OPTIMAL SENSOR POSITIONING FOR ISAR

IMAGING

IGARSS 2010HONOLULU, HAWAII, JULY 2010

Marco Martorella

Motivation

•In ISAR, long data recordings are often needed in order to form an image with desired characteristics (useful for target classification)

•Such image characteristics depend of both the target motions and the sensor position

•Since the target is often non-cooperative, only the sensor position can be used as a degree of freedom to drive the outcome towards the desired result

Need of a simple tool that provides the means for predicting the optimal sensor position: this will minimise the time on target and maximise the probability of obtaining a desired image

Outline•Background•ISAR imaging•Image Projection Plane (IPP)

•Sensor position as a degree of freedom

•Signal model

•IPP constraints•Front, Side, Top and Composite target views

•Cross-range resolution constraint

•Numerical results

•Conslusions

ISAR Imaging

x1

x2x3

θa

θe

Differently from SAR,

•ISAR imaging is a processing that enables a radar system to produce focussed e.m. images of non-cooperative targets

•In ISAR, the knowledge about the radar-target geometry and its dynamics are not known a priori and cannot be controlled

•Autofocusing techniques are always needed and they work based on the only use of the received data (no a priori knowledge, no ancillary data)

•The target image “quality” strongly depends on the target orientation and dynamics, which are not known a priori

•the ISAR image interpretation is harder due to the dependence of image parameters (resolution, image projection plane, etc) on the target motions

ISARgeometry

ISARimage

iLOS

Image Projection Plane (IPP)

iLOS

Ω

Ωeff

icr , ir( ) = iLOS × Ωeff , iLOS( )

Ωeff = iLOS × Ω × iLOS( )•Effective rotation vector

•Image projection plane

•The image projection plane is a plane orthogonal to the effective rotation vector

•The image projection plane depends on the effective rotation vector and the radar-target Line of Sight

•The target plays a role in this since its own motions strongly contribute to the target’s rotation vector and hence to the effective rotation vector

•The IPP becomes very important when dealing with ISAR image interpretation (target’s projection onto the image plane), which can be seen as a first step towards target classification and recognition

Sensor position

x1

x2x3

θa

θe

iLOS•The position of the sensor is given by means of two angles: azimuth and elevation

•The IPP is defined once the target’s motion and the relative position of the sensor with respect to the target are given

•In some ISAR applications, the position of the sensor can be controlled by the operator

•In ISAR system design, the position of the sensor becomes one of the system parameters that has to be defined to optimise the imaging system

•We can see the sensor position as the only degree of freedom if we want to have some control over the IPP

•As a criterion for ISAR imaging system optimisation, we will use the concept of desired IPP

•Typical desired IPPs are: front view, side view, top view and composite front/side view

Signal model (1)

RADAR

Ideal Scatterers =

Tob=1.5 s

•The cross-range image formation can be seen as a Doppler analysis

•Scatterers in different position along the cross-range direction produce different Doppler and therefore are mapped in different cross-range positions in the image

•The Doppler induced by a scatterer positioned at x can be calculated analytically

fd (t) =2λ

[Ωeff (t)× x]

Signal model (2)

fd (t) =2λ

[Ωeff (t)× x] =2λ

ΩT (t)Lx

•The Doppler frequency can also be calculated by using a matrix notation

•where L is a 3x3 matrix with elements equal to

L11 = L22 = L33 = 0L12 = −L21 = sin θe

L31 = −L13 = cos θe sin θa

L23 = −L32 = cos θa cos θe

Rotation vectorEffective rotation vector

•The Doppler frequency can therefore be rewritten as the sum of three contributions

fd (t) = L1 (t) x1 + L2 (t) x2 + L3 (t) x3

L1 (t) = Ω2 (t) L21 + Ω3 (t) L31

L2 (t) = Ω1 (t) L12 + Ω3 (t) L32

L3 (t) = Ω1 (t) L13 + Ω2 (t) L23

where are the Doppler Generating Factors (DGF)

Sensor positionrelated matrix

Scatterer’s position

Li t( ) i = 1,2,3

Desired IPP (1/4)

3D Target

Front view Side view Top view

Composite front/side view

Desired IPP (2/4)

•A desired IPP can be obtained by acting on the sensor position

•For front, side and top views, this can be done by constraining

•one DGF •one of the two angles that define the sensor position

•For a composite front/side view, this can be done by constraining

•two DGFs •none of the angles that define the sensor position

Desired IPP (3/4) Front view

•The contribution relative to the coordinate x1 must be forced to zero •The sensor must be located in the plane formed by x2 and x3

Side view

•The contribution relative to the coordinate x2 must be forced to zero •The sensor must be located in the plane formed by x1 and x3

L2 (t) = Ω1 (t) sin θe − Ω3 (t) cos θa cos θe = 0,subject to θa = 0

L1 (t) = −Ω2 (t) sin θe + Ω3 (t) cos θe sin θa = 0subject to θa = π

2

θe (t) = arctanΩ3 (t)Ω1 (t)

θe (t) = arctanΩ3 (t)Ω2 (t)

Desired IPP (4/4) Top view

•The contribution relative to the coordinate x3 must be forced to zero •The sensor must be located in the plane formed by x1 and x2

Composite front/side view

•The contribution relative to the coordinates x1 and x2 must be forced to zero

L3 (t) = −Ω1 (t) cos θe sin θa + Ω2 (t) cos θa cos θe = 0subject to θe = 0

θa (t) = arctanΩ2 (t)Ω1 (t)

θa (t) = arctan

Ω2(t)Ω1(t)

θe (t) = arctan

Ω3(t)√Ω1(t)+Ω2(t)

L1 (t) = −Ω2 (t) sin θe + Ω3 (t) cos θe sin θa = 0L2 (t) = Ω1 (t) sin θe − Ω3 (t) cos θa cos θe = 0

Cross-range Resolution Constraint

δcr =c

2f0ΩeffTob

Ω · iLoS = cos θa cos θeΩ1 + sin θa cos θeΩ2 + sin θeΩ3 = 0

θe = − arctanΩ1 cos θa + Ω2 sin θa

Ω3

•The cross-range resolution can be determined in the case of constant target rotation vector

•Given a target rotation vector, the sensor position that minimises the cross-range resolution can be obtained by constraining the inner product between the radar LoS and the target rotation vector to zero

•There are an infinite number of solutions. The generic solution can be written as

•The solution of the problem of obtaining a desired IPP may produce an image with poor cross-range resolution

Ωeff = iLOS × Ω × iLOS( )•Note: the effective rotation vector can be small even when the target rotation vector is large because of a bad choice of the sensor position

Cross-range Resolution Constraint

•Note: generally, the solution of the minimum resolution problem does not coincide with the solution of the desired IPP

•When the minimum cross-resolution constraint is not applied

δcr =c

2f0ΩeffTob≥ δmin =

c

2f0ΩTob

•Criterion of optimality

•Define the desired IPP

•Set a maximum cross-range resolution loss, i.e. accept a desired IPP solution as an optimal solution only if the cross-range resolution does not exceed a pre-set value

•Maximum cross-range resolution loss

δmax = Kδmin K ≥ 1

Mapping target motion distribution onto optimal sensor position distribution

Non-cooperative target motions

•are not known a priori and in a general case cannot be predicted with sufficient accuracy

•depend on several parameters: both internal (e.g. target’s maneuvers) and external (e.g. sea conditions for a ship)

Statistical distribution of target motions

•derived from models

•derived from measurements

•For each target motion, there exist an optimal sensor position that can be determined by applying the desired IPP and cross-resolution constaints

•We can see the result as a map that transforms elements from the target motion space onto the sensor position space

fΩ ω( )→ fΘ θa ,θe( )

Numerical results (1/3)

8 6 4 2 0 2 4 60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16Pitch rate

Degrees/s

Prob

abilit

y

20 15 10 5 0 5 10 15 200

0.05

0.1

0.15

0.2

0.25Roll rate

Degrees/s

Prob

abilit

y

6 4 2 0 2 40

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Yaw rate

Degrees/s

Prob

abilit

y

DATA SET

•Pitch, roll and yaw motions of a small boat have been measured by using an Inertial Measurement Unit (IMU)

•3500 samples at a rate of 0.2 sample/s

Normalised histograms of Pitch, roll and yaw

•We can interpret the histograms as approximation of Probability Density Functions

Numerical results (2/3)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Histogram Elevation Effective pure side view a = 0 degrees

e

prob

abilit

y

L2 (t) = Ω1 (t) sin θe − Ω3 (t) cos θa cos θe = 0,subject to θa = 0

θe (t) = arctanΩ3 (t)Ω1 (t)

Side View

8 6 4 2 0 2 4 60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16Pitch rate

Degrees/s

Prob

abilit

y

6 4 2 0 2 40

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Yaw rate

Degrees/s

Prob

abilit

y

K = 3

Numerical results (3/3)

Probability of effective mixed front/side view

a

e

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60

70

80

90

0.005

0.01

0.015

0.02

0.025

0.03

Composite Front/Side View

θa (t) = arctan

Ω2(t)Ω1(t)

θe (t) = arctan

Ω3(t)√Ω1(t)+Ω2(t)

L1 (t) = −Ω2 (t) sin θe + Ω3 (t) cos θe sin θa = 0L2 (t) = Ω1 (t) sin θe − Ω3 (t) cos θa cos θe = 0

K = 3

8 6 4 2 0 2 4 60

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16Pitch rate

Degrees/s

Prob

abilit

y

20 15 10 5 0 5 10 15 200

0.05

0.1

0.15

0.2

0.25Roll rate

Degrees/s

Prob

abilit

y

6 4 2 0 2 40

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Yaw rate

Degrees/s

Prob

abilit

y

Conclusions

•Definition of optimality criteria for ISAR sensor positioning

•Mathematical derivation of a tool for predicting the optimal sensor position

•Useful for placement of static sensors given the surveillance scenario

•Useful for route planning of moving sensors

•Useful for predicting the probability of obtaining a desired IPP given a scenario of interest and the position of the sensor

•Can be extended to bistatic and multistatic scenarios (please check the proceedings of next EURAD conference)