Search Radar Detection and Track with the Hough Transform Part I: System Concept

B. D. CARLSON, Senior Member, IEEE

E. D. EVANS, Member, IEEE

S. L. WILSON M.I.T. Lincoln Laboratory

A method of viewing search radar signals and data is described

and analyzed in which the image processing technique of the

Hongh transform is used to extract detections and simultaneous

tracks from multi-dimensional data maps. System design

concepts are considered and simulation examples arc given that

iUustrate the concept The technique offers many advantages when

compared with more traditional techniques. These advantages

include improved detecti�n, a solution to the range walk problem,

f1exibiUty of implementation, elimination of slow scan-ratc latency

and automatic track acquisition without revisit. The concept is similar to track-before-delect algorithms that use preliminary

information from previous scans to aid in target declarations. The

companion paper, Part II: Detection Statistics, gives a detailed

analysis of the prohability-of-detection and false-alarm rate

for this Hough detector. The third paper, Part III: Detection

Performance with Binary Detection, describes and analyzes a

variation of this technique that has more rohn.t performance in

the presence of multiple targets with high dynamic range.

Manuscript received May 26, 1992; revised November 3, 1992.

IEEE Log No. T-AES!30/1/13043-A.

This work was supported by the Department of the Navy under Air Force Contract FI9628-8S-C-0002.

Authors' address: M.LT. Lincoln Laboratory, 244 Wood Street, P.O. Box 73, Lexington, MA 02173.

0018-92511941$4.00 @ 1994 IEEE

I. INTRODUCTION

This paper introduces a method of improved target detection and simultaneous track acquisition for search radars. The technique combines data from previous search scans, multiple beams and multiple Doppler bins into one large multi-dimensional data map. Thrgets appear as curves or "features" in this data space. The Hough transform [1-3] is a feature detector, often used in image processing, that can be used to extract these detection-tracks from the data. This technique has the advantage of using data from previous scans and multiple beams in the detection process and not just for track determination. The track is defined by the curve that is detected, eliminating the need to revisit a detected target for track acquisition. The literature on the Hough transform is extensive and a paper by Illingworth and Kittler [4], is a survey of the Hough transform and provides a chronological reference list of 136 papers in the field.

It is well known that the detectability of a target is directly related to the amount of radar energy on the target. Traditionally, search radars spread their encrgy over the coverage volume in some desirable shape and the detectability of the target follows directly as a consequence. A range-azimuth-elevation cell return is compared against a threshold and a dctcction decision is made. Once this is done, the information for that cell is discarded and other cells are considered. It is clearly wasteful to discard this data which still may contain target returns even if no dctcction was declared. A method of making use of this information is needed. There is an assumption here for the traditional radar that the target remains in the cell for the duration of the energy dwell or integration period. If a target moves from one cell to another during the integration period, reduced detectability results.

This new method uses scan-to-scan processing and has the advantage of not discarding past scan information which may have weak undetected targets and is, therefore, still useful. The method enables one to do noncoherent integration from scan to scan, from beam to beam (azimuth and elevation) and from range-cell to range-cell with stationary, moving, or accclerating targets. The improved detectability results from better use of old energy and spatially separated energy rather than the addition of more energy.

The Hough transform is used here to detect target tracks in a multi-dimensional data space that is filled with return data from the radar. Much more data is stored and used in this technique than is contained in a single scan of a traditional radar. This data map can be thought of as an image which may contain target trajectories. Much of this work focuses on a 2D data space since the concepts are more clearly described, demonstrated, and explained with limited dimensions. The advantages and problems of extensions to higher

102 IEEE lRANSACTlONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 30, NO. 1 JANUARY 1994

dimensions are also discussed. Higher dimensionality has the potential to providc improved performance, but it requires more processing and data storage. Extensions to higher dimensions will be easier for future radars because of the growth of computer processing technology.

Section II explains the details of this method. Section III contains a simulation example demonstrating the concept. Section IV contains a discussion of the technique and the example. The discussion includes possible problems, modifications, and extensions. Section V contains a summary and conclusions.

The first companion paper, Part II: Detection Statistics [5], contains a dctailed analysis of the probability-of-detection and the false-alarm rate for this method with an example radar. This analysis is confirmed with Monte Carlo simulations. Other topics are also discussed such as the Hough parameter space granularity. Improved performance over traditional search radars is shown.

The third paper, Part III: Detection Performance with Binary Integration [6], contains an alternative approach to the problem using binary integration in the Hough transform. Binary integration offers some advantages over straight noncoherent integration when multiple targets appear in the data or when the detector receives signals with a wide range of power. This alternate approach is described and developed along with analysis and Monte Carlo simulations. Improved performance over traditional search radars is shown.

II. SEARCH RADAR HOUGH DETECTOR

Consider a search radar that provides range, azimuth, elevation and Doppler information as a function of time. Time, here, is quantized by the scan period or search frame time. The search radar Hough detector technique can be used with either a rotating antenna or a phased array, but since phased arrays are more f lexible than other antennas and scan methods they are the focus of this description. Traditional radars often dwell in a particular beam position using a set of coherent processing intervals (CPls), each at a different frequency. The coverage sector is searched by sequentially looking in all beam directions. When this is done, the beam sequence is started over, seconds later. With an M of N detector, a target is declared if the returns for a certain number M of the N CPls for a beam position and range gate are above a threshold. It is possible to spread these CPIs out evenly in time (rather than dwelling on a beam with many adjacent CPls) thereby reducing the frame time. A traditional radar would have problems with this because moving targets could move from one detection cell to another during the frame time. If the power return data from the radar is arranged as a multi-dimensional discrete

0---

-101

-25 -30 �

-40t , ,

---2-0 -4i)"-6C;cO---;;80' 100 120 140 160 \ 180 Range m Km

Fig. 1. 1tack of Mach 3 target in range-time space.

data map (with 5 dimensions of range, azimuth, elevation, Doppler, and time), a target will appear as a curve whose intensity depends on the p ower level of the returns. If this curve can be detected, it contains all of the current information about the detected target as well as a complete history of its trajectory. A target with only a constant radial velocity appears as a straight line. A projection of this 5D data space onto the range-time plane is a convenient way to view this curve since not much of interest is happening in the other 3 dimensions for this simple but important target. In an actual implementation, one might choose to process each beam in azimuth-elevation separately and each Doppler bin (if Doppler information is available) separately.

Fig. 1 shows an example of a Mach 3 radial velocity target in the range-time plane for a particular beam and Doppler bin. The slope of this line is determined by the velocity of the target. A stationary target or point clutter would appear as a vertical line in this space. All moving targets would have some finite slope approaching zero slope for infinitely fast targets. The rangc axis extends from the minimum range of the radar to its maximum. The time axis runs from the past up to the current time of zero. It is convenient to truncate the past at some reasonable level (-48 s in this case) for practical data processing reasons as well as because data older than this is probably not much help in detecting moving targets. The actual radar data space is assumed to contain white thermal noise that is Rayleigh distributed in amplitude in all of the cells. The problem now becomes one of pulling this line out of a plane of noise.

The Hough transform is a feature detection method frequently used in image processing and is well suited to locating lines in a plane of noise. Other arbitrary shapes can be detected too [3], but for now, only straight lines are considered. Fig. 2 shows several data points which determine a line in the range-time data space. The line can be defined by the angle (} of its perpendicular from the origin and the distance p from the origin to the line along the perpendicular. The Hough transform maps points in the range-time (r - t)

CARLSON ET AL.: SEARCH RADAR DETECTION AND TRACK WITH THE HOUGH TRANSFORM 103

Rho

\ 'c '_ " I -heta \ .

----�--- ---------------- --�

Range In Km Fig. 2. Geometry in range-time space showing three data points

and defining parameters of line through them.

data space into curves in a p -() space or Hough parameter space by

p = rcos() + (sin().

The rand t in this equation are measured from the origin of the p and () system in the lower left. Thc mapping can be viewed as stepping through () from

(1)

0° to 180° and calculating the corresponding p. It can be shown through trigonometric manipulation that (1) is equivalent to

p = Jr2 + t2sin (() + arctan�) . (2)

The mapping in (2) results in a sinusoid with an amplitude and phase dependent on the r - t value of the data point that is mapped. The maximum value for Ipl is equal to the length of the diagonal across the r - t data space. Equation (1) is the simpler version that is actually used for the mapping. Fig. 3 shows a view of the Hough transform of the points in Fig. 2. A single p - 9 point in Hough parameter space corresponds to a single straight line in r - t data space with that p and () value. Any one of the sinusoidal eurves in Hough parameter space corresponds to the set of all possible lines in the data space through the corresponding data point. If a line of points does exist in , - t data space, this line is represented in Hough parameter space as the point of intersection of all of the mapped sinusoids.

The range-time space is divided into cells equal to the number of range gates times the number of scans in the history. A low primary threshold is set and any range-time cell with a value exceeding this thrcshold is mapped into parameter spacc. The parameter space is quantized in the p and () dimensions (a discussion of the issues involved in how to quantize the parameter space is provided in Part II, [5]). When a primary threshold-crossing is mapped into parameter space from the r - t cell, its power is added into the p - () cells that intersect the corresponding sinusoidal curve in parameter space. In this way, the accumulator cell at

- Rhomax r-------------- -----i

l' 0 0:

- Rhomax L---______________ � ______________ _ 90 180

Fig. 3. Hough parameter space showing three sinusoids

corresponding to three data points of Fig. 2. Point of intersection defines rho and theta of line in data space through the three data

points.

the intersection of several sinusoids will reach a high value. A secondary threshold applied to the parameter space can now be used to declare detections. This is a way of noncoherently integrating the returns from a moving target over a long time period. The p and () of the detected point in parameter space can now be mapped back into, - t space to show the time history and current position of the detccted target.

This mapping from, - t data space into parameter space is easily implemented by a simple matrix multiplication. A data matrix D can be defined from any data crossing the primary threshold as

['1

D= tl (3)

where the columns are the range and time values of all of the I primary threshold crossings. A transformation matrix H, composed of the sines and cosines from (1) can be defined as [ sin()1

sin 92 H=

sin9Nr

COS()I 1 cos 92

COS

:

9NT

(4)

where the 9 values are the NT discrete values from 0° to 180° for the cells of parameter space.

The multiplication of D and H produces an NT by I matrix, R, containing the needed values of p. The subscripts of the p values represent tlte index of the primary threshold crossing data point and the angle, 9, used in the discrete map [ PI,O]

R= :

PI,ONT

PI.O] 1 : = HD.

PI.ONT

(5)

Each column of R contains the p values for one of the parameter space sinusoids. It is clear that the more

104 lEER TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 30, NO.1 JANUARY 1994

190

-48

Fig. 4. Power returns from Mach 3 target in range-time space.

primary threshold crossings that occur, the bigger D and R will be, resulting in more computations. The matrix H is precomputed and its size depends only on the parameter space quantization.

III. SIMULATION EXAMPLE

For an example, consider a radar with 256 range gates and a time history composed of 128 scans with the search frame time being 0.375 s. This gives a total time history of 48 s and the dimension of r - t space are 256 by 128 for a total of 32,768 data cells. This search frame time comes from taking a typical traditional search radar scan time of 12 s and dividing by a typical number of CPIs (32) used in an M of N detector. If the range cell size is 750 m, then this data map covers the range from 0 to 192 km. For this example an approaching target with a radial velocity of Mach 3 is currently at 120 km range. This is shown without noise in the mesh plot of the r - t data space in Fig. 4. The target is not statistically f luctuating, but the returns are higher at the closer ranges. Fig. 5 shows a mesh plot of the same target in the r - t plane, but now, with the addition of white noise which is Rayleigh distributed in amplitude or exponential in power. At the time t = 0, the signal-to-noisc-ratio (SNR) is 6 dB. Note that this target has a SNR of 0 dB at the beginning of the time history (-48 s) due to its greater range at that time. It is now almost impossible to see the target. With a traditional radar it would be difficult to set a threshold which would allow this to be detected without false alarms. Longer dwells would have to be used to integrate up to a highcr SNR. This would result in longer scarch frame times and the problems of intercell target motion. If this data is viewed as an image then a low level curve becomes much easier to see. Fig. 6 is a gray-scale plot of thc power in the data plane for this simulation scenario (dark is higher p ower). The target track is more visible in the recent past where it" SNR is higher, but it still can be seen at the bottom where the SNR approaches o dB. Fig. 6 shows that the eye is a very good line detector.

190

Fig. 5. Mesh plot of range-time data space for exampl e. Mach 3 target (0 to 6 dB SNR) with additive white noise.

u . � , (j) c dl �

. , � .. ". <c,

<D ' � '---- '-,--, 0 Range in Km 190

Fig. 6. Gray-scale plot of range-time data space for example.

Mach 3 target (0 to 6 dB SNR) with additive white noise.

A primary threshold can be set in this image which allows most of the target returns through along with some of the largest noise spikes. A primary threshold of 7 dB is used in this example. This is above the level of any of the target returns, but with the addition of the random noise, the threshold is crossed 250 times in the plane. Fig. 7 shows the range-time space after primary thresholding. It is clear that most of thc target information has been preserved and much of the noise has been eliminated. Each of these crossings can now be transformed into parameter space. The parameter space chosen for this example is 256 by 256 for the p and B dimensions. Fig. 8 is a mesh plot of the Hough parameter space showing the 250 'inusoids. The mesh plot shows a clear peak or convergence point. A secondary threshold of 22 dB in parameter space produces a single point detection of the peak that is well above the rest of the p - B plane. For lower secondary threshold levels there may be multiple detections of adjacent cells ncar the peak for a single target. Clustering algorithms could be used to recognize this as a single target. The p and B for this detection can now be mapped back into a range-time track, as shown in Fig. 9, where the complete time history of this target can be seen. The primary and secondary thresholds used in this example are not necessarily optimum for detection. They can be manipulated to achieve the desired false-alarm rate or probability-of-detection. They were chosen in this example to give a clear picture of the mechanics of the method. The Hough transform is quite good at detecting constant velocity radial targets

CARLSON ET AL.: SEARCH RADAR DETECTION AND TRACK WITH THE HOUGH TRANSFORM 105

o

F' . �-. . -.-:----:;�

. ". ".. . �

�I; . .' .. . .

V) • c " E ···· ;::

�l_" �_'._." o Range in Km 190

Fig. 7. Range-time data space for example after threshold of 7 dB.

PARAMETER SPACE

rhomax

1\"1£11>­\deg)

180

Fig. 8. Hough parameter space for the example showing 250 sinusoids and peak corresponding to Mach 3 target detection -track.

·15

-::i � -20 o

E -251 ;:: -30f

·35 � J 45 � 0- 2O-�40--60 '--;Sc,;-0--;1""OO--;1""'20c---'140 160 . 180

Range in Km

Fig. 9. Detected target track for example mapped back into range-time space.

that would be missed by traditional radars. The amount of improvement is quantified in the accompanying papers, Part II and Part III, along with analysis of how to set the primary and secondary thresholds.

IV. DISCUSSION

A. Other Target Motions

The Hough transform has been shown to be an effective technique for detecting targets represented by a straight line in the data space. Not all targets will appear that way. If a radial target is changing its speed by a constant radial acceleration over the data history,

-:1 I -10 � 15 f

-20 f -25 t -3� � ·35 .40 f ·45 f I

() 20 40 60 , --------'----_ .. 80 100 120 140 Range in Km

lS0

Fig. 10. Detected target tracks for a 1 g accelerating target with a c onstant velocity line detector.

it will appear as a quadratic curve represented by the following equation:

r = rO + vot + ! at2

where ro is some initial reference range, Vo is the velocity at the reference range, and a is the acceleration. It takes three parameters to represent this curve in the range-time plane. The Hough transform can easily be used to detect this type of three parameter acceleration curve. The data points

(6)

in the 2D data plane can be mapped into a 3D Hough parameter space with dimensions of ro, vo, and a. Other parameters can also be chosen to represent the three dimensions. The 2D parameter space in the example could have been implemented with other parameters such as slopc and intercept. Slope and intercept are not a particularly good choice because the infinite slope of a vertical line presents problems. It is clear that the computational load goes up as more complicated tracks are considered and the dimensionality of the parameter space is increased.

It is not absolutely essential that a quadratic detector be used for a quadratic track. A simple line detector works fairly well on a quadratic track. The example above has been redone with the addition of a 1 g radial acceleration. At first no detection was declared because the velocity vector was not along any particular straight line long enough to integratc up to the threshold. When the threshold was lowered, however, multiple detections occurred. These detections were mapped back into tracks in the data plane and arc shown in Fig. 10. Each of these lines represents a section along the actual curved path. The target is tangent to these lines along the inside edge. The uncertainty in the target position at the current time (t = 0) is only a range of about 5 km. Lowering the threshold enough to find these lines did not cause any false alarms for this set of random numbers. While some uncertainty does result from this mismatched system, it is not extreme and the system is much simpler to implement because only two dimensions are being used.

106 IEEE JRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 30, NO.1 JANUARY 1994

Another departure from a straight line in the data space occurs if a constant velocity target is not radial. This can be thought of as coordinate acceleration. A passing target has a range which decreases to a minimum and then increases. This situation is not of great concern for most applications since the minimum range of the radar prevents the mapping of targets close to the coordinate origin where the problem is worst. The limited time history and narrow beamwidth of most radars also helps in this respect. A straight line is a very good approximation.

Targets may also pass from one heam position to another in azimuth or elevation. For many radar targets, passing from one elevation beam to another happens relatively infrequently due to the geometry of long range targets. Targets will cross azimuth beams more frequently than elevation beams. This can be handled in several ways. The effect could be ignored by processing each beam separately. This is what most radars do now. Another option is to usc a higher dimensional data space. A range-time-azimuth data volume could be used. A constant velocity target is still approximated by a straight line, but now it takes four parameters to define the line. The mapping would be from a 3D data space to a 4D parameter space. This will take more processing and may not be worth the added effort required. The processing for all dimensions may not be equal. That is, the quantization in different data dimensio ns will he different, for example, there may be a small number of beam positions. The quantization in parameter space can also be different for each dimension. A 3D data space of this type has the benefit of smoothing over beam scalloping losses and adding to the availahle integration time for beam-crossing targets. In many applications, the target moves so slowly through the heams that this expansion is not desirable.

B. False-Alarm Control

The Hough transform in the example is designed to detect target data points that constitute a straight line. Two or more large noise spikes in the data plane can also constitute a straight line and can create false alarms. The control of false alarms begins with the primary threshold in the data plane. A line of noise below this threshold will not be passed on for further processing. Raising this threshold too high will also limit target deteetability as in a traditional radar. Secondary thresholding in the parameter space also serves to limit both false alarms and target detections. Part II and Part III contain a detailed analysis of the statistics and tradeoffs involved with setting these two thresholds.

Further false-alarm control, not included in the statistical analysis, is also obtained by the clustering of secondary threshold crossings. If detections occur in neighboring parameter cells, chances arc good that

there is only one target. Clustering can reduce this "false-alarm" case to a single detection. Clustering also has an effect on accelerating targets with quadratic trajectories. The clustering algorithm chooses a single track-line out of the fan shown in Fig. 10. The line chosen is the one with the strongest straight-line returns. This will probably be the most recent returns for an approaching target. This may be good, but the fan-like detection of Fig. 10 would not be secn.

Probably the most effective method of false-alarm control not considered in the statistical analysis of Part II and Part III is to use logic. If Doppler velocity information is available and a data space is created for each Doppler bin, then targets will occur in a particular bin with a limited range of slopes due to the velocity presorting. If a detection occurs in a Doppler bin with the wrong slope then this can be discarded as a false alarm. This will add considerable improvement to the false-alarm statistics when compared with a radar with no Doppler information. This type of logic can also be used with higher dimensional data spaces. Certain trajectories are physically impossible for the targets of interest. These can be thrown out.

Another form of logic which will help with false-alarm control is to look at the data which constituted a declared detection. If the detection came from only a few spikes separated widely by many seconds, this may be due to only noise spikes. A target detection is more likely to be formed from a steadier stream of low returns. Looking at the variance of the constituent data may allow for sorting. One must be careful with this since a highly fluctuating target may also have a high variance.

Clutter mapping is a technique which is very easily implemented in a Hough detector. The detector can find the stationary clutter as a vertical line in the data space. If desired, these range points can then be excluded from any new data that is added to the data space. 'Pdrgets can then f ly right through the clutter regions and only a few small returns will he missed.

C. M ultisensor Integration

Multisensor integration or data fusion is the concept of combining the data from several dissimilar or separated sensors in order to do a combined detection. This could easily be implemented by allowing multiple radars to "vote" for lines in a common Hough parameter space. If the sensors are widely separated, appropriate data transformations could be done to put the data into a common coordinate system. If the sensors are collocated this would not be necessary. Differences in radar parameters such as range gate size and beamwidths would not be a major problem in the transformation, but they would have to be considered when designing

CARLSON ET AL.: SEARCH RADAR DETECTIOl\ AND TRACK WITH THE HOUGH TI{ANSFOI{M 107

the common Hough parameter space granularity and dimensions.

Another form of data integration could easily be implemented with this type of data radar. If a radar could receive both vertical and horizontal polarizations, even if only one polarization is being transmitted, then two separate data maps could be created for the one radar. Small or complex targets can strongly depolarize radar signals. Large amounts of energy may be in the cross-pol mode. Both polarizations could vote in a common Hough parameter space. This would add energy for detectability and also provide some immunity to fades in one polarization or the other for f luctuating targets. In this situation, there would be no difference in radar parameters for the two maps. This would amount to the noncoherent addition of the two data maps, but with separate thresholding in each map.

V. SUMMARY AND CONCLUSIONS

A method has been presented of using the Hough transform to perform noncoherent integration in the multi-dimensional data space of a search radar. Improved detectability results from using data that would be thrown out in a traditional search radar. This method integrates signals from targets with inter-cell motion, i.e., range-walk is no longer a problem. Flexibility is inherent in the radar design since once the data is collected it can be shaped many different ways in data maps and can grow as future processing technology grows. The long latency in waiting for a search beam to return to a point

can be eliminated by spreading the CPIs evenly over time. This is possible since range-walk is no longer a problem. This detection method not only declares targets, but tracks as well, without the need to revisit and acquire. The completeness of the track depends on the dimensionality of the chosen data space. Part II and Part III contain a much more mathematical analysis of the statistics of this method as well as further discussion.

REFERENCES

[I] Hough, P. V. C. (1962) Methods and means for recognizing complex patterns. V.S. Patent 3,069,654, 1%2.

[2] Duda, R. 0., and Hart, P. E. (1972) Vse of the Hough transformation to dctcct lines and curves in pictures. Communication a/ the ACM, 15, 1 (Jan. 1972), 11-15.

[3] Ballard, D. H. (1981) Generalizing the Hough transform to detect arbitrary shapes. Pallern Recognition, 13, 2 (1981), 111-122.

[4] Illingworth, J., and Kittler, J. (1988) A survey of the Hough transform.

Computer Vision, Graphics, and Image Processing, 44 (1988), 87-116.

[5] Wilson, S. L., Carlson, B. D., and Evans, E. D. (1994) Search radar detection and track with the Hough transform, Part II: Detection statistics. IEEE Transactions on Aerospace and Electronic Systems, this issue 109-115.

[6] Evans, E. D., Wilson, S. 1.., and Carlson, B. D. (1994) Search radar detection and track with the Hough transform, Part III: Detection performance with binary integration. IEEE Transactions on Aerospace and Electronic Systems, this issue 116-125.

Author photographs and biographies will be found on page 125 in this issue.

108 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 30, NO.1 JANUARY 1994