial lineal con coeficientes constantes

P (D) :=Xk

akDk F−→ P (ω) =

Xk

ak ωk

(3)

Transformada inversa de Fourier

P (D)u(x) =1

(2π)d

ZRddy

ZRddωe

i(x−y)ωP (ω)u(y)

Dk

:= (−i∂1)k1 . . . (−i∂d)kd(4)

Operador pseudodiferencial

P (x,D)u(x) =1

(2π)d

ZRddy

ZRddωe

i(x−y)ωP (x, ω)u(y) (5)



Fourier Integral Operators

Propagan singularidades (sistemas hiperbólicos)

La funciones de fase satisfacen relaciones de homogeneidad

Φ(x, λω) = λΦ(x, ω) (6)

Ecuación del calor, ecuación de Schrödinger no son consideradas OIF

porque dispersan y atenúan los frentes de onda



Fluido de densidad variable 1

Índice de refracción. Por determinar: f(x)

n2(x) = n

20(x) + f(x), n0 se conoceOperador diferencial y su perturbación

L0 := ∇2x + k2n

20, L1 := k

2f(x)

Campo incidente debido a una fuente puntual

(∇2x + k2n

20)v

in(x, η) = δ(x− η)

Approximación óptica geométrica

vin

(x, η) ≈ e−i(π/2)(d+1)/2k(d−3)/2Ain(x, η)eikφin(x,η)



Fluido de densidad variable 2

Función fase y la ecuación eikonal

(∇xφin)2 = n20(x)Amplitud y la ecuación de transporte

Ain∇2xφ

in+ 2A

in · ∇xφin = 0Intercambio de fuentes por receptores

vout

(x, ξ) ≈ e−i(π/2)(d+1)/2k(d−3)/2Aout(x, ξ)eikφout(x,ξ)

Representación integral del campo dispersado

vds

(k, ξ, η) = (−ik)d−1Z

Ω

eikφout(x,ξ)

eikφin(x,η)

Aout

(x, ξ)Ain

(x, η)f(x)dx



Ondas en un medio dispersivo: Precursores (Brillouin,1914)

x

x

t = 0

t > 0Precursor

x = b x = a

Signal Wavefront

E(t) =

(0, t < 0,

sin(ωt), t ≥ 0.

x = 0

E(x, t) = Re

1

2π

Z ∞−∞

F (k)e−i[ω(k)t−kx]

dk

ff→ ω(k) ≈ ω0 + (k − k0)

dω0

dk0

E(x, t) ≈ E(x− (dω0/dk0)t, 0) exp„−i»ω0 − k0

dω0

dk0

–t

«|E|2 = |E(x− (dω0/dk0)t, 0)|2 → vg =

dω0

dk0



Modelo dieléctrico de Fung-Ulaby en follaje

Permitividad efectiva relativa (tipo Debye):

�r,eff(ω) = vl�l + (1− vl),

�R(ω) = 5,5 +em − 5,51 + τ2ω2

,

�I(ω) =(em − 5,5)τω

1 + τ2ω2,

donde �l = �R(ω) + i �I(ω) y em = 5 + 51,56vm.

vl volumen fraccional ocupado por hojas.

vw volumen fraccional de agua en “hoja t́ıpica”.

τ tiempo de relajación del material (a 20 ◦C, τ ≈ 10.1 ps).



Matched Filtering (North, 1943)

El “matched filter” es un filtro lineal óptimo que maximiza la razón de

señal a ruido en presencia de ruido aditivo.

sr(t) = st(t) + η(t)

so(t) = h(t) ∗t sr(t)„S

N

«salida

=|so(t)|2

〈η2(t)〉.

Teorema. En presencia de ruido, con densidad de potencia N0 watts porhertz, la máxima razón de potencia de señal a ruido se alcanza por medio

del filtro

h(t) = Ks∗r(t0 − t)

La respuesta impulsiva es una “imagen” retrasada del conjugado complejo

de la señal.


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