F.K. Knight, S.R. Kulkarni, R.M. Marino, and J.K. Parker

Tomographic Techniques Applied toLaser Radar Reflective Measurements

Methods of tomography are applied to laser radar reflective measurements to studyremote imaging of macroscopic objects. Techniques to produce 2-D images from I-Ddata and 3-D images from 2-D data are described, and examples are shown. The dataare the received signals from laser radars, resolved in either I-D (range or Doppler) or2-D (angle-angle) and taken from many viewing directions. Examples are presented ofreconstructed images of laboratory test objec1s obtained with infrared and visible laserradars. Each reconstructed image depicts the object's geometric features. Prospects forfuture applications are discussed.

This article describes the application of to­mographic Image-reconstruction techniques tomeasurements made with laser radar remotesensors. Tomographic methods are used to re­construct an image from a set of its projectionsand have been applied to many fields, e.g., radioastronomy and medical imaging [1,2]. Here weuse a well-developed technique of tomographyto combine laser radar reflective measurementstaken from many viewing directions. The resultis an image ofthe illuminated object. We discussthe reconstruction of 2-D images from f-D dataand the reconstruction of 3-D images from 2-D

'< data. This article reviews work reported in more(''c detail elsewhere [3-5].

As inX-ray absorption CAT scans, the goal of2-D transmission tomography is to estimate thespatial dependence of the absorption of a pene­trating radiation-based on a series of 1-D pro­jections of a slice of an object. Transmissiontomography utilizes a line of detectors to resolvethe absorption characteristics of the objectalong an axis perpendicular to the line of sight(LOS) of the detector. The signal from eachdetector is the integrated absorption along theLOS through the object, so that a line of detec­tors produces a 1-D absorption projection oftheobject. The absorption at each point in the slicecan be estimated from a series of such projec­tions measured in angular increments aroundthe object.

In the laser radar measurements discussed

The Lincoln Laboralory Journal. Volume 2. Number 2 (1989)

here, the object is resolved in either range,Doppler (velocity), or angle. The signal in eachresolution cell represents the energy reflected offthe corresponding illuminated surface of theobject. A series of signals along the resolutioncoordinate produces a reflective projection oftheobject. The goal of reflective tomography is toestimate object surface features based on a setof reflective projections that are measured inangular incre,ments around the object.

While the' two types of measurements­transmission and reflective-are fundamentallydifferent, as shown in Fig. 1, there are similari­ties in the collected data. The sensor geometriesassociated with looking around the respectiveobjects can be similar. Also, radar reflectivemeasurements can be interpreted as weightedprojections of the object's radar cross sectionalong the direction in which the object is re­solved. For these reasons, techniques of trans­mission tomography can be applied to reflectivemeasurements to yield information about thesurface features of the object 13-13]. (See thebox, "Transmission and Reflective Tomogra­phy," for an illustration of reflective and trans­mission tomography.)

This article describes three types of radardata (range-resolved, Doppler-resolved, andangle-angIe-resolved) and a standard method ofimage reconstruction from projections (filteredback-projection). Laboratory and field measure­ments that use the three radar types are pre-

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Knight et aI. - Tomographic Techniques Applied toLaser Radar Rejlective Measurements

(b)

Fig 1-A diagram of two types of tomography distribution.(a) In transmission tomography, transmission through theobject is used to reconstruct the interior mass. (b) Inreflective tomography, light reflected off the surface of theobject is used to reconstruct surface features.

sented. Doppler-resolved projections of an ob­ject are measured with a narrowband infraredlaser radar; range-resolved and angle-angle­resolved reflective projections are measuredwith a short-pulse visible laser radar. Thesemeasurements serve as examples of data setstaken from many viewing directions and areused as input to the tomographic reconstructionalgorithms.

Laser Radar Measurements

Laser radars can be designed to provide a

144

OJOlCellc:

Fig. 2-A range-resolving radar views an object whosedepth is greater than the range resolution L'>R. The receiverdetects a signal (blue), which is a continuous function ofrange, and produces a histogram (red) with a cell size L'>R.The magnitude of the received signal depends on thesurface area and orientation in each cell, and on thematerial reflectance.

variety of data from remote objects. With suffi­cient angular resolution, an object can be im­aged in angle to yield a 2-D angle-angle sig­nature. With sufficient range resolution, theobject's reflective Signature can be resolvedin a 1-0 range dimension. Similarly, with rela­tive rotation between the object and the sen­sor, and sufficient Doppler resolution, the ob­ject's reflective Signature can be resolved in a1-0 Doppler dimension. For any laser radar,the received signal represents informationabout the surface of the object illuminated bythe radar from a given LOS.

Consider an object illuminated by a range­resolving radar, either with short pulses or withfrequency modulation, as shown in Fig. 2. Thereceived signal is separated into time or fre­quency cells, each corresponding to a rangeextent tl.R. We are interested in the case inwhich tl.R is less than the projected depth of the

The Lincoln Laboratory Joumal. Volume 2. Number 2 (1989)

Velocity

Fig. 3-A Doppler-resolving radar views a rotating objectwhose velocity spread is greater than the velocity resolutiont:.V. The receiver detects a signal (blue), which is acontinuous function of velocity, and produces a histogram(red) with a cell size t:.V. The magnitude of the receivedsignal depends on the surface area and orientation in eachvelocity cell and on the material reflectance.

object, so that the object is resolved in range.Each cell that receives a signal from the objectcontains reflected radiation from all unshad­owed portions ofthe object's surface in a slice ofrange extent ,....R. For a radar pulse of duration -ror a bandwidth B = 1/-r,

t:.R = cr = ~2 2B

where c is the speed of light. The visible radarused for range-resolved measurements hasT = 250 ps, which yields a range resolution of

The Lincoln Laboratory Journal. Volume 2. Number 2 (J 989)

Knight et aI. - Tomographic Techniques Applied toLaser Radar Reflective Measurements

about 4 cm. This resolution is sufficient to showdetails on meter-sized objects.

Alternatively, consider an object illuminatedby a Doppler-resolving radar, as shown in Fig. 3.The frequency of the received signal is shifted byan amount proportional to the object's velocitycomponent along the radar LOS. If the object isspinning, the received signal is spread in fre­quency because of the variation of the LOScomponent of velocity across the object's sur­face. The received signal is separated into fre­quency cells; each cell corresponds to an inter­val of projected velocity L1V. We are interested inthe case in which L1V is less than the projectedvelocity spread ofthe object, so that the object isresolved in velocity. The Doppler-resolved mea­surement for a spinning object represents across-range projection of the object weighted bythe reflectance of the surface. Velocityresolution L1 V can be achieved with a narrow­band waveform of carrier frequency V andduration or, where

t:.v=-.£....2vr

The infrared Doppler radar used for the Doppler­resolved measurements is a continuous-waveradar sampled with a time window -rof 1 ms. Thewavelength of 10.6 11m or a frequency of V = 28THz yields a velocity resolution of 0.52 cm/s.For a I-m diameter cone, spinning at 1 rpm, ori­ented perpendicular to the LOS, the spread ofprojected velocity is ±5.2 cm/s, which yields 20cells across the cone.

Two-dimensional images can be obtained di­rectly with a laser radar that resolves the objectin angle-angle dimensions, as shown in Fig. 4. Inan angle-angle measurement, the object is re­solved in angular pixels across the object field.The resolution of angle-angle imaging is deter­mined by the diameter D of the optics, thewavelength A., and the size of the detector ele­ments. For diffraction-limited performance, theresolution is proportional to A./ D. The visibleradar used for the angle-angle measurementshas a 1V camera with 300 x 300 angular pixelsover a field ofview of 17° diameter. The pixels on

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Knight et a1. - Tomographic Techniques Applied toLaser Radar Reflective Measurements

Transmission and Reflective Tomography

__rL

contain the same information. Al­though the Doppler-resolved pro­jections don·t resemble the trans­mission projections. tomographictechniques can still be applied toyield reconstructions that showobject features but not necessar­ily outlines.

P(U3'¢3)

P(U2'¢2)+P(U3.r/J3)~

Fig. 8-Three range-resolved reflective projections of a hol­low diffuse box. Notice that the combined projection p(u, ¢)+ p(-u, ¢ + 180") has a shape similar to the transmissiveprojections of Fig. A.

Fig. A-Three transmission projections of a hollow box. Noticethat the projections from ¢2 and from ¢3 (= ¢2 + 180") contain thesame information.

reconstruct an image resemblingthe projected outline ofan opaqueobject.

Figure C illustrates an opaquebox with Doppler-resolved reflec­tive projections taken from thesame three directions. Againcolinear measurements takenfrom opposite directions do not

A simple example illustratesthe functional sirnilartties be­tween transmission tomographyand reflective tomography. FigureA shows an empty rectangularbox of uniform wall mass densityand thickness. Three separatetransmission projections p(u. ¢)of the empty box. taken fromviewing directions ¢!' ¢2' and ¢3(= ¢2 + 180°), are shown. The twopeaks of each projection are dueto the illcreased absorption fromthe two walls that have normalsperpendicular to the LOS for eachmeasurement. otice that theprojections of ¢2 and 1/>3 containthe same information. In general.measurements made from 180°to 359° contain no more informa­tion than those made from 0° to179°.

Figure B shows range-re­solved reflective projectionsp(u, ¢) measured from the sameobject with a laser radar. To thelaser radar the walls of the objectare opaque. and the strong reflec­tion from the face nearest thesensor, which is normal to theWS for viewing directions ¢!' ¢2'and ¢3 (= ¢2 + 180°). dominateseach projection. Unlike the trans­mission case, the projections of¢2 and ¢3 do not measure thesame features (ill this case, thesame surfaces). Thus. measure­ments over a full 360° are re­quired for nonsymmetric or un­known objects. otice the similarshape of the range-resolvedprojection P(~, ¢2) + P(U:3' ¢3)with those of the transmissionprojections. The sirnilartty leadsto an illteresting observation. Aseries of transmission projectionstaken from a hoUow object can re­construct the outline of the ob­ject, whereas a series of range­resolved reflective projections can

146 The Lincoln Laboratory Journal. Volume 2. Number 2 (1989)

p(u. ep) = Jg(x. y) dsL u .¢

--1....-----..,L

Fig. C-Three Doppler-resolved reflective projec­tions of a rotating hollow diffuse box. Notice thatthese projections do not clearly resolve the facesof the box.

an object at a range of 10 m are 1 cm x 1 cm.These dimensions. although not diffraction­limited. are sufficient to show details on meter­sized objects.

Reconstruction Method

A number of methods for transmission to­mography have been developed and described inRefs. 1 and 2. We have considered and comparedsome of the methods as applied to laser radardata [5J. This section describes one standardtomographic method known as filtered back­projection.

First. notation and definitions are required.Let g(x. y) denote the image to be reconstructed.and let Lu . ¢ denote the line u = x cos </J + Y sin </J.

Let p(u. </J) denote the integral of g(x. y) alongL '" Le ..u. ",.

where s represents arc length along L u. ¢ (Fig. 5).For a fixed </J, p(u, </J) as a function of u is calledthe projection of g(x. y) in the direction </J.

The Lincoln Laboratory Journal. Volume 2. Number 2 (1989)

Knight et aI. - Tomographic Techniques Applied toLaser Radar Reflective Measurements

Fig. 4-An angle-angle radar illuminates the object of Figs.2 and 3 from the same line ofsight. The perspective view isfrom the side and not the top as in Figs. 2 and 3. The angle­angle signal is a digital photograph of the object; pixelboundaries are shown in red. For comparison, the signal(green) from the range-resolving radar of Fig. 2 is shown inperspective.

The simplest algorithm for image reconstruc­tion estimates the image g(x. y) by spreading(back-projecting) the values of individualprojections p(xcos </J. + y sin </J. </J.) back along the

:J l l l

Fig. 5-Notation for transmission tomography.

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Knight et aI. - Tomographic Techniques Applied toLaser Radar Reflective Measurements

Fig. 6-Projections of a playground slide at many angles for a range-resolving radar form the input to the tomographicreconstruction. The signal versus range for three selected angles appears in two forms: a line plot and a color-coded bar.The bars can be compressed and combined into a color plot of range versus angle, shown at the right.

viewing direction, and then summing over allprojection angles 1/>;- This elementary methodgenerally produces a starlike pattern of streakartifacts, which results in a low-quality image.To provide a better reconstruction, each projec­tion can be modified before back-projection by afiltering operation in which the magnitude ofeach Fourier component of each projection isincreased in proportion to the magnitude of itsspatial frequency I kl. The image 9

PB(x, yl. re­

constructed by using filtered back-projection, isgiven by

m

gFB(x, y) = I. q(x cos ¢i + Y sin cfli' ¢di: I

where the modified projections q(u, 1/» are givenby

'T denotes the Fourier transform with respect tou

148

u, while 'TkO) denotes the inverse Fourier trans­form with respect to /e The angle of the ithprojection is 1/>;, and m is the number ofprojections.

In the application of tomography to laserradar data, the reflective projections (range,Doppler, or angle-angle) serve as input for therespective reconstruction algorithms. In thecase of range or Doppler measurements, theprojections p(u, I/>i)' i= I, ... , m are processed asdiscussed above. The angle-angle measure­ments are processed with a modified version ofback-projection described in a later section. Forall three measurements, it is assumed that theviewing directions and a common point of refer­ence for the projections are known or can beestimated.

For the range or Doppler measurementsdescribed, the large amount of input data isconveniently displayed, as shown in Fig. 6. Thisfigure depicts a set of projections measured in 10

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increments around 3600 for a range-resolvingradar. The data are displayed by color-encodingintensity versus range for the 360 angles, asshown at the right ofFig. 6. From the display onecan follow discrete reflectors as their projectedrange varies sinusoidally with angle. The ampli­tude of the sine wave is the distance of thereflector from the axis of rotation, while thephase represents the polar angle at the origin ofthe reflector with respect to a common point ofreference. The next three sections present labo­ratory and field measurements taken over arange of viewing directions with three laserradars.

Reflective Tomography fromRange-Resolved Measurements

This section describes range-resolved mea­surements, using a visible, short-pulse laser, of

ControlComputer

Knight et aI. - Tomographic Techniques Applied toLaser Radar Rejlective Measurements

test objects on an indoor ground range. Two re­ceivers are utilized: a photomultiplier tube con­nected to a transient digitizer, and a streakcamera connected to a vidicon. The former hasbetter sensitivity, while the latter has betterrange resolution. More details on the experi­mental setup and the results are described inRefs. 3 and 4.

Each receiver views objects up to 2 m inlength on a 10-m indoor range, as shown in Fig.7. The illumination comes from a frequency­doubled, Quantel Nd:YAG, pulsed laser thatproduces 532-nm pulses of 26 mJ with pulselengths =dOO ps FWHM. The laser pulses arediverged by a ground glass to produce uniformillumination and millimeter-sized speckles onthe object. Objects are mounted on a single­axis rotator with a vertical axis of rotation andoriented so that the projections lie in the de­sired plane. Two calibration plates are used

Rotator

ReceiverLens

Pulsed,VisibleLaser

I~

Diffuse Target«2 m)

10 m

Calibration Plates

~I

Fig. 7-The laboratory setup for range-resolved and angle-angle-resolved measurements. The pulsed laser illuminates theobject on the rotator. The reflected light is received by one of two time-resolving (range-resolving) detectors, oran angle-angledetector, and then processed.

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Knight et aI. - Tomographic Techniques Applied toLaser Radar Rejlective Measurements

Fig. B-Photographs (left), range-resolved data (center), and reconstruction (right) of an aluminum cone. The range­resolved data have a range extent of2.5 m along the abscissa andangles from fY' at the bottom to 36fY' at the top alongthe ordinate. Intensity is coded in color. The vertical bars at the edges of the range-resolved data are the returns fromthe stationary calibration plates.

as stationary range markers and intensitycalibrators.

Each of the two receivers records the timevariation of optical input signals. The importantcharacteristics are time resolution, time win­dow, and sensitivity. In the first receiver, thereflected light is detected by a HamamatsuR2083 photomultiplier tube and recorded every100 ps with a Tektronix 7250 transient digitizer.The detector is photon-noise-limited and has atime resolution of 750 ps in a window of 400 ns,which yields a range resolution of 12 cm in arange window of 64 m. In the second receiver,the reflected light is detected by an E.G. & G.Energy Measurements streak camera, attachedto an lIT 40-mm image intensifier and anRCA4804 vidicon. An 8-bit AID converter di­gitizes the output. This receiver is not pho-

150

ton-noise-limited but has a time resolution of250 ps in a window of 25 ns, which yields arange resolution of 4 cm in a range window of4m.

In a typical experiment, the object is viewed atmany directions around the rotator's verticalaxis. At each direction the receiver is triggeredonce, and a computer stores the data. Mterobject data at all the desired viewing directionsare acquired, the entire data set is processed.

For a first example, an object with a simpleshape was imaged. The left and center panels ofFig. 8 show photographs and the data for an alu­minum cone of 170-cm length and 53-cm di­ameter. The right panel of Fig. 8 demonstratesthe feasibility of reconstructing an image of thecone with reflected light. The reconstructedimage is the outline of the cone as viewed from

The Lincoln Laboratory Journal. Volume 2. Number 2 (1989)

Knight et aI. - Tomographic Techniques Applied toLaser Radar Reflective Measurements

Fig. 9-Photograph and reconstructions of a model of a OSP satellite.

above. along the axis perpendicular to the planeof rotation. The image is an accurate represen­tation in scale and geometry of the object whenviewed in green laser light.

For a second example. Fig. 9 shows labora­tory images of an object with a more complexstructure: a model of a DSP satellite (an early­warning satellite launched in 1971). The model,constructed of styrofoam and wood and paintedsilver with aluminum solar panels. was posi­tioned with the parabolic antennas and the bodyaxis in the horizontal plane. so that the projec­tion image shows the same view as the photo­graph. The green light reflected from the modelhas a wide dynamic range because of the brightspecular solar panels and the dim diffuse woodbody. The effects of enhancing portions of thedynamic range by data scaling are shown in thefour reconstructions. From left to right. the datascalings are (1) linear scaling. (2) logarithmicscaling. (3) linear added to logarithmic withoutthresholding. and (4) linear added to logarith­mic with thresholding. Linear scaling withoutthresholding makes the specular reflec­tions dominate. The fainter diffuse reflectionsappear brighter with the logarithmic scaling.The sums (3) and (4) show the detail containedin the images and indicate that an outlinecould be extracted. perhaps for input to anobject recognition algorithm.

Finally, we imaged a specular object withmany contours: a five-foot toy rabbit. The mate-

The Lincoln Laboratory Journal. Volume 2. Number 2 (1989)

rial is smooth plastic with a variety ofcolors. Therabbit is mounted face up in the horizontal planeto provide the best cross section to image. Fig­ure 10 shows the rabbit, the range-resolveddata. and the reconstruction. which has highcontrast and is easily recognizable by the eye.

Reflective Tomography fromDoppler-Resolved Measurements

This section demonstrates the use of 1-DDoppler-resolved projections to form a 2-D to­mographic image of a rotating object. TheFirepond 1O.6-,um CO

2narrowband laser radar

[14) and the 5.4-km ground range were used tomake Doppler-resolved measurements overtime (Doppler-time-intensity. or DTI) of a scalemodel of a Thor-Delta rocket body [51. Figure 11shows a diagram of the experimental setup. Themodel was rotated at approximately 1 rpmaround an axis perpendicular to the sensor LOS.The return energy was detected by a heterodynereceiver and digitized at 256 kilosamplesjs.Data were recorded over 360° in 1° angular in­crements and averaged over 5° to reducespeckle. Tomographic reconstruction usingthese data produces a 2-D image ofthe model asprojected onto a plane perpendicular to the axisof rotation.

Figure 12(a) shows a photograph of the alu­minum model ofthe Thor-Delta rocket body. Themodel was rotated around an axis perpendicu-

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Knight et aI. - Tomographic Techniques Applied toLaser Radar Rejlective Measurements

Fig. 10-Photograph (left), range-resolved data (center), and reconstruction (right) of a child's toy rabbit. Therange-resolved data have a range extent of about 1 m along the abscissa and angles from (f' to 36(f' along theordinate.

lar to the axis that passed through the center ofthe model body; this object motion is analogousto end-over-end tumbling. Figure 12(b) showsthe DTI data of the object. Notice that theDoppler signature of the object is added to thezero-Doppler clutter (center stripe), and that thesignature is complicated. In addition, the alumi­num is specular near normal incidence, whichcauses bright reflections when sections of themodel are normal to the sensor LOS. In general,individual specular reflections do not persist asthe object rotates. Figure 12(c) shows how thefiltered back-projection technique leads to thereconstruction of the Doppler image of the Thor­Delta model. With the optimal geometry chosenfor these measurements, the image appearsas a projection onto a plane that is perpendicu­lar to the axis ofrotation. Preprocessing the databefore image reconstruction can enhance the

152

desired features.Unlike a reconstruction from range-resolved

projections, a Doppler-resolved reconstructionis not a simple outline of the object. This resultis due to the reflective properties of typicalmaterials as well as to the geometry of range­resolved versus Doppler-resolved measure­ments. Materials typically have high reflectanceat normal incidence and low reflectance at graz­ing incidence. This variation in reflectance re­sults in bright edges in the reconstruction withrange-resolved measurements, since an edge ofthe object is reconstructed from the projectionsthat view the edge at normal incidence. On theother hand, with Doppler-resolved measure­ments an edge of the object is reconstructedfrom the projections that view the edge at graz­ing incidence, which results in lOW-intensityedges in the reconstruction.

The Lincoln Laboratory Journal. Volume 2. Number 2 (1989)

Knight et al. - Tomographic Techniques Applied toLaser Radar Reflective MeasuremenLs

Quick­Lock

1 Hz to 300 kHz

'6 f ± Doppler Frequency

SignalAnalyzerHP3562A

Receiver Detector

10 MHz + '6f

1-kW Amp Transmit Optics 4Jr-tf----Ir----iL----~O

2=======-,Thor-Delta

Model

Ground II- Range ---~

(5.4 km)M

Detector

'.

LO

MO

10-MHz ReferenceOscillator

Fig. 11-Experimental setup of the 5.4-km narrowband ground range at Firepond. This system measured theDoppler-time-intensity projections of the aluminum Thor-Delta rocket model.

Fig. 12-A photograph (left) ofan aluminum Thor-Delta rocket model. The axis of rotation passes through the centerof the model and is perpendicular to the model's cylindrical axis. The 1O.6-J..1m Doppler-resolved data (center) of theThor-Delta rocket model. The zero-Doppler clutter reconstructs as a bright spot in the center of the image. Areconstructed image (right) of the Thor-Delta model.

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Reflective Tomography from Angle­Angle Measurements

The previous two sections describe combin­ing 1-0 data to form 2-0 projection images; thissection describes combining 2-D data to form3-D projection images. The 3-D projection im­age-the tomographic reconstruction of the 2-Dimages-approximates the size and shape of theobject. We describe the laboratory setup, theresults for one object, and some trade-offs con­cerning the parameters of the experiment.

The laboratory data come from a digitalangle-angle camera that records reflected lightfrom an object. The object is mounted on a

turntable, so that many views can be imaged.The beam of a doubled, pulsed, Nd:YAG laser isdiffused to illuminate objects of 30-cm-to-200­cm length on an indoor range of 10m. Theobjects reflect the green laser light (532 nm)diffusely, so that light is collected over wideangles. Unless the test object is polished tooptical quality, diffuse reflections (typical forvisible wavelengths) yield sufficient signal atmany viewing directions. At each viewing direc­tion, the reflected light is imaged onto a TVcamera, digitized, and stored for later recon­struction. Typically we take 5° steps around agreat circle and use at least 100 x 100 pixels,which provides data that are more than ade-

Fig. 13-The creation of a 3-D image with 2-D angle-angle images and tomography. In this method, a threshold, a back­projection, and a second threshold are applied to the input 2-D data to form the 3-D reconstruction. The reconstruction canbe displayed as 2-D slices (as in a CA T scan) or as a perspective view.

154 The Lincoln Laboratory Joumal. Volume 2. Number 2 (1989)

2-D Binary Images

3-D Reconstruction in Perspective

Fig. 14- Laboratory 2-0 images and tomography create a3-0 image of two interlocking toroids. The input data weretaken in SO steps around a great circle, although only one­quarter of the input 2-0 images are shown. The 3-0 re­construction is drawn in perspective and is a solid model ofthe toroids.

quate to produce the 3-D image. Figure 7 illus­trates the laboratory setup.

Figure 13 shows the process of reconstruc­tion. Variation in intensity is eliminated in each2-D image by selecting a threshold level toproduce a black-on-white silhouette: each im­age pixel is called black (or 1 in the computer)and each background pixel is called white (or 0in the computer). The thresholding is used tolimit the dynamic range on the 3-D reconstruc-

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Knight et al. - Tomographic Techniques Applied toLaser Radar Reflective Measurements

tion. Sufficient signal to noise of the 2-D imagesis required, so that the process of threshold­ing produces accurate silhouettes. The tomo­graphic reconstruction uses back-projection tobuild an image in a reconstruction volume. Theimage built up by summing the projections fromall aspect angles is a solid model of the object.

As displayed, the process is really two steps:the first step reconstructs slices of the object,and the second step combines slices into a 3-Dsilhouette. This process is similar to medicalCAT scans. To extract 3-D information, a secondthreshold is selected and applied to the set ofslices. Generally, the level of the second thresh­old is equal to the number of views; in otherwords, the image is the set of points that havecontributions from all the views. The final resultcan be displayed as a set of slices or as aperspective view of the 3-D model. These resultscan be generalized to views from arbitrarydirections.

Figure 14 illustrates the process of imagereconstruction for two interlocking toroids (ac­tually two worn auto tires) mounted at the centerof our turntable and rotated through 3600 in 50steps. The black tires reflect enough light toproduce clean binary images. The reconstruc­tion, which is formed by using a threshold equalto the number of views, yields a recognizablesolid model.

The reconstruction above uses a large num­ber of views (72), high angular resolution(==100 x 100 pixels), high signal to noise (pulsedlaser with a low-light-Ievel TV detector), and adiffuse test object. All of these conditions can berelaxed, depending on the ultimate use ofthe data. For a high-fidelity 3-D image, a highdensity of views is needed. For quantitativeshape information, mass moments of the imagecan be calculated to compare the image to amodel or another object whose moments areknown [15). For crude orientation estimates,using mass moments of a tall cone, our workindicates that only a few views are needed (aminimum of three equally spaced around agreat circle), provided enough angular pixels areavailable (a minimum of 30 along the cone axis).The pulsed laser gives high signal to noise, but

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Knight et aI. - Tomographic Techniques Applied toLaser Radar Rejlective Measurements

similar 2-D images can be obtained from brightambient light. Lower signal to noise introducesa higher probability of false pixels in the 2-Dbinary images (which leads to holes in the image,or spurious background); one compensation islowering the second threshold. A scanning laserradar can be employed as long as the 2-D imagesare registered in angle. Finally, we have re­stricted our views to a great circle, but arbitraryviewing directions from anywhere on the sphereare possible.

Discussion

This paper describes the use of tomographictechniques to process three types of laser radardata. For each type, the data are taken frommany viewing directions to form a set of projec­tions of the object. When processed, they pro­duce a visual image (with an additional dimen­sion) that can be used to estimate the object'sstructure. Each example of range-resolved datashows a 2-D image that accurately representsthe size and projected shape of the test object,whereas, in the input data, size and shape arenot readily apparent. The example of Doppler­resolved data shows a 2-D image that consti­tutes true narrowband imaging. The example ofangle-angle-resolved data shows a 3-D imagethat reveals the size and shape of the object andprovides an estimate of its 3-D structure. Ineach of the three examples, the image recon­struction presents a more complete picture ofthe object: each reconstruction reveals charac­teristics not apparent in the input data. Allthree examples are, however, ideal situationsbecause they describe laboratory setups withhigh resolution, high signal to noise, and a largenumber of views. This section discusses severalquestions motivated by the initial experiments.

How do resolution, number oj views, andregistration affect the data? For both 2-D and3-D tomographiC imaging (and higher dimen­sions), the resulting image resolution is gov­erned by the resolution and quality of the indi­vidual laser radar measurements, the density(or number) of individual views, and the accu­racy with which the individual measurements

156

can be registered or related to a common point.These issues must be addressed in detail beforethe reconstruction techniques can be applied toa specific problem, for example, the orientationofa tall cone ofthe previous section. To minimizethe data collected (to reduce acquisition time orto save storage space) few views at low resolutionare needed. Registration is necessary only to thewidth of a resolution cell, not a wavelength as incoherent addition of data.

How do complex motions degrade the results?In laboratory measurements, both a commonorigin and the viewing directions for all theprojections are known. In a practical scenario,the objects of interest may execute complexmotions with unknown rotation rates. In gen­eral, to apply these tomographiC techniques therelative dynamics between the object and sen­sors must be known. In some cases, the dynam­ics may be derived from the measurementsthemselves. In the case when multiple range-re­solving sensors take data simultaneously, noknowledge of dynamics is reqUired. Further­more, the independent measurements need tobe aligned only to a range-resolution cell and notto a wavelength. This alignment puts fewerrequirements on the single-sensor and themultiple-sensor scenarios, and with propermultiple-sensor geometry allows imaging ofboth rotating and nonrotating objects.

How do the results reported here compare tothe results from microwave radar? The problemof reconstructing images from reflective micro­wave radar data by using tomographic tech­niques has been addressed by many research­ers in the radar community [6-9). In general, thediffuse scattering characteristics of a materialincrease with decreasing wavelength. Our re­sults indicate that better image definition andimproved resolution come from surfaces with asignificant diffuse reflectance component. Im­proved images can thus be expected at laserwavelengths.

Can tomography be applied to range-Dopplermeasurements? Two-dimensional images canbe obtained directly with laser radars that re­solve the object in range-Doppler dimensions.As with 2-D angle-angle images, applying to-

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mographic techniques to 2-D range-Dopplerdata to obtain 3-D reconstructions may bepossible. In addition. an interesting applicationof tomographic imaging exists in the range­Doppler domain. It is possible to sample a single1-0 projection of the 2-D range-Doppler imagewith a single linear FM-chirp waveform for anobject that has both range and Doppler extent[16. 17). Projections at different angles can beobtained by changing the slope of the transmit­ted linear FM chirp. Since these data are pro­jections in the range-Doppler space. 2-D range­Doppler images can be generated by applyingtomography. Images formed in this way over­come some of the limitations of :traditionalrange-Doppler imaging that uses repetitivewaveforms.

What are some potential applications oJtheseimage-reconstruction techniques? New develop­ments for remote sensing and imaging haveapplications in surveillance. medicine. machinevision. toxic environments. manufacturingquality control, and fusion plasma diagnostics.

For example. in robotics a detailed image ofan object may be needed to grasp the object. Byusing the surface in the image. tangent linescould be drawn analytically to define graspingpoints.

As a second example. for transient eventsmany sensors can be positioned at importantaspect angles and triggered simultaneously toprovide instantaneous input to the reconstruc­tion algorithm [181. The result is a snapshot intime to record the event for subsequent study.

As a third example. space surveillance couldbe performed by using laser radars from ground­based or airborne platforms. A single sensor onthe ground could view spin-stabilized satellitesor tumbling bodies in low earth orbit. in a widerange of directions. Other satellites, such asgravity-gradient-stabilized satellites that al­ways point down. or geosynchronous sateIlitesthat remain fixed over one location on the equa­tor. would offer fewer viewing directions for asingle ground-based or space-based sensor. Insuch cases an array of ground-based receiversor active sensors could be used to extend therange of viewing aspects. The technologies re-

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Knight et al. - Tomographic Techniques Applied LoLaser Radar RejlecUve MeasuremenLs

qUired for using tomographic imaging tech­niques with laser radars are currently available.

Summary

We have applied tomographic image-recon­struction techniques to three types of laserradar reflective projections: range. Doppler. andangle-angle. Each reconstruction is an image ofthe object illuminated by the laser radar: a 2-Dimage formed from 1-0 data or a 3-D imageformed from 2-D data. As in all tomography. theinput data must be taken from many viewing di­rections. The images are formed from laboratorymeasurements by using standard tomographictechniques. The reconstructions reveal detailedcharacteristics of the objects. and indicate thathigh-resolution images can be formed by usingcurrent laser radars.

Acknowledgments

Many individuals contributed to efforts ontomography in the Laser Radar MeasurementsGroup. On the basis of a suggestion from W.E.Keicher. RE. Knowlden originally conSidered indetail the use of tomography for combiningDoppler measurements. D.R Cohn assembledmaterial on tomography in medicine and plasmaresearch and presented a group seminar. W.E.Keicher suggested a novel reconstruction tech­nique. E.F. Breau. and J.R Senning helpedstudy and implement a number of algorithms tocombine existing Doppler-resolved measure­ments and computer-generated range-resolveddata. L.J. SuIlivan. D.G. Kocher. RN. Capes.L.W. Swezey. J.A Daley. RA Westberg. J.M.Anderson. E.J. Christiansen. G.L. Peck. AS.Ruscitti. and H.A Weigel helped make the infra­red Doppler measurements on different objects.OJ. Klick. B.K. Tussey. AS. Lele. AM. Beck­man. J.R Theriault. Jr.. K. Whittingham. andE.F. Breau helped obtain and process visiblerange-resolved measurements of a cone. D.1.Klick. D.P. Ryan-Howard. B.K. Tussey. J.RTheriault. Jr.. and AM. Beckman helped makevisible range-resolved and angle-angIe-resolvedmeasurements on many objects. AS. Lele

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worked on angle-angIe-resolved measurements.K.I. Schultz and M.F. Reiley helped comparelaser radar tomography and microwave radar

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tomography. We thank Kenny Tussey for theloan of the toy rabbit.

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References

1. RA. Brooks and G. DiChiro, "Principles of ComputerAssisted Tomography in Radiographic and Radioiso­topic Imaging..' Phys. Med. BioI. 21. 689 (1976).

2. G.T. Herman. Image Reconstruction Jrom Projections(Academic Press. New York. 1980).

3. J.K. Parker. E.B. Craig. D.1. Klick. F.K Knight. S.RKulkarni, RM. Marino, J.R Senning. and B.K Tussey."Reflective Tomography: Images from Range-ResolvedLaser Radar Measurements'" App1. Opt. 27. 2642(1988).

4. F.K Knight. D.1. Klick. D.P. Ryan-Howard, J.RTheriault. Jr.. B.K Tussey, and AM. Beckman."Two-Dimensional Tomographs Using RangeMeasurements," SPIE 999. 269 (1988).

5. RM. Marino. RN. Capes. W.E. Keicher, S.R Kulkarni.J.K Parker, L.W. Swezey. J.R Senning. M.F. Reiley.and E.B. Craig. "Tomographic Image Reconstructionfrom Laser Radar Reflective Projections." SPIE999. 248(1988).

6. N.H. Farhat. T.H. Chu. and C.L. Werner. "Tomographicand Projective Reconstruction of 3-D Image Detail inInverse Scattering," Proc. oj IEEE 10th Int. OpticalComputing Con] IEEE Cat. 83CHI880-4. Cambridge,MA. 6-8 Apr. 1983, p. 82.

7. Y. Das and W.M. Boerner, "On Radar Target ShapeEstimation Using Algorithms for Reconstruction fromProjections'" IEEE Trans. Antennas Propag. AP-26. 274(1978).

8. D.C. Munson. J.D. O·Brien. and W.K Jenkins. "A To­mographic Formulation of Spotlight-Mode Synthetic

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Aperture Radar..' Proc. IEEE 71, 917 (1983).9. D.L. Mensa, S. Halevy. and G. Wade. "Coherent Dop­

pler Tomography for Microwave Imaging." Proc. IEEE71. 254 (1983).

10. AM. Androsov, V.G. Vygon, and N.D. Ustinov. "Recon­struction of Images of Rotating Bodies of ArbitraryAngular Dimensions. I. Structure of Doppler Spectraand Reconstruction of Images from Projections,,' Sov. J.Quantum Electron 15, 168 (1985).

11. AM. Androsov. V.G. Vygon. and N.D. Ustinov. "Recon­struction of Images of Rotating Bodies of ArbitraryAngular Dimensions. II. Doppler Interferometry'" Sou.J. Quantum Electron. 15, 172 (1985).

12. AM. Androsov. V.G. Vygon. and N.D. Ustinov. "Recon­struction of Images of Rotating Bodies of ArbitraryAngular Dimensions. Ill. DynamiC Doppler Approach'"Sou. J. Quantum Electron. 15, 174 (1985).

13. KI. Schultz, L.W. Swezey. and M.F. Reiley. "Applica­tions of Reconstruction from Projection Techniques toTime-Varying Autodyne Signatures'" SP1E 999. 216(1988).

14. L.J. Sullivan. "Infrared Coherent Radar." SPIE 227:CO2 Laser Devices and Applications. p. 148 (1980).

15. M.K Teague. "Image Analysis via the General Theory ofMoments'" J. Opt. Soc. Am. 70, 920 (1980)

16. H.P. Raabe. "Graphical Interpretation of Chirp Echoesfrom Complex Targets." IEEE Trans. Aerosp. Electron.Syst. AES-12. 140 (1976).

17. M. Bernfeld. "CHIRP Doppler Radar," Proc. IEEE 72.540 (1984).

18. M. Wilke and N.S.P. King. "Two-Dimensional Time­Dependent Imaging Utilizing Tomographic Concepts'"SPIE 648, 24 (1986).

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FREDERICK K. KNIGHT is astaff member in the LaserRadar MeasurementsGroup. He received his B.A.in mathematics from Carle­

ton College. and his Ph.D. degree in physics from theUniversity of California at San Diego. Before coming toLincoln Laboratory in 1985. Fred worked at the Harvard/Smithsonian Center for Astrophysics. His research inter­e ts are in visible laser radar and gamma-ray astronomy.

RICHARD M. MARINO is astaff member in the Las­er Radar MeasurementsGroup. He received a B.S.cum laude in physics from

Cleveland State University, and an M.S. in physics and aPh.D. in experimental particle physics from Case WesternReserve University. His research interests include novelapplications of laser radars and systems analysis. Rich re­ceived the Best Paper award from the IRIS Active SystemsGroup for both 1986 and 1987. He has been at LincolnLaboratory since 1985. and is currently leader of the SensorSystem Engineering Project.

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SANJEEV R. KULKARNI is astaff member in the LaserRadar MeasurementsGroup. His focus of research

- is in machine vision andsignal processing. He received a B.S. degree in mathemat­ics and electrical engineering from Clarkson University, andan M.S. degree in electrical engineering from Stanford Uni­versity. Sanj has been at Lincoln Laboratory since 1985.and he is currently pursuing a Ph.D. in electrical engineer­ing at MIT in the Center for Intelligent Control Systems.

JEFFREY K. PARKER is astaff member in the las­er Radar MeasurementsGroup. His research inter-ests are in radar imaging

and radar operations. He received a B.A. in physics fromReed College. and a Ph.D. in experimental plasma physicsfrom MIT. Before coming to Lincoln Laboratory in 1986. Jeffworked in the Advanced Technologies Group at VarianAssociates. At Lincoln. he managed the development pro­gram of the FIREFLY instrumented laser radar target. andhe will be participating in the upcoming FIREFLY soundingrocket experiments. Jeff also has a commercial pilot'slicense, and he owns his own plane, a 1946 Luscombe 8A.

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