ISAR Signal Formation and Image Reconstruction as Complex Spatial Transforms

Inverse aperture synthesis in the radar theory is a recording of the complex reflective pattern (complex microwave hologram) of a moving target as a complex signal. The trajectory of moving target limited by the radar’s antenna pattern or time of observation is referred to as inverse synthetic aperture, and radar using the principle of inverse aperture synthesis is inverse synthetic aperture radar (ISAR). The spatial distribution of the reflectivity function of the target referred to as a target image can be retrieved from the received complex signals by applying image reconstruction techniques.


Introduction
Inverse aperture synthesis in the radar theory is a recording of the complex reflective pattern (complex microwave hologram) of a moving target as a complex signal. The trajectory of moving target limited by the radar's antenna pattern or time of observation is referred to as inverse synthetic aperture, and radar using the principle of inverse aperture synthesis is inverse synthetic aperture radar (ISAR). The spatial distribution of the reflectivity function of the target referred to as a target image can be retrieved from the received complex signals by applying image reconstruction techniques.
Conventional ISAR systems are coherent radars. In case the radars utilize a range-Doppler principle to obtain the desired image the range resolution of the radar image is directly related to the bandwidth of the transmitted radar signal, and the cross-range resolution is obtained from the Doppler frequency gradient generated by the radial displacement of the object relative to the radar.
A common approach in ISAR technique is division of the arbitrary movement of the target into radial displacement of its mass centre and rational motion over the mass centre. Radial displacement is compensated considered as not informative and only rotational motion is used for signal processing and image reconstruction. In this case the feature extraction is decomposed into motion compensation and image reconstruction (Li et al., 2001). Multiple ISAR image reconstruction techniques have been created, which can be divided into parametric and nonparametric methods in accordance with the signal model description and the methods of a target features extraction. (Berizzi et al., 2002;Mrtorella et al., 2003;Berizzi et al., 2004). The range-Doppler is the simplest non parametric technique implemented by two-dimensional inverse Fourier transform (2-D IFT). Due to significant change of the effective rotation vector or large aspect angle variation during integration time the image becomes blurred, then motion compensation is applied, which consist in coarse range alignment and fine phase correction, called autofocus algorithm. It is performed via tracking and polynomial approximation of signal history from a dominant or well isolated point scatterer on the target (Chen & Andrews, 1980), referred to as dominant scatterer algorithm or prominent point processing, a synthesized scatterer such as the centroid of multiple scatterers (Wu et al., 1995), referred to as multiple scatterer algorithm. Autofocus technique for random translational motion compensation based on definition of an entropy image cost function is developed in (Xi et al., 1999). Time window technique for suitable selection of the signals to be coherently processed and to provide a focused image is suggested in (Martorella Berizzi, 2005). A robust autofocus algorithm based on a flexible parametric signal model for motion estimation and feature extraction in ISAR imaging of moving targets via minimizing a nonlinear least squares cost function is proposed in (Li et al., 2001). Joint time-frequency transform for radar range-Doppler imaging and ISAR motion compensation via adaptive joint time-frequency technique is presented in (Chen  Qian, 1998;Qian , Chen 1998).
In the present chapter assuming the target to be imaged is an assembly of generic point scatterers an ISAR concept, comprising three-dimensional (3-D) geometry and kinematics, short monochromatic, linear frequency modulated (LFM) and phase code modulated (PCM) signals, and target imaging algorithms is thoroughly considered. Based on the functional analysis an original interpretation of the mathematical descriptions of ISAR signal formation and image reconstruction, as a direct and inverse spatial transform, respectively is suggested. It is proven that the Doppler frequency of a particular generic point is congruent with its space coordinate at the moment of imaging. In this sense the ISAR image reconstruction in its essence is a technique of total radial motion compensation of a moving target. Without resort to the signal history of a dominant point scatterer a motion compensation of higher algorithm based on image entropy minimization is created.

Kinematic equation of a moving point target
The Doppler frequency induced by the radial displacement of the target with respect to the point of observation is a major characteristic in ISAR imaging. It requires analysis of the kinematics and signal reflected by moving target. Consider an ISAR placed in the origin of the coordinate system (Oxy) and the point A as an initial position with vector R(0) at the moment t = 0, and the point B as a current or final position with vector R(t) at the moment t ( Fig. 1). Assume a point target is moving at a vector velocity v, and then the kinematic vector equation can be expressed as (1) which in matrix form can be rewritten as are the coordinates of the initial position of the target (point A); is the module of the initial vector;  is the initial aspect angle; .cos are the coordinates of the vector velocity; v is the module of the vector velocity and  is the angle between vector velocity and Ox axis.
The time dependent distance ISAR -point target can be expressed as is the angle between position vector (0) R and vector velocity v , defined by the equation Then Eq. (3) can be rewritten as where (0) R is the distance to the target at the moment t  0, measured on OA, the initial line of sight (LOS).
The radial velocity of the target at the moment t is defined by differentiation of Eq. (5), i.e.
If t  0, the radial velocity (0)  . The time variation of the radial velocity of the target causes a time dependent Doppler shift in the frequency of the signal reflected from the target.

Doppler frequency of a moving point target
Assume that the ISAR emits to the target a continuous sinusoidal waveform, i.e. www.intechopen.com Then the current angular frequency of the reflected signal can be determined as   the Doppler frequency is time dependent during the aperture syntheses, coherent processing interval (CPI), but only one value has a meaning for ISAR imaging, the value defined at the moment of imaging, which will be proven in subsection 3.3.

Example 1
Assume that the point target is moving at the velocity v =29 m/s and illuminated by a continuous waveform with wavelength  = 3.10 -2 m (frequency It is worth noting that the current signal frequency decreases during CPI due to the alteration of the value and sign of the Doppler frequency varying from -3 to 3 Hz. At the moment t = 717 s the Doppler frequency is zero. The time instance where Doppler changes its sign (zero Doppler differential) can be regarded as a moment of target imaging.
Computational results of the imaginary and real part of ISAR signal reflected by a point target with time varying radial velocity are presented in Figs. 3, (a), and (b). It can be clearly seen the variation of the current frequency of the signal due to the time dependent Doppler frequency of the point target. The existence of wide bandwidth of Doppler variation in the signal allows multiple point scatterers to be potentially resolved at the moment of imaging.
www.intechopen.com (a) Imaginary part of an ISAR signal (b) Real part of an ISAR signal.

Example 2
It is assumed that the point target moves at the velocity v =29 m/s and is illuminated with continuous waveform with wavelength  = 10 -2 m (frequency It can be seen that the current signal frequency has two constant values during CPI due to the constant Doppler frequency with two signs, -5.8 Hz and +5.8 Hz. At the moment t = 1.04 s the Doppler frequency alters its sign. The time instance where Doppler changes its sign (zero Doppler differential) can be regarded as a moment of point target imaging that means one point target can be resolved.
www.intechopen.com (a) Imaginary part of an ISAR signal (b) Real part of an ISAR signal.

3-D ISAR geometry and kinematics
The basic characteristic in ISAR imaging is the time dependent distance between a particular generic point from the target and ISAR. Consider 3-D geometry of ISAR scenario with radar and moving target in the coordinate system Oxyz (Fig. 6). The target is located in a regular grid, defined in the coordinate system ' OX Y Z. The generic point scatterer g from the target area is specified by the index vector (, , ) ijk , i.e. (, , ) ijk  g . The position vector () ijk p R of the ijk th generic point scatterer in the coordinate system Oxyz at the moment p is described by the following vector equation The projection angles , The projection of the vector equation (14) then the distance between the generic point and ISAR can be expressed as Eq. (20) is used in calculation of the time delay of the signal reflected by a particular generic point scatterer from the target area while signal modeling.

Short pulse ISAR signal formation
Consider 3-D ISAR scenario (Fig. 6) and a generic point g from the target illuminated by sequence of short monochromatic pulses, each of which is described by www.intechopen.com () () 1, 0 1, rect 0, otherwise.
is the time delay of the signal, g stands for the discrete vector coordinate that locates the generic point scatterer in the target area G, a g stands for the magnitude of the 3-D discrete image function, mod p tt T   is the slow time, p denotes the number of the emitted pulse, p T is the pulse repetition period, where k is the number of range bin, where the ISAR signal is placed. The demodulated ISAR signal from the target area is The expression (23) is a weighted complex series of finite complex exponential base functions. It can be regarded as an asymmetric complex transform of the 3-D image function a g ,  gG , defined for a whole discrete target area G into 2-D signal plane (,) spk .

Image reconstruction from a short pulse ISAR signal
Eq. (23) where p is the number of emitted pulse, N is the full number of emitted pulses during CPI.
Because (,) spk is a 2-D signal, only a 2-D image function â g can be extracted. Eq. (25) is a symmetric complex inverse spatial transform or inverse projective operation of the 2-D signal plane (,) spk into 2-D image function â g , and can be regarded as a spatial correlation where r g , v g , a g and h g is the distance, radial velocity, acceleration and jerk of the generic point, respectively at the moment of imaging.

www.intechopen.com
Due to range uncertainty of generic points placed in the kth range resolution cell, r g can be assumed constant, and (25) Eq. (27) stands for a procedure of total motion compensation of every generic point from kth range resolution cell. The range distance r g does not influence on the image reconstruction and can be removed from the equation (27) For each kth range cell the term 2 v  g stands for the Doppler frequency whereas terms as 2 a  g , 2 h  g …., denote the higher order derivations of the time dependent Doppler frequency, defined at the moment of imaging.
If the Doppler frequency of generic points in the kth range cell is equal or tends to constant during CPI the equation (28) The equation (30) stands for an IFT of   , s p k for each kth range resolution cell and can be considered as phase and/or motion compensation of first order. where ˆ(,) a p k g denotes the complex azimuth image of the target, p denotes the unknown index of the azimuth space coordinate equal to the unknown Doppler index of the generic point scatterer from the target at the moment of imaging. The polynomial coefficients m a , m  2, 3, are calculated iteratively via applying image quality criterion, which will be discussed in subsections 4.4.
Eq. (32) can be interpreted as an ISAR image reconstruction procedure implemented through inverse Fourier transform (IFT) of a phase corrected ISAR signal into a complex azimuth image ˆ(,) a p k g for each kth range cell. In this sense the ISAR signal (,) spk can be referred to as a spatial frequency spectrum whereas ˆ(,) a p k g can be referred to as a spatial image function defined at the moment of imaging. Based on Eq. (32) two steps of image reconstruction algorithm can be outlined.
Step 1 Compensate the phases, induced by higher order radial movement, by multiplication of   Step 2 Compensate the phases induced by first order radial displacement of generic points in the kth range cell by applying IFT (extract complex image), i.e.
Complex image extraction can be implemented by inverse fast Fourier transform (IFFT). The algorithm can be implemented if the phase correction function () p  is preliminary known.
Otherwise only IFT can be applied. Then non compensated radial acceleration and jerk of the target still remain and the image becomes blurred (unfocused). In order to obtain a focused image motion compensation of second, third and/or higher order has to be applied, that means coefficients of higher order terms in () p  have to be determined. The definition and application of these terms in image reconstruction is named an autofocus procedure accomplished by an optimization step search algorithm (SSA) which will be discussed in subsection 4.4.

LFM waveform
Consider 3-D ISAR scenario (Fig. 6) and a target illuminated by sequence of LFM waveforms, each of which is described by where a g is the reflection coefficient of the gth generic point scatterer, a 3-D image function; is the round trip time delay of the signal from gth generic point scatterer; which in discrete form can be expressed as Eq. (41) can be interpreted as a spatial transform of the 3-D image function a g into 2-D ISAR signal plane ˆ(,) spk by the finite transformation operator, the exponential term Formally the 3-D image function a g should be extracted from 2-D ISAR signal plane by the inverse spatial transform but due to theoretical limitation based on the number of measurement parameters only a 2-D image function may be extracted, i.e.
Extraction of the image function is a procedure of complete phase compensation of the signals reflected by all point scatterers from the object that means total compensation of target movement during CPI. The argument of the exponential term ( Eq. (48) can be considered as an image reconstruction computational procedure, which does reveal the 2-D discrete complex image function(,) ap k g .

LFM ISAR image reconstruction algorithm
Based on the previous analysis the following image reconstruction steps can be defined.
Step 1 Compensate phase terms of higher order by multiplication of complex matrix ˆ(,) spk by a complex exponential function The aforementioned algorithm is feasible if the phase correction function (,) p k  is a priory known. Otherwise, a focused image is impossible to extract. In this case taking into account the linear property of computational operations in (48)

Autofocusing phase correction by image entropy minimization
If the image obtained by only range (50) and azimuth (51) The procedure is repeated until the global minimum value of the entropy s H is acquired.

Numerical experiment
To verify the properties of the LFM ISAR signal model and to prove the correctness of the image reconstruction algorithm a numerical experiment is carried out. Assume the target, Mig-35, detected in 3-D coordinate system ' OX Y Z is moving rectilinearly in a coordinate system Oxyz .  www.intechopen.com

PCM waveform
Consider 3-D ISAR scenario (Fig. 6)  The real and imaginary part of the complex Barker's PCM ISAR signal is presented in Fig.  10, the final image -2-D space image function -in Fig. 11, and entropy evolution in Fig.  12.
(a) Real part of ISAR signal (b) Imaginary part of ISAR signal

Conclusion
In the present chapter a mathematical description and original interpretation of ISAR signal formation and imaging has been suggested. It has been illustrated that both of these operations can be interpreted as direct and inverse spatial complex transforms, respectively. It has been proven that the image extraction is a threefold procedure; including phase correction, range compression performed by IFT in case LFM waveforms and by crosscorrelation in case PCM waveforms, and azimuth compression performed by IFT in both cases. It has been underlined that the image reconstruction is a procedure of total motion compensation, i.e. compensation of all phases induced by the target motion. Only phases proportional to the distances from ISAR to all point scatterers on the target at the moment of their imaging still remain. These phases define a complex character of the ISAR image. The drawback of the proposed higher motion compensation algorithm is the existence of multiple local minimums in entropy evolution in case the target is fast maneuvering. In order to find out a global minimum in the entropy and optimal values of the polynomial coefficients the computation process has to be enlarged in wide interval of their variation. The subject of the future research is the exploration of the image reconstruction algorithm with higher order terms and cross-terms of the phase correction polynomial while the target exhibits complicated movement.

Acknowledgement
This chapter is supported by NATO Science for Peace and Security (SPS) Programme: NATO: ESP. EAP. CLG 983876.