# Patent

## Title
Method for developing and using an image reconstruction algorithm for multipath scattering

## Abstract

Described herein is an implementation of the IRAMS processing based upon a multi-delay-resolution framework applied to SAR image data measured at different aspect angles. The power of this new embodiment of IRAMS is that it produces a good separation of immediate response scatterer and delayed response scatterer data for the case of anisotropic scattering events, i.e., those in which the scattering intensity depends upon the aspect angle. Two sources of delayed response scattering include multiple reflection scattering events and delayed responses arising from the physical material composition of the scatterer. That is, this multi-delay-resolution IRAMS processing separates immediate response and delayed response scattering for cases in which there exist delayed response scattering data in the original SAR image data at some aspect angles, but the intensity of these delayed response scattering data is weak or non-existent at different aspect angles. Thus, this IRAMS embodiment provides the additional information of the particular aspect angles at which delayed response scattering effects are most important, thereby improving both the estimates of the delayed response scattering data and the immediate response scattering data.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
This application is a continuation of and claims priority to U.S. Pat. application No. 10/967,298 entitled METHOD FOR DEVELOPING AND USING AN IMAGE RECONSTRUCTION ALGORITHM FOR MULTIPATH SCATTERING filed Oct. 19, 2004, now U.S. Pat. No. 7,259,715, which is a continuation-in-part of and claims priority to U.S. application Ser. No. 10/954,218 entitled PROCESS FOR MAPPING MULTIPLE-BOUNCE GHOSTING ARTIFACTS FROM RADAR IMAGING DATA filed Oct. 1, 2004, now U.S. Pat. No. 7,173,562, which is a continuation of U.S. application No. 10/631,712 entitled PROCESS FOR MAPPING MULTIPLE-BOUNCE GHOSTING ARTIFACTS FROM RADAR IMAGING DATA filed Aug. 1, 2003, now U.S. Pat. No. 6,812,886, which is a continuation of U.S. application Ser. No. 10/059,416 entitled PROCESS FOR MAPPING MULTIPLE-BOUNCE GHOSTING ARTIFACTS FROM RADAR IMAGING DATA filed Jan. 31, 2002, now U.S. Pat. No. 6,646,593, which claims priority to U.S. Provisional Patent Application No. 60/345,639, entitled SPOTLIGHT SAR IMAGE FORMATION WITHOUT MULTIPLE-BOUNCE GHOSTING ARTIFACTS filed Jan. 8, 2002, all of which are incorporated herein by reference in their entireties.


Embodiments of the invention may have been conceived or first actually reduced to practice in the performance of work under a Government contract. As a result, the Government may have certain rights in those inventions.


BACKGROUND OF THE INVENTION
1. Field of the Invention
Generally, the invention involves a process for forming radar images. More specifically, the invention involves a process for mapping multiple bounce ghosts (MBGs) and for forming radar images without the deleterious effects of MBGs.
2. Description of the Related Art
Radar, at its most basic application, is used to measure the range to a target. With knowledge of the speed of propagation of the wave, i.e., electromagnetic wave, that is transmitted toward the target, it is possible to resolve in a first dimension, the distance to the target, based on the received reflected wave or echo. In order to use radar as an imaging tool, it is necessary to collect information about the cross-range of the target, in addition to the first dimension information. This cross-range information is about a second dimension perpendicular to the first dimension.
Synthetic aperture radar (SAR) can be used to collect data in both the first and second dimensions, through a process wherein the reflected waves are measured at different angles with respect to an object-of-interest. This process is referred to in the art as collecting radar measurements over a synthetic (as opposed to a literal) aperture. By taking various measurements of the object-of-interest from varying aspect angles, it is possible to determine approximate distance to the scattering centers within an object-of-interest in the first dimension and location of these scattering centers within the object-of-interest in the second, cross-range dimension. This process of two-dimensional imaging is commonly referred to as reflection tomography.
SAR systems take advantage of the long-range propagation characteristics of radar signals and the complex information processing capability of modern digital electronics to provide high-resolution imagery. SAR imaging is not restricted by time of day or atmospheric conditions due to its operative frequencies. Consequently, SAR imaging supplements other photographic and optical imaging techniques in order to facilitate environmental monitoring, earth-resource mapping, and military operations which may require broad-area imaging at high resolutions. More specifically, SAR technology provides detailed terrain information to geologists for mineral exploration, environmentalists for determination of oil spill boundaries, navigators for sea state and ice hazard mapping, and the military for reconnaissance and targeting information.
Other systems using reflection data, also referred to as projection measurements, are fault inspection systems using acoustic imaging, submarine sonar for imaging underwater objects and the like, seismic imaging system for tunnel detection, oil exploration, geological surveys, etc., and medical diagnostic tools such as sonograms and echocardiograms.
There have been two basic types of processing techniques used in the field of reflection tomography to reconstruct single-bounce (SB) reflection data. First, the frequency-domain projection-slice theorem takes the measured phase history from the reflection data taken at different aspect angles and generates the reconstruction of an image using Fourier transforms. This reconstruction technique is often used for reconstructing SAR image data in order to minimize the computational load that results from necessarily complex processing. A second technique, more prevalent in the medical imaging community, is based on the time-domain back projection techniques. Both of these techniques are discussed in U.S. Pat. No. 5,805,098 to McCorkle which is incorporated herein by reference in its entirety.
SUMMARY OF THE INVENTION
Summary of the Problem
The reflection data processing techniques described in the related art assume that the impinging wave reflects a single time off of the object of interest before returning back to the receiver. This assumption neglects the situation wherein the wave actually reflects off of multiple portions of the object of interest, cascading for any number of times, before returning to the receiver. All prior art on reflection tomography assumes only one bounce. That is, conventional tomography does not include the effects of multiple-bounce (MB) scattering events wherein the mediating waveform first scatters off of one portion of the extended object-of-interest, which then scatters in a cascade fashion off of one or more other regions of this same extended object-of-interest or off of other objects before scattering into the receiver. These facts motivate the need for the development of a modified image formation process that applies for cases in which the measured reflection data also identifies and accounts for MB scattering events.
A particular imaging scenario wherein the prior art process is insufficient includes determining the physical distribution of the scattering centers for cases in which multiple-bounce echoes occur. For example, prior art processes would have a difficult time determining from three (3) received echoes that instead of three scattering centers for an object-of-interest, the image actually contained two targets, wherein the third echo was between the two scattering centers as opposed to a third scattering center. The problem is exacerbated in a case where there are many scattering centers within the image, with the possibility for multiple bounces between the scattering centers. Further, in the case of SAR imaging, the process becomes even more complicated by the fact that the object (or region)-of-interest radar information is only gathered over a very small aspect angle range, for example, up to approximately 15 degrees. Current processing schemes place the multiple-bounce echoes at incorrect (ghost) locations due to fundamental assumptions implicit in the processing.
Summary of the Solution
Described herein are frequency-domain back-projection processes for forming, e.g., spotlight SAR images that are not corrupted by the effects of multiple-bounce ghosting artifacts. These processes give an approximately exact reconstruction of the multiple bounce reflectivity function (MBRF) ƒ(x,y,γ). As shown in the embodiments herein, the reconstruction process is not affected to any greater extent by noise and interference than in the case of the single bounce processing. Specifically, the evaluation of ƒ(x,y,γ) in the γ=0 plane gives an approximately exact reconstruction of the true object scattering centers which is uncorrupted by multiple-bounce contributions to the phase history data G(ξ, θ). In addition, the non-zero dependence of ƒ(x,y,γ) upon the MB coordinate γ can be used to facilitate the identification of features-interest within the imaged region.
In an embodiment of the present invention, a process is described for incorporating the effects of multiple bounces obtained using the back projection imaging process referred to herein, in order to obtain a cleaned up image that has removed the misleading effects of the multiple bounces, commonly referred to as multiple bounce ghosts (MBGs). The process identifies the MBGs and maps the MBG reflections into a metrically-correct image plane, also referred to as, a delay image plane or higher dimensional image plane, creating auxiliary images, which are useful in providing further information about the viewing region, including additional discriminating information that could assist in object recognition and identification. The imaging information, including both single and multiple bounce information, is only required for a narrow range of measurement angles.
The processes of the present invention are useful in three dimensional imaging, providing images that are free of multi-path effects, and extending the field-of-use of the technology beyond SAR imaging to all areas of reflection tomography. Because the technology incorporates a physical model that explicitly allows for multi-bounce effects in image formation, it is applicable to any image formation technology where that phenomenon is found such as, but not limited to, real aperture radar imaging, synthetic aperture radar (SAR) imaging, inverse SAR (ISAR) imaging, active sonar underwater acoustic mapping, active geographic acoustic exploration and ultrasonic medical imaging.
In an embodiment of the present invention, a process for reconstructing a scene from scattering data is described. The process includes: collecting multiple images of a scene, the multiple images containing scattering data from the scene representing (a) single reflection data and (b) multiple reflection data represented within the multiple images by at least one of amplitude data and phase data, wherein (a) and (b) are located in any one of image planes, 0 to P, wherein P is greater than 0; accounting for a change in the at least one of amplitude data and phase data representative of (b) from one of the multiple images of the scene to another of the multiple images of the scene; and separating (a) and (b) from the scattering data according to their respective locations within image planes 0 to P utilizing the multiple images.
In another embodiment of the present invention, a process for analyzing scattering data from an area-of-interest is described. The process includes: collecting a first image of the area-of-interest at a first aspect angle, the first image containing first scattering data comprised of first single bounce data and first multiple bounce data, which includes at least one of amplitude data and phase data; collecting a second image of the area-of-interest at a second aspect angle, the second image containing second scattering data comprised of second single bounce data and second multiple bounce data, which includes at least one of amplitude data and phase data; determining a change in the at least one of amplitude data and phase data of the first multiple bounce data with respect to the second multiple bounce data; accounting for the change in the at least one of amplitude data and phase data, in order to differentiate the first and second single bounce data from the first and second multiple bounce data; and analyzing the differentiated first and second single bounce data and the first and second multiple bounce data to determine characteristics of the area-of-interest.
In a further embodiment of the present invention, a method for imaging an object-of-interest is described. The method includes: collecting phase history scattering data from the object-of-interest over a range of aspect angles, the scattering data including single point scattering data and multiple point scattering data, which includes at least one of amplitude data and phase data; identifying the multiple point scattering data within the scattering data over the range of aspect angles according to a change in at least one of amplitude data and phase data; and comparing the identified multiple point scattering data with the collected scattering data in order to identify the single point scattering data; and imaging the object-of-interest using the identified single point scattering data.
In a further embodiment of the present invention, a process for analyzing scattering data from an area-of-interest is described. The method includes: collecting a first image of the area-of-interest at a first aspect angle, the first image containing first scattering data comprised of first immediate data and first delayed data, which includes at least one of amplitude data and phase data; collecting a second image of the area-of-interest at a second aspect angle, the second image containing second scattering data comprised of second immediate data and second delayed data, which includes at least one of amplitude data and phase data; determining a change in the at least one of amplitude data and phase data of the first delayed data with respect to the second delayed data; accounting for the change in the at least one of amplitude data and phase data, in order to differentiate the first and second immediate data from the first and second delayed data; and analyzing the differentiated first and second immediate data and the first and second delayed data to determine characteristics of the area-of-interest.
In still a further embodiment of the present invention, a process for reconstructing a scene from scattering data is described. The process includes: collecting multiple images of a scene, the multiple images containing scattering data from the scene representing (a) immediate response data and (b) delayed response data represented within the multiple images by at least one of amplitude data and phase data, wherein (a) and (b) are located in any one of image planes, 0 to P, wherein P is greater than 0; accounting for a change in the at least one of amplitude data and phase data representative of (b) from one of the multiple images of the scene to another of the multiple images of the scene; and separating (a) and (b) from the scattering data according to their respective locations within image planes 0 to P utilizing the multiple images.



BRIEF DESCRIPTION OF THE DRAWINGS
The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee.
In the Figures:
FIGS. 1(a)-1(b) illustrate scattering events according to embodiments of the present invention;
FIG. 2(a)-2(d) illustrate true scattering point data from multiple depth planes, according to a first embodiment of the present invention;
FIGS. 3(a)-3(b) illustrate measurement functions for scattering data according to a first embodiment of the present invention;
FIG. 4 illustrates scattering point data for an object-of-interest utilizing single bounce reconstruction, according to a first embodiment of the present invention;
FIGS. 5(a)-5(d) illustrate scattering point data for an object-of-interest utilizing multiple bounce reconstruction according to a first embodiment of the present invention;
FIG. 6(a)-6(d) illustrate true scattering point data from multiple depth planes, according to a second embodiment of the present invention;
FIGS. 7(a)-7(b) illustrate measurement functions for scattering data according to a second embodiment of the present invention;
FIG. 8 illustrates scattering point data for an object-of-interest utilizing single bounce reconstruction, according to a second embodiment of the present invention;
FIGS. 9(a)-9(b) illustrate scattering point data for an object-of-interest utilizing multiple bounce reconstruction according to a second embodiment of the present invention;
FIG. 10 illustrates a processing diagram for a multi-delay resolution IRAMS tree according to a first embodiment of the present invention;
FIG. 11 illustrates a truth plot containing five actual scattering centers and the simulated multipath ghosts resulting therefrom according to an embodiment of the present invention;
FIG. 12 illustrates a SAR image corresponding to the truth plot of FIG. 11, showing the response of both the direct and multipath scattering event;
FIGS. 13a and 13b illustrates the predicted cross-range drift of multipath scattering ghosts with respect to changing aspect angle;
FIG. 14 illustrates a first result of a direct path image at the fourth level of processing (see FIG. 10) in the multi-delay-resolution IRAMS approach applied in accordance with an embodiment of the present invention;
FIG. 15 illustrates a second result of a direct path image at the fourth level of processing (see FIG. 10) in the multi-delay-resolution IRAMS approach applied in accordance with an embodiment of the present invention;
FIG. 16 illustrates a final direct path image at the fifth level (see FIG. 10) containing primarily the effects of the true scattering centers while suppressing the multipath ghost effects (see FIG. 11);
FIG. 17 illustrates a final multipath image at the fifth level (see FIG. 10) containing primarily the multipath ghosts while suppressing the effects of the direct scattering centers (see FIG. 11);
FIG. 18 maps images corresponding to the last three levels of the multi-delay-resolution IRAMS tree of FIG. 3, including the individual images from FIGS. 14 through 17 corresponding to the example using idealized point scattering data presented in FIG. 11;
FIG. 19 illustrates the difference between the final direct-path image (see FIGS. 16 and 17) at the highest level and the integrated sum of all of the multipath images in the multi-resolution tree (see FIGS. 10 and 19);
FIG. 20a illustrates a truth plot containing five actual scattering centers and corresponding delayed material responses according to an embodiment of the present invention and 20b illustrates a SAR image corresponding to the truth plot of FIG. 20a, showing the response of both the immediate and delayed responses; and
FIG. 21 illustrates a processing diagram for a multi-delay resolution IRAMS tree according to a second embodiment of the present invention.



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS OF THE PRESENT INVENTION
In a preferred embodiment of the present invention, the imaging process assumes the condition that a scattered waveform has the same characteristics as the incident waveform. In the context of SAR imaging, the measured range to target (i.e., assumed object-of-interest) of a given multiple bounce (MB) scattering event at some aperture angle is identical to that of an equivalent single bounce (SB) scattering event. FIG. 1a presents an example of such equivalent scattering events. In this example, the transmitter is co-located with the receiver (both shown as 10) at a large distance from the region-of-interest, so that the effects of a curved wave front can be neglected. One skilled in the art recognizes that the process described herein may be adjusted for non co-located transmitter and receiver configurations, as well as for distances from the region of interest which result in curved wavefronts. Additionally, multiple receivers may be used in order to introduce additional degrees of freedom in the process of data collection and analysis. FIG. 1a shows that waveform traverses a total distance of 2L in propagating from the transmitter at 10 to scattering point (SP) 1 and back to the co-located receiver at 10. This distance represents the measured range-to-target from the co-located transmitter-receiver to SP 1 based upon waveform transmission and the corresponding reception of the its echo.
The specifics of the transmitter and receiver configurations usable with the embodiments described herein are known to those skilled in the art. Further, one skilled in the art recognizes that the mode of collection of the SAR data is not limited to any single mode, but may include, among others, spotlight mode, strip map mode, range Doppler profiler (RDP) mode, air-to-air mode, and the like.
Still referring to FIG. 1a, the multiple scattering event due to SPs 2 and 3 are considered. Here, the transmitted waveform reflects off of SP 2, which then reflects off of SP 3 prior to reception. If the sum of the line segments A+B+C is equal to 2L in FIG. 1a, then the measured range-to-target of the single-scattering event at SP 1 at the shown measurement angle θ is identical to, and therefore indistinguishable from, the double-scattering event due to the cascaded reflection from SP 2 to SP 3. In addition, multiple-scattering events involving three or more scattering points can have this same range-to-target, as shown in FIG. 1b. 
Referring to FIG. 1b, the sum of the two-point scattering event, A+B+C, cascaded from SP 2 and SP 3 and measured at the receiver 10, is equivalent to the sum of the three-point scattering event, D+E+F+G, cascaded from SP 4, SP 5, and SP 6 measured at the receiver 10, at the shown measurement angle θ.
As described in the Summary of the Problem, conventional SAR theory assumes that all scattering events which affect the measured phase history data (i.e., range-to-target projections) are due to SB effects, as occurs with SP 1 in FIG. 1a. This is not always the case. When MB events, such as SPs 2 and 3 in FIG. 1a, also significantly impact the measured phase history data, the application of the conventional SAR image formation techniques yield imperfect reconstructions of the region-of-interest.
According to an embodiment of the present invention, the best image of the true scattering centers is obtained by taking into account the MBG data retrieved via imaging through application of an image reconstruction algorithm for multi-bounce scattering (IRAMS). This process for applying the IRAMS is developed through examination of the properties of the MB scattering event shown in, for example, FIG. 1a. The data, retrieved via the transmitter and receiver configurations 10, is processed according to the IRAMS via a processor (not shown). Processors are well known in the art and will not be discussed further herein. Let (xi, yi) denote the two-dimensional (2-D) Cartesian coordinates of the spatial position of SP 2 with which the incident waveform initially interacts. Define (xf, yf) to be the spatial position of SP 3 with which the waveform that was initially reflected off of SP 2 interacts before being reflected into the receiver. Let the variable {tilde over (γ)} denotes the effective path length that the waveform traverses between the initial (xi, yi) and final (xf, yf) scattering centers. This path length {tilde over (γ)} is not required to be equal to the distance between the initial and final scattering centers, i.e., √{square root over ((xf−xi)2+(yf−yi)2)}{square root over ((xf−xi)2+(yf−yi)2)}, as occurs for scattering events involving only two scattering centers. For example, triple-point scattering events are possible, as shown in FIG. 1b. FIG. 1b shows a waveform scattering off of the initial point (xi, yi) of SP 4 to an intermediate location (x5, y5) at SP 5, before reflecting off of the final scattering center (xf, yf) at SP 6 and subsequently being collected into the receiver.
Under the definitions above, all single-reflection scattering events are characterized by {tilde over (γ)}=0. Double-reflection scattering events have {tilde over (γ)}=√{square root over ((xf−xi)2+(yf−yi)2)}{square root over ((xf−xi)2+(yf−yi)2)}, whereas N-point scattering events with N>2 are characterized by {tilde over (γ)}=√{square root over ((xf−xi)2+(yf−yi)2)}{square root over ((xf−xi)2+(yf−yi)2)}. Again, the goal of this analysis is to remove the effects of all MB scattering events, which are described by {tilde over (γ)}>0, in order to form an improved spotlight SAR image.
FIGS. 1a and 1b reveal that the measured range-to-target of a scattering event relative to that of the coordinate origin (x=0, y=0) depends upon the observation angle θ via

s(θ)=[{xi+xf} cos(θ)+{yi+yf} sin(θ)+{tilde over (γ)}]/2.  (1)

The division by 2 arises since the range-to-target is equal to one-half of the round-trip distance that the waveform traverses from the transmitter to the object and back into the co-located receiver. The range-to-target spatial coordinate s is related to the waveform time-of-flight t via





  
    
      
        
          t
          =
          
            
              2
              ⁢
              s
            
            c
          
        
        ,
      
    
    
      
        (
        2
        )
      
    
  




with c equal to the waveform speed. Given the MB range-to-target measurements s(θ) collected over some extent of angles θ for an isolated scattering event, it is clear from Eq. (1) that it is possible to determine only the mean position of the initial and final scattering centers, i.e., {(xi+xf)/2, (yi+yf)/2}, and the total path length {tilde over (γ)} between these two points. No other information can be unambiguously gleaned.

Since only the mean position of the initial and final scattering centers can be determined uniquely, the measured range-to-target for a N-point scattering event with respect to the selected coordinate origin can be expressed in the form

s(θ)=x cos(θ)+y sin(θ)+γ,  (3)

in terms of the parameters {x≡(xi+xf)/2, y≡{yi+yf}/2}, γ≡{tilde over (γ)}/2} of Eq. (1). Every N-point scattering event is characterized by the parameters (x,y,γ) and traces a sinusoid in range-angle (s, θ) that is shifted in the s-direction by a distance of γ. In contrast, each SB scattering event is characterized by the parameters (x,y; γ=0) and traces an unshifted sinusoid in range-angle space.

Equation (3) shows that for any generic N-point scattering event with N>1, range-to-target measurements can yield only the values (x,y,γ). This result applies even if there are no other scattering events present. In fact, many individual scattering events having one, two, or more bounces can have identical values of (x,y,γ). Specifically, all scattering events having the same average position for the initial and final scattering centers, i.e., x≡(xi+xf)/2 and y≡{yi+yf}/2}, and having the same total path length {tilde over (γ)}≡2γ between the initial and final scattering centers contribute to the same (x,y,γ)-point in the MB reflectivity function (MBRF) density ƒ(x,y,γ). That is, the density ƒ(x,y,γ) at a given point in (x,y,γ)-space is formed by the integral of all N-point scattering events characterized by the same values of x, y, and γ. In general, the MBRF ƒ(x,y,γ) is permitted to be a complex function, as in the modeling of the single-bounce reflectivity function ƒSB(x,y).
Consider the two different scattering events presented in FIG. 1b. the range-to-target of the transmitted waveform is considered with respect to the parallel wave front denoted by the thin dashed lines in this figure. The line segments have been drawn so that the following relations hold:

A+B+C=D+E+F+G,  (4)

x2+x3=x4+x6  (5)

y2+y3=y4+y6  (6)

Thus, the 2-point scattering event of SPs 2 and 3, and the 3-point scattering event of SPs 4, 5, and 6 have the same values of x, y, and γ within the MBRF ƒ(x,y,γ). The MBRF is the most basic representation of the region-of-interest which can be reconstructed based upon phase history measurements that include contributions due to MB scattering events.

The measured MB projection function g(s, θ) is determined by the MBRF ƒ(x,y,γ) via




  
    
      
        
          
            g
            ⁡
            
              (
              
                s
                ,
                θ
              
              )
            
          
          ≡
          
            
              ∫
              
                -
                ∞
              
              ∞
            
            ⁢
            
              
            
            ⁢
            
              
                d
                x
              
              ⁢
              
                
                  ∫
                  
                    -
                    ∞
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    d
                    y
                  
                  ⁢
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        d
                        γ
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        f
                        ⁡
                        
                          (
                          
                            x
                            ,
                            y
                            ,
                            γ
                          
                          )
                        
                      
                      ⁢
                      
                        δ
                        ⁡
                        
                          (
                          
                            
                              x
                              ⁢
                              
                                
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  θ
                                  )
                                
                              
                            
                            +
                            
                              y
                              ⁢
                              
                                
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  θ
                                  )
                                
                              
                            
                            +
                            γ
                            -
                            s
                          
                          )
                        
                      
                    
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        7
        )
      
    
  




with δ( . . . ) the standard Delta function defined by A(s)≡∫ds′ A(s′)δ(s′−s). That is, Eq. (7) is merely the integral of all range-to-target projections of MB scattering events, as determined by Eq. (3). The integral over dγ in Eq. (7) applies from −∞ to ∞, provided that the MBRF ƒ(x,y,γ) is constrained to be equal to zero for γ<0. If the MBRF density contains only single-bounce effects, i.e., ƒSB(x,y)≡ƒ(x,y; γ=0), then the MB transformation equation (7) reduces to the conventional Radon transform





  
    
      
        
          
            g
            SB
          
          ⁡
          
            (
            
              s
              ,
              θ
            
            )
          
        
        ≡
        
          
            ∫
            
              -
              ∞
            
            ∞
          
          ⁢
          
            
          
          ⁢
          
            
              d
              x
            
            ⁢
            
              
                ∫
                
                  -
                  ∞
                
                ∞
              
              ⁢
              
                
              
              ⁢
              
                
                  d
                  y
                
                ⁢
                
                  
                
                ⁢
                
                  
                    f
                    SB
                  
                  ⁡
                  
                    (
                    
                      x
                      ,
                      y
                    
                    )
                  
                
                ⁢
                
                  δ
                  ⁡
                  
                    (
                    
                      
                        x
                        ⁢
                        
                          
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            θ
                            )
                          
                        
                      
                      +
                      
                        y
                        ⁢
                        
                          
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            θ
                            )
                          
                        
                      
                      -
                      s
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




Define the 1-D Fourier transform of the function g(s, θ) with respect to s, applied independently for each observation angle θ, i.e.,




  
    
      
        
          G
          ⁡
          
            (
            
              ξ
              ,
              θ
            
            )
          
        
        ≡
        
          
            ∫
            
              -
              ∞
            
            ∞
          
          ⁢
          
            
          
          ⁢
          
            
              d
              s
            
            ⁢
            
              
            
            ⁢
            
              exp
              ⁡
              
                (
                
                  
                    -
                    j2π
                  
                  ⁢
                  
                    
                  
                  ⁢
                  s
                  ⁢
                  
                    
                  
                  ⁢
                  ξ
                
                )
              
            
            ⁢
            
              
                g
                ⁡
                
                  (
                  
                    s
                    ,
                    θ
                  
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




The notation g(s, θ)G(ξ, θ) expresses this relationship. The measurements for G(ξ, θ) in Eq. (9) are collected over some finite interval with respect to the spatial frequency variable, i.e., ξmin≦ξ≦ξmax, and some finite interval in the observation angle, i.e., θmin≦θ≦θmax. The spatial frequency coordinate ξ is related to the waveform temporal frequency coordinate τ via





  
    
      
        
          τ
          =
          
            
              c
              2
            
            ⁢
            ξ
          
        
        ,
      
    
    
      
        (
        10
        )
      
    
  




so that the waveform temporal bandwidth is





  
    
      
        
          Δ
          ⁢
          
            
          
          ⁢
          τ
        
        =
        
          
            c
            2
          
          ⁢
          
            
              {
              
                
                  ξ
                  max
                
                -
                
                  ξ
                  min
                
              
              }
            
            .
          
        
      
    
    
      
        (
        11
        )
      
    
  




Inserting Eq. (7) into Eq. (9) and interchanging the order of integration gives





  
    
      
        
          G
          ⁢
          
            (
            
              ξ
              ,
              θ
            
            )
          
        
        ≡
        
           
          
            
              ∫
              
                -
                ∞
              
              ∞
            
            ⁢
            
              
            
            ⁢
            
              
                d
                x
              
              ⁢
              
                
                  ∫
                  
                    -
                    ∞
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    d
                    y
                  
                  ⁢
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        d
                        γ
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        f
                        ⁡
                        
                          (
                          
                            x
                            ,
                            
                              
                            
                            ⁢
                            y
                            ,
                            
                              
                            
                            ⁢
                            γ
                          
                          )
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          ∫
                          
                            -
                            ∞
                          
                          ∞
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            d
                            s
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            exp
                            ⁡
                            
                              (
                              
                                
                                  -
                                  j
                                
                                ⁢
                                
                                  
                                
                                ⁢
                                2
                                ⁢
                                π
                                ⁢
                                
                                  
                                
                                ⁢
                                s
                                ⁢
                                
                                  
                                
                                ⁢
                                ξ
                              
                              )
                            
                          
                          ⁢
                          
                            
                              δ
                              ⁡
                              
                                (
                                
                                  
                                    x
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      cos
                                      ⁡
                                      
                                        (
                                        θ
                                        )
                                      
                                    
                                  
                                  +
                                  
                                    y
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      sin
                                      ⁡
                                      
                                        (
                                        θ
                                        )
                                      
                                    
                                  
                                  +
                                  y
                                  -
                                  s
                                
                                )
                              
                            
                            .
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        12
        )
      
    
  




Evaluation of the delta function yields the forward MB transformation between the MBRF ƒ(x,y,γ) and the phase history measurements G(ξ, θ):




  
    
      
        
          
            
              G
              ⁢
              
                (
                
                  ξ
                  ,
                  θ
                
                )
              
            
            ≡
          
           
        
        ⁢
        
           
          
            
          
          ⁢
          
            

          
          ⁢
          
            
              ∫
              
                -
                ∞
              
              ∞
            
            ⁢
            
              
            
            ⁢
            
              
                d
                x
              
              ⁢
              
                
              
              ⁢
              
                
                  ∫
                  
                    -
                    ∞
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    d
                    y
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        d
                        γ
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        f
                        ⁡
                        
                          (
                          
                            x
                            ,
                            y
                            ,
                            γ
                          
                          )
                        
                      
                      ⁢
                      
                        exp
                        (
                        
                          
                            -
                            j
                          
                          ⁢
                          
                            
                          
                          ⁢
                          2
                          ⁢
                          
                            
                              π
                              [
                              
                                
                                  x
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      θ
                                      )
                                    
                                  
                                
                                +
                                
                                  y
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    sin
                                    ⁡
                                    
                                      (
                                      θ
                                      )
                                    
                                  
                                
                                +
                                
                                  

                                
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  
                                     
                                    γ
                                    ]
                                  
                                  ⁢
                                  ξ
                                
                              
                              )
                            
                            .
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        13
        )
      
    
  




This equation implicitly assumes that the MBRF ƒ(x,y,γ) is invariant over the extent of measurement angles θ and spatial frequencies ξ under consideration. In a similar fashion, the omission of MB effects yields the corresponding SB result:





  
    
      
        
          
            
              G
              SB
            
            ⁡
            
              (
              
                ξ
                ,
                θ
              
              )
            
          
          ≡
          
            
              ∫
              
                -
                ∞
              
              ∞
            
            ⁢
            
              
            
            ⁢
            
              
                d
                x
              
              ⁢
              
                
                  ∫
                  
                    -
                    ∞
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    d
                    γ
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      f
                      SB
                    
                    ⁡
                    
                      (
                      
                        x
                        ,
                        y
                      
                      )
                    
                  
                  ⁢
                  
                    exp
                    ⁡
                    
                      (
                      
                        
                          -
                          j
                        
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          π
                          ⁡
                          
                            [
                            
                              
                                x
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    θ
                                    )
                                  
                                
                              
                              +
                              
                                y
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    θ
                                    )
                                  
                                
                              
                            
                            ]
                          
                        
                        ⁢
                        ξ
                      
                      )
                    
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        14
        )
      
    
  




with GSB(ξ, θ)gSB(s, θ).

There are two different classes of techniques for inverting Eq. (14) in order to obtain the SB reflectivity function density ƒSB(x,y) given the measured projections GSB(ξ, θ). The first class of methods invokes the projection-slice theorem. However, it is not clear that these projection-slice techniques can be utilized to invert the particular MB transformation equation (13) of the present embodiment. In particular, the measurements G(ξ, θ) are collected by varying the values of only two variables, i.e., ξ and θ. In contrast, the three-dimensional (3-D) Fourier transform of the desired MBRF density f(x,y,γ) requires an appropriate set of projection measurements over three dimensions, whereas there exist only two in the current example. Therefore, the application of projection-slice methods is not applied to invert the MB transformation equation (13) in the present embodiment. One skilled in the art recognizes that the application of the projection-slice method may be used as an alternative to inversion in an alternative embodiment, wherein the data consists of three-dimensional projection measurements.
The second class of inversion methods used to reconstruct the SB reflectivity function ƒSB(x,y) utilizes back-projection functions. Back projection functions reconstruct ƒSB(x,y) based upon the measured function-domain projection function gSB(s, θ). Alternatively, a frequency-domain back-projection technique may be used to reconstruct ƒSB(x,y). The present embodiment applies back-projection techniques in order to reconstruct the MBRF ƒ(x,y,γ) based upon the phase history measurements G(ξ, θ).
Alternatively, for a continuous-domain solution, the inversion of the MB projection equation begins with the following definition of the back-projection operation applied to the MB measured transform-domain projections G(ξ, θ):




  
    
      
        
          b
          ⁡
          
            (
            
              x
              ,
              y
              ,
              γ
            
            )
          
        
        ≡
        
          
            ∫
            
              θ
              ⁢
              
                
              
              ⁢
              min
            
            
              θ
              ⁢
              
                
              
              ⁢
              mzx
            
          
          ⁢
          
            
          
          ⁢
          
            
              d
              θ
            
            ⁢
            
              
                ∫
                
                  ξ
                  ⁢
                  
                    
                  
                  ⁢
                  min
                
                
                  ξ
                  ⁢
                  
                    
                  
                  ⁢
                  max
                
              
              ⁢
              
                
              
              ⁢
              
                
                  d
                  
                    ξG
                    ⁡
                    
                      (
                      
                        ξ
                        ,
                        θ
                      
                      )
                    
                  
                
                ⁢
                
                  
                    exp
                    ⁡
                    
                      (
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          π
                          ⁡
                          
                            [
                            
                              
                                x
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    θ
                                    )
                                  
                                
                              
                              +
                              
                                y
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    θ
                                    )
                                  
                                
                              
                              +
                              γ
                            
                            ]
                          
                        
                        ⁢
                        ξ
                      
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        15
        )
      
    
  




The limits of integration in Eq. (15) is chosen to be consistent with the collected transform-domain measurements, i.e., ξmin≦ξ≦ξmax and θmin≦θ≦θmax, as described above. Substituting Eq. (13) for G(ξ, θ) into the back projection function b(x,y,γ) of Eq. (15) and interchanging the order of integration gives





  
    
      
        
          b
          ⁢
          
            (
            
              x
              ,
              y
              ,
              γ
            
            )
          
        
        ≡
        
          

        
        ⁢
        
           
          
            
          
          ⁢
          
            
              ∫
              
                -
                ∞
              
              ∞
            
            ⁢
            
              
            
            ⁢
            
              
                d
                
                  x
                  ′
                
              
              ⁢
              
                
                  ∫
                  
                    -
                    ∞
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  
                    d
                    
                      y
                      ′
                    
                  
                  ⁢
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      ∞
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        d
                        
                          γ
                          ′
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        f
                        ⁡
                        
                          (
                          
                            
                              x
                              ′
                            
                            ,
                            
                              
                            
                            ⁢
                            
                              y
                              ′
                            
                            ,
                            
                              
                            
                            ⁢
                            
                              γ
                              ′
                            
                          
                          )
                        
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          ∫
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            min
                          
                          
                            θ
                            ⁢
                            
                              
                            
                            ⁢
                            mzx
                          
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            d
                            θ
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            
                              ∫
                              
                                ξ
                                ⁢
                                
                                  
                                
                                ⁢
                                min
                              
                              
                                ξ
                                ⁢
                                
                                  
                                
                                ⁢
                                max
                              
                            
                            ⁢
                            
                              d
                              
                                ξλ
                                (
                                
                                  
                                
                                ⁢
                                
                                  x
                                  -
                                  
                                    
                                       
                                      
                                        

                                      
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      
                                        
                                          x
                                          ′
                                        
                                        ,
                                        
                                          y
                                          -
                                          
                                            y
                                            ′
                                          
                                        
                                        ,
                                        
                                          γ
                                          -
                                          
                                            γ
                                            ′
                                          
                                        
                                      
                                      )
                                    
                                    .
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        16
        )
      
    
  




in terms of the kernel function λ(x,y,γ) is defined by





  
    
      
        
          λ
          ⁡
          
            (
            
              x
              ,
              y
              ,
              γ
            
            )
          
        
        ≡
        
          
            ∫
            
              θ
              ⁢
              
                
              
              ⁢
              min
            
            
              θ
              ⁢
              
                
              
              ⁢
              mzx
            
          
          ⁢
          
            
          
          ⁢
          
            
              d
              θ
            
            ⁢
            
              
                ∫
                
                  ξ
                  ⁢
                  
                    
                  
                  ⁢
                  min
                
                
                  ξ
                  ⁢
                  
                    
                  
                  ⁢
                  max
                
              
              ⁢
              
                
                  d
                  
                    ξexp
                    ⁡
                    
                      (
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        
                          π
                          ⁡
                          
                            [
                            
                              
                                x
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    θ
                                    )
                                  
                                
                              
                              +
                              
                                y
                                ⁡
                                
                                  (
                                  θ
                                  )
                                
                              
                              +
                              γ
                            
                            ]
                          
                        
                        ⁢
                        ξ
                      
                      )
                    
                  
                
                .
              
            
          
        
      
    
    
      
        (
        17
        )
      
    
  




Thus, the MB back-projection function b(x,y,γ) can be expressed in terms of the MBRF ƒ(x,y,γ) via

b(x, y, γ)=λ(x, y, γ)*ƒ(x, y, γ).  (18)

In this equation, the symbol * denotes the 3-D convolution operation. The 3-D Fourier transform operator F3 applied to both sides of Eq. (18) gives the following inversion technique for obtaining the MBRF ƒ(x, y, γ), given the transform-domain measurements G(ξ, θ):





  
    
      
        
          f
          ⁡
          
            (
            
              x
              ,
              y
              ,
              γ
            
            )
          
        
        =
        
          
            F
            3
            
              -
              1
            
          
          ⁢
          
            
              {
              
                
                  
                    F
                    3
                  
                  ⁢
                  
                    {
                    
                      b
                      ⁡
                      
                        (
                        
                          x
                          ,
                          y
                          ,
                          γ
                        
                        )
                      
                    
                    }
                  
                
                
                  
                    F
                    3
                  
                  ⁢
                  
                    {
                    
                      λ
                      ⁡
                      
                        (
                        
                          x
                          ,
                          y
                          ,
                          γ
                        
                        )
                      
                    
                    }
                  
                
              
              }
            
            .
          
        
      
    
    
      
        (
        19
        )
      
    
  




Here, the operator F3−1 denotes the inverse Fourier transform in three dimensions. In Eq. (19), the back-projection function b(x,y,γ) is computed by inserting the transform-domain measurement function G(ξ, θ) into Eq. (15). A solution for ƒ(x,y,γ) exists provided that the 3-D Fourier transform of the kernel λ(x,y,γ) has no zeros in the region-of-interest. This constraint involved in obtaining the MBRF solution is more transparent in the discrete-domain, as shown in the following subsection.

SAR image reconstruction requires that the MBRF ƒ(x,y,γ) be determined on a discrete grid in (x,y,γ) space, i.e., {xl}l=1, . . . , L, {ym}m=1, . . . , M, and {γn}n=1, . . . , N. This grid is not required to have equal intervals between grid points in each of the three dimensions of (x,y,γ)-space. However, the assumption of a uniform grid will be shown to reduce the required computation time, even if each of the three dimensions has a different resolution. This analysis generates an estimate of the MBRF matrix f(l,m,n), which is the discrete approximation of the MBRF density ƒ(x,y,γ).
The measurements arising from the coherent processing of the received radar waveform are transduced at a finite set of discrete spatial frequencies {ξp}p=1, . . . , P and a finite set of discrete observation angles {θq}q=1, . . . , Q relative to the defined coordinate system. This grid in (ξp, θq)-space is not required to be uniform, just as the (xl,ym,γn)-space of the MBRF matrix f(l,m,n) can be non-uniform. Thus, the measured phase history data are determined by the discrete domain measurement vector G(p,q) as an approximation of the density G(ξ, θ). Given the definitions for the vectors f(l,m,n) and G(p,q), the MB transformation equation (13) has the discrete form




  
    
      
        
          
            G
            ⁡
            
              (
              
                p
                ,
                q
              
              )
            
          
          =
          
            
              ∑
              
                l
                =
                1
              
              L
            
            ⁢
            
              
                ∑
                
                  m
                  =
                  1
                
                M
              
              ⁢
              
                
                  ∑
                  
                    n
                    =
                    1
                  
                  N
                
                ⁢
                
                  
                    H
                    ⁡
                    
                      (
                      
                        l
                        ,
                        m
                        ,
                        n
                        ,
                        p
                        ,
                        q
                      
                      )
                    
                  
                  ⁢
                  
                    f
                    ⁡
                    
                      (
                      
                        l
                        ,
                        m
                        ,
                        n
                      
                      )
                    
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        20
        )
      
    
  




in terms of the matrix of phase functions

H(l,m,n,p,q)=exp(−j2π[xlcos(θq)+ymsin(θq)+γn]ξp).  (21)

Equation (20) can be expressed in the basic form of the algebraic reconstruction problem

G=HF,  (22)

where F is the vector of unknowns of length U≡L×M×N, the variable G is a vector of measurements of length V≡P×Q and H is the U×V matrix of the vectors giving the projection of an (l,m,n) object space point onto (p,q) measurement space. The requirement that V≧U must be satisfied in order to obtain a solution for F; otherwise, Eq. (22) is undetermined.

A method for obtaining a solution of Eq. (22) is via the pseudo-inverse

F={HHH}−1HHG,  (23)

with the superscript H denoting the Hermitian transpose. The pseudo-inverse gives the unique least-squares solution provided that the square matrix HHH is non-singular. For cases in which the matrix HHH is ill-conditioned, either diagonal loading or singular-value decomposition can be applied. In addition, related techniques, known to those skilled in the art, can be applied for cases of additive noise.

The matrix HHH is of size U×U, so that the computation of the matrix inverse in Eq. (23) is of order O(U3), which is the rate-determining step. However, techniques of order O(U×V) can be applied of the desired MBRF vector f(l,m,n) is computed on a uniform grid in (xl,ym,γn)-space. To develop these techniques, the back-projection vector is defined via




  
    
      
        
          b
          ⁡
          
            (
            
              l
              ,
              m
              ,
              n
            
            )
          
        
        ≡
        
          
            ∑
            
              p
              =
              1
            
            P
          
          ⁢
          
            
              ∑
              
                q
                =
                1
              
              Q
            
            ⁢
            
              
                
                  H
                  H
                
                ⁡
                
                  (
                  
                    l
                    ,
                    m
                    ,
                    n
                    ,
                    p
                    ,
                    q
                  
                  )
                
              
              ⁢
              
                
                  G
                  ⁡
                  
                    (
                    
                      p
                      ,
                      q
                    
                    )
                  
                
                .
              
            
          
        
      
    
    
      
        (
        24
        )
      
    
  




Inserting Eq. (20) for G(p,q) into Eq. (24) and interchanging the order of the summation gives





  
    
      
        
          
            b
            ⁡
            
              (
              
                l
                ,
                m
                ,
                n
              
              )
            
          
          =
          
            
              ∑
              
                l
                =
                1
              
              L
            
            ⁢
            
              
                ∑
                
                  m
                  =
                  1
                
                M
              
              ⁢
              
                
                  ∑
                  
                    n
                    =
                    1
                  
                  N
                
                ⁢
                
                  
                    
                      Λ
                      _
                    
                    ⁡
                    
                      (
                      
                        l
                        ,
                        
                          l
                          ′
                        
                        ,
                        m
                        ,
                        
                          m
                          ′
                        
                        ,
                        n
                        ,
                        
                          n
                          ′
                        
                      
                      )
                    
                  
                  ⁢
                  
                    f
                    ⁡
                    
                      (
                      
                        
                          l
                          ′
                        
                        ,
                        
                          m
                          ′
                        
                        ,
                        
                          n
                          ′
                        
                      
                      )
                    
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        25
        )
      
    
  




in terms of the matrix kernel





  
    
      
        
          
            
              
                Λ
                _
              
              ⁡
              
                (
                
                  l
                  ,
                  
                    l
                    ′
                  
                  ,
                  m
                  ,
                  
                    m
                    ′
                  
                  ,
                  n
                  ,
                  
                    n
                    ′
                  
                
                )
              
            
            ≡
            
              
                ∑
                
                  p
                  =
                  1
                
                P
              
              ⁢
              
                
                  ∑
                  
                    q
                    =
                    1
                  
                  Q
                
                ⁢
                
                  
                    
                      H
                      H
                    
                    ⁡
                    
                      (
                      
                        l
                        ,
                        m
                        ,
                        n
                        ,
                        p
                        ,
                        q
                      
                      )
                    
                  
                  ⁢
                  
                    H
                    ⁡
                    
                      (
                      
                        l
                        ,
                        m
                        ,
                        n
                        ,
                        p
                        ,
                        q
                      
                      )
                    
                  
                
              
            
          
          =
          
            
              ∑
              
                p
                =
                1
              
              P
            
            ⁢
            
              
                ∑
                
                  q
                  =
                  1
                
                Q
              
              ⁢
              
                exp
                ⁡
                
                  (
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    2
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        Ω
                        _
                      
                      ⁡
                      
                        (
                        
                          l
                          ,
                          
                            l
                            
                              ′
                              ⁢
                              
                                
                              
                            
                          
                          ,
                          m
                          ,
                          
                            m
                            ′
                          
                          ,
                          n
                          ,
                          
                            n
                            ′
                          
                        
                        )
                      
                    
                  
                  )
                
              
            
          
        
        ,
      
    
    
      
        (
        26
        )
      
    
  




with

 Ω(l,l′,m,m′,n,n′)≡[{xl−xl′}cos(θq)+{ym−ym′}sin(θq)+{γn−γn′}]ξp,  (27)

For the case of a uniform grid in (xl, ym, γn)-space, i.e., {xl=lδx}l=1, . . . , L, {ym=mδy}m=1, . . . , M, and {γn=nδγ}n=1, . . . , N, with δx, δy, δγ, all constants, Eq. (27) reduces to

 Ω(l,l′,m,m′,n,n′)=[{l−l′}δxcos(θq)+{m−m′}δysin(θq)+{n−n′}δγ]ξp,  (28)

Thus, Eq. (25) has the form

b(l,m,n)=Λ(l,m,n)*f(l,m,n),  (29)

with





  
    
      
        
          
            Λ
            ⁡
            
              (
              
                l
                ,
                m
                ,
                n
              
              )
            
          
          ≡
          
            
              ∑
              
                p
                =
                1
              
              P
            
            ⁢
            
              
                ∑
                
                  q
                  =
                  1
                
                Q
              
              ⁢
              
                exp
                ⁡
                
                  (
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    2
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Ω
                      ⁡
                      
                        (
                        
                          l
                          ,
                          m
                          ,
                          n
                          ,
                          p
                          ,
                          q
                        
                        )
                      
                    
                  
                  )
                
              
            
          
        
        ,
      
    
    
      
        (
        30
        )
      
    
  




and the phase matrix

Ω(l, m, n, p, q)≡[lδxcos(θq)+mδysin(θq)+nδγ]ξp.  (31)

Here, the symbol “*” denotes the 3-D discrete convolution. The convolutional structure of Eq. (29) permits the inversion of f(l,m,n) to be computed in FFT-time of order O(U log2(U)). Thus the rate-determining step is the computation of the back-projection vector b(l,m,n) given the measurement phase history vector G(p,q) via Eq. (24), which is of order O(U×V). Alternatively, function-domain back-projection operations can be performed in order O(U log2(U)), such that in an alternative embodiment of the present invention, the transform-domain back-projections applied to MB image reconstruction described herein, can also can be implemented in order O(U log2(U)).

The SB analog of Eq. (29) has the form

bSB(l,m)=ΛSB(l,m)*fSB(l,m),  (32)

with





  
    
      
        
          
            Λ
            SB
          
          ⁡
          
            (
            
              l
              ,
              m
            
            )
          
        
        ≡
        
          
            ∑
            
              p
              =
              1
            
            P
          
          ⁢
          
            
              ∑
              
                q
                =
                1
              
              Q
            
            ⁢
            
              
                exp
                ⁡
                
                  (
                  
                    j
                    ⁢
                    
                      
                    
                    ⁢
                    2
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        Ω
                        SB
                      
                      ⁡
                      
                        (
                        
                          l
                          ,
                          m
                          ,
                          p
                          ,
                          q
                        
                        )
                      
                    
                  
                  )
                
              
              .
            
          
        
      
    
    
      
        (
        33
        )
      
    
  




and

ΩSB(l, m, n, p, q)≡[lδxcos(θq)+mδysin(θq)]ξp.  (34)

Here, the symbol “*” denotes two-dimensional (2-D) discrete convolution. In Eq. (32), bSB(l,m) is determined by





  
    
      
        
          
            
              b
              SB
            
            ⁡
            
              (
              
                l
                ,
                m
              
              )
            
          
          =
          
            
              ∑
              
                p
                =
                1
              
              P
            
            ⁢
            
              
                ∑
                
                  q
                  =
                  1
                
                Q
              
              ⁢
              
                
                  
                    H
                    SB
                    H
                  
                  ⁡
                  
                    (
                    
                      l
                      ,
                      m
                      ,
                      n
                      ,
                      p
                      ,
                      q
                    
                    )
                  
                
                ⁢
                
                  G
                  ⁡
                  
                    (
                    
                      p
                      ,
                      q
                    
                    )
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        35
        )
      
    
  




in terms of the matrix of SB phase functions

HSB(l,m,p,q)=exp(−j2πΩSB(l, m, p, q)).  (36)

Since the measurements G(p,q) are determined by the MB transformation Eq. (20), the SB reconstruction technique of computing f(l, m) based upon Eqs. (32)-(36) will contain undesirable artifacts unless contributions due to MB scattering events are absent in the measurement phase history data G(p,q).

The previous paragraphs have developed a technique for MB image reconstruction by deriving the 3-D convolution equation (29), which is equivalent to the matrix form of Eq. (25). The existence of a well-behaved solution for the MBRF f(l, m, n) requires that the 3-D Hermitian-Toeplitx matrix Λ(l,l′,m,m′,n,n′)=Λ(l−l′, m−m′, n−n′) be non-singular. The constraint is satisfied if the vectors hl,m,n(p,q)≡H(l,m,n,p,q) are all linearly independent functions of the phase history measurement indices p and q. This constraint is satisfied if one or more of the elements of the normalized phase vectors al,m,n(p,q)≡Ω(l,m,n,p,q) of Eq. (31) change by unity or greater between adjacent cells in image-space (l,m,n). The normalized phase vectors al,m,n(p,q) satisfy this requirement, and thus yield a well-behaved solution to Eq. (29) or (25), if the following conditions apply.
First, consider the fact that spotlight SAR phase history data is typically collected over a narrow extent of measurement angles Δθ≡θmax−θmin<<1, here considered about θ=0 without loss in generality. For a power-series expansion about θ=0 through only first order in θ, the factor cos(θq) in the normalized phase of Eq. (31) is approximately unity. Thus, the dependence of the along-range coordinate x in Eq. (31) is identical to that of the MB coordinate γ, so that the effects of x and γ cannot be differentiated through first order. Therefore, a well-behaved MB reconstruction requires that the extent of measurement angles Δθ be sufficiently large that second order effects become significant. The required second order expansion of Eq. (31) has the form:




  
    
      
        
          
            a
            
              l
              ,
              m
              ,
              n
            
          
          ⁡
          
            (
            
              p
              ,
              q
            
            )
          
        
        ≡
        
          Ω
          ⁡
          
            (
            
              l
              ,
              m
              ,
              n
              ,
              p
              ,
              q
            
            )
          
        
        ≅
        
          
            [
            
              
                
                  {
                  
                    
                      l
                      ⁢
                      
                        
                      
                      ⁢
                      
                        δ
                        x
                      
                    
                    +
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      
                        δ
                        y
                      
                    
                  
                  }
                
                ⁢
                m
                ⁢
                
                  
                
                ⁢
                
                  δ
                  y
                
                ⁢
                
                  θ
                  q
                
              
              -
              
                
                  1
                  2
                
                ⁢
                l
                ⁢
                
                  
                
                ⁢
                
                  δ
                  x
                
                ⁢
                
                  θ
                  q
                  2
                
              
            
            ]
          
          ⁢
          
            ξ
            p
          
        
      
    
    
      
        (
        37
        )
      
    
  




In order to give a desired image-space resolution (δx, δy, δγ), the extent of measurements in spatial frequency Δξ and in angle Δθ must be selected to be large enough so that the zeroth, first, and second order terms in Eq. (37) all change by unity or greater between adjacent image-space cells (l,m,n). The change in the zeroth order phase term {lδx+nδγ}ξp of Eq. (37) exceeds unity between adjacent cells in both the x and γ dimensions over the extent of spatial frequencies if

Δξ≳max {δx−1,δγ−1},  (38)

which reduces to the well-known SB relation Δξ ̃δx−1.

The change in the first order phase term mδyθqξp of Eq. (37) exceeds unity between adjacent cells in the y dimension over the extent of measurement angles if

Δθ≳{ξcfδy}−1,  (39)

with ξcf equal to the center spatial frequency. This relation is identical to that of the SB case. The change in the second order phase term−(1/2)lδxθq2ξp of Eq. (37) exceeds unity between adjacent cells in the x dimension over the extent of measurement angles if

Δθ≳√{square root over (2{ξcfδx}−1)},  (40)

Thus, the first and second order constraints of Eqs. (39) and (40) are combined to yield

Δθ≳max {{ξcfδy}−1,√{square root over (2{ξcfδx}−1)}}  (41)

For the case of identical image-space resolutions in each dimension, i.e., δ≡δx=δy=δγ, the required measurement extents reduce to

Δξ≳δ−1,  (42)

Δθ≳√{square root over (2{ξcfδ}−1)}.  (43)

Therefore, the extent of measurement frequencies required for the MB case is unmodified from that of the SB case. However, the extent of measurement angles is greater for the MB case than for the SB case.

In order to derive an expression for the measurement intervals δξ and δθ required for a desired image-space extent (Δx≡Lδx, Δy≡Mδy, Δγ≡Nδγ, consider the normalized phase of Eq. (37) with θq=qδθ and ξp=ξcf+pδξ, i.e.,




  
    
      
        
          Ω
          ⁡
          
            (
            
              l
              ,
              m
              ,
              n
              ,
              p
              ,
              q
            
            )
          
        
        ≅
        
          
            
              (
              
                
                  l
                  ⁢
                  
                    
                  
                  ⁢
                  
                    δ
                    x
                  
                
                +
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  
                    δ
                    y
                  
                
              
              }
            
            ⁢
            
              {
              
                
                  ξ
                  cf
                
                +
                
                  p
                  ⁢
                  
                    
                  
                  ⁢
                  
                    δ
                    ξ
                  
                
              
              }
            
          
          +
          
            m
            ⁢
            
              
            
            ⁢
            
              δ
              y
            
            ⁢
            q
            ⁢
            
              
            
            ⁢
            
              δ
              θ
            
            ⁢
            
              {
              
                
                  ξ
                  cf
                
                +
                
                  p
                  ⁢
                  
                    
                  
                  ⁢
                  
                    δ
                    ξ
                  
                
              
              }
            
          
          -
          
            
              1
              2
            
            ⁢
            l
            ⁢
            
              
            
            ⁢
            
              δ
              x
            
            ⁢
            
              q
              2
            
            ⁢
            
              δ
              θ
              2
            
            ⁢
            
              
                {
                
                  
                    ξ
                    cf
                  
                  +
                  
                    p
                    ⁢
                    
                      
                    
                    ⁢
                    
                      δ
                      ξ
                    
                  
                
                }
              
              .
            
          
        
      
    
    
      
        (
        44
        )
      
    
  




The change in the normalized phase of Eq. (44) must be unity or less between adjacent measurement samples δξ and δθ in order to obtain a well-behaved solution. The change in the zeroth order term of Eq. (44) is unity or smaller over the image-space extents in the x and γ dimensions if

δξ≲{Δx+Δy}−1.  (45)

The first order term of Eq. (44) is unity or smaller over the image-space extents in the y dimension if

δθ≲{ξcfΔy}−1,  (46)

whereas the second order term does so for

δθ≲√{square root over (2{ξcfΔx}−1)}.  (47)

The combination Eqs. (46) and (47) gives

δθ≲{{ξcfΔy}−1, {square root over (2{ξcfΔx}−1)},  (49)

For equal extents in the x and y spatial dimensions, i.e., Δ≡Δx=Δy, Eq. (48) reduces to

δθ≲{ξcfΔ}−1,  (49)

which is identical to that of the SB case.

The SAR image reconstruction processes developed above are applied to simulated MB phase history data with the following results. FIG. 2a presents an example containing a number of assumed scattering center reflectivities. The co-located transmitter and receiver (not shown) are located at the top of each of the figures, transmitting and receiving down and up the paper in a lengthwise direction. This plot corresponds to the γ=0 plane of the MBRF ƒ(x,y,γ). In addition, it is assumed that there exists a MB coupling within the four sets of closely-spaced scattering centers located in the four corners 12, 14, 16, and 18 of FIG. 2a. For example, there is a contribution to the MBRF ƒ(x,y,γ) corresponding to the two scattering centers at (x,y)=(12, 3) and (12, 5) located in corner 12 of the truth plot in FIG. 2a. Specifically, the γ=1 plane of the MBRF shown in FIG. 2b contains a contribution at the mean position (x,y)=(12, 4) between these two scattering centers, in accord with the distance 2γ=2 between these two scattering centers. Likewise, the truth scattering centers at (x,y)=(12, 13) and (14, 13) in the lower right corner 14 of FIG. 2a yield the γ=1 MB contribution at (13, 13) in FIG. 2b. Similarly, the truth scattering centers at (x,y)=(3, 5) and (5, 3) in the upper left corner 16 of FIG. 2a have a MB coupling length of 2γ=2√{square root over (2)}, thus giving linearly-interpolated contributions at (4, 4) in the γ=1 and γ=2 planes of FIGS. 2b and 2c. 
The three coupled scattering centers at (x,y)=(3, 12), (5, 12) and (5, 14) in the upper right corner 18 of FIG. 2a give three different double-bounce contributions corresponding to (x,y,γ)=(4, 12, 1), (5, 13, 1), and (4, 13, √{square root over (2)}), similar to the double-bounce couplings in the other three corners 12, 14, and 16 of FIG. 2a. In addition, the upper right corner 18 scattering centers yield various triple-bound events. For example, the scattering sequence (5, 14)→(5, 12)→(3, 12) gives the MB contribution at (x,y,γ)=(4, 13, 2) in FIG. 2c, since the total path length between the initial point (5, 14) and the final point (3, 12) is 2γ=4. Likewise, the scattering sequence (5, 14)→(3, 12)→(5, 12) gives a contribution for (x,y,γ)=(5, 13, √{square root over (2)}+1), and the sequence (3, 12)→(5, 14)→(5, 12) yields (x,y,γ)=(4, 12, √{square root over (2)}+1).
The MBRF ƒ(x,y,γ) for γ≠0 in FIGS. 2b-2d is based upon the true path lengths between the scattering centers of FIG. 2a. However, the coupling strengths between the scattering centers were selected in an ad hoc manner to be equal to that of the truth scattering centers themselves. Nevertheless, the MB techniques described herein apply generically for any specific quantitative rules governing the MB coupling strengths.
The true values of the MBRF ƒ(x,y,γ) shown in FIGS. 2a-2d are applied within Eq. (20) in order to simulate the measurement function G(ξ, θ) shown in FIGS. 3a and 3b. The measurement functions shown in FIGS. 3a and 3b are coherent measurements, resulting from a quadrature demodulation of the received waveform, i.e., the reflected waveform. The x-axis is the measure of angle in radians, while the y, axis is spatial frequency. The measurement functions of FIGS. 3a and 3b, when inverted using the single bounce theory, give the result shown in FIG. 4. These same measurement functions, when inverted using the process of the present invention, give the results discussed below with reference to FIGS. 5a-5d, creating not only an identical replica of the desired image but, also creating higher delay planes or multiple bounce image planes that contain the information rich echo portions of the reflected wave form. The signal-to-noise ratio in this embodiment is approximately 40 dB. As stated above, noise and interference play no greater a role in the IRAMS processing than in the single bounce processing. Consequently, one skilled in the art recognizes the standard signal-to-noise limitations that are applicable to IRAMS processing.
The results of Eq. (37)-(49) are used to determine the required sampling intervals and extents of both the spatial frequency ξ and the angular coordinate θ which are necessary to reconstruct the truth MBRF ƒ(x,y,γ) of FIGS. 2a-2d. The desired image-space resolution is δ≡δx=δy=δγ=1, so that Eq. (42) implies that a spatial frequency bandwidth of Δξ=1/δ is sufficient for a well-behaved MB reconstruction. SAR measurements are collected over a narrow extent of measurement angles, which is chosen to be Δθ≅0.2 rad≅11.5° in this particular example. Then the spatial center frequency is selected to be ξcf=2/Δ2/θ≅45, consistent with δ=1 and Δθ≅0.2 rad in Eq. (43). The desired reconstruction extents are given by Δx=Δy=16 and Δγ=4 for this example. Thus, Eq. (45) implies that the required sampling interval in spatial frequency is δξ=1/20, and Eq. (49) gives a required angular sampling interval of δθ≅1/720 rad≅0.08°.
FIGS. 5a-5d show the application of the MB inversion of Eqs. (24) and (29) in order to estimate the MBRF ƒ(x,y,γ) based upon the simulated phase history data G(ξ, θ) of FIGS. 3a and 3b. As shown, the γ=0, γ=1, γ=2, and γ=3 planes of FIGS. 5a-5d give an approximately identical reconstruction of the desired scattering center locations and strengths of FIG. 2a. Further, the extracted MB scattering events located within the γ≠0 planes can be used to augment the feature vectors applied to, for example, target recognition analysis. For the sake of comparison, the desirable MB image reconstruction of FIGS. 5a-5d is compared with that based upon the assumption of only SB scattering events. FIG. 4 presents the result of the SB inversion technique of Eqs. (32) and (35) applied to the simulated MB phase history data of FIGS. 3a and 3b. Clearly, the SB technique reveals a significant degradation in the estimation of the object scattering centers of FIG. 2a. Although the object locations and strengths of the various scattering centers of FIG. 2a are well-reproduced, FIG. 4 contains many additional spurious scattering centers that arise from the MB contributions to the simulated phase history data of FIGS. 3a and 3b. Whereas the MB inversion technique accurately reconstructs a given MB scattering event characterized by (x,y,γ), the SB inversion method incorrectly projects such a point to (x+γ, y) for phase history data measured over a narrow angular extent about θ=0. For example, the MB scattering event of (x,y,γ)=(12, 4, 1) of FIG. 2b is incorrectly projected to the point (x,y)=(13, 4) within the SB estimate of FIG. 4 and similar incorrect results apply to the other MB scattering events.
The ability to form a well-behaved MB inversion depends entirely upon the coordinate grids {xl, ym, γn} and {ξp, θq} selected for the desired MBRF f(l,m,n) and the measurements G(p,q). That is, the specific number of scattering centers and their MB coupling strengths within the true MBRF are irrelevant to the inversion process, so that examples containing a much higher density of scattering events can also be generated. The particular example presented herein was selected in order to demonstrate the most salient effects in the MB and SB image reconstructions. The embodiment described above may be extended to include data from a higher number of spatial dimensions. Define r to be a vector in a N-dimensional vector space. Likewise, define k to be a unit vector in this N-D vector space. The N=2 case presented above characterizes the spatial vector via r≡[x,y]T, with the superscript denoting vector transpose. Similarly, the unit vector for N=2 is expressed in terms of the polar coordinate angle θ, i.e., k≡[cos(θ), sin(θ)]T. Likewise, the N=3 case can be developed by defining the spatial vector r≡[x,y,z]T and the corresponding unit vector k≡[cos(θ) sin(ø), sin(θ) sin(ø), cos(ø)]T in terms of the spherical coordinate angles ø and θ.
The generic N−D form of the MB projection equations (7) is

g(s, k)≡∫dr∫dγ ƒ(r, γ)δ(kTr+γ−s).  (50)

The corresponding N−D form of the MB transform-domain projection equation (13) becomes

G(ξ, k)=∫dr ƒ(r, γ) exp(−j2π{kTr+γ}ξ).  (51)

In a similar fashion, the N−D form of the MB back-projection equation (15) is

b(r, γ)≡∫dk G(ξ, k) exp(j2π{kTr+γ}ξ).  (52)

This N−D form of the back-projection function b(r, γ) can be expressed in the following convolutional form

b(r, γ)=λ(r, γ)*ƒ(r, γ),  (53)

in terms of the kernel function

λ(r, γ)≡∫dk ∫dξ exp(j2π{kTr+γ}ξ)  (54)

Thus, the inversion for the MBRF yields





  
    
      
        
          f
          ⁡
          
            (
            
              r
              ,
              γ
            
            )
          
        
        =
        
          
            F
            
              N
              =
              1
            
            
              -
              1
            
          
          ⁢
          
            
              {
              
                
                  
                    F
                    
                      N
                      +
                      1
                    
                  
                  ⁢
                  
                    {
                    
                      b
                      ⁡
                      
                        (
                        
                          r
                          ,
                          γ
                        
                        )
                      
                    
                    }
                  
                
                
                  
                    F
                    
                      N
                      +
                      1
                    
                  
                  ⁢
                  
                    {
                    
                      λ
                      ⁡
                      
                        (
                        
                          r
                          ,
                          γ
                        
                        )
                      
                    
                    }
                  
                
              
              }
            
            .
          
        
      
    
    
      
        (
        55
        )
      
    
  




A similar set of generic N−D equations can be developed in the discrete domain.

The multiple-bounce inversion technique yields excellent reconstruction of the multiple-bounce reflectivity function ƒ(x,y,γ). In fact, these multiple-bounce inversion methods will not degrade the reconstructed image for cases in which the phase history measurements G(ξ, θ) contain only single-bounce contributions. In contrast, the single-bounce inversion methods of conventional spotlight SAR theory yield spurious ghosting artifacts in the image reconstruction ƒSB(x,y) for cases in which multiple-bounce events contribute to the measured phase history data G(ξ, θ).
In an alternate embodiment of the present invention, FIGS. 6a-6d refer to truth reflectivity functions identical to those shown in FIGS. 3a-3d. FIGS. 7a-7b, similar to FIGS. 4a-4b, show the measurement data for G(ξ, θ) corresponding to the given ƒ(x,y,γ). The measurement data shown in FIGS. 7a-7b differs from that shown in FIGS. 4a-4b, in that the data is collected at low spatial frequencies from about ξ=0.5 to ξ=1.5 as opposed to the much higher spatial frequencies in FIGS. 4a-4b which is from ξ=45.1 to ξ=46.1. Inversion of the measurement data using the MB reconstruction process described herein results in a virtually identical replica depicted in FIGS. 9a-9d of the truth reflectivity function shown in FIGS. 6a-6d. Whereas, the inversion of the measurement data using the SB reconstruction process shown in FIG. 8 contains significant degradation due to the appearance of ghost scattering centers. The signal-to-noise ratio in this embodiment is approximately 40 dB. As stated above, noise and interference play no greater a role in the IRAMS processing than in the single bounce processing. Consequently, one skilled in the art recognizes the standard signal-to-noise limitations that are applicable to IRAMS processing.
The IRAMS process described herein is also applicable to near-field collection data, wherein the wavefronts are curved. For the 2-D problem above, assume that the radius and angle (in polar coordinates) of the sensor are given by R(η) and φ(η), respectively, in terms of a scalar parameter η which can vary in any fashion with regard to the attribute of the scene of interest. Then the measurement range-to-target with respect to the coordinate origin is




  
    
      
        
          s
          ⁡
          
            (
            
              
                R
                ⁡
                
                  (
                  η
                  )
                
              
              ,
              
                φ
                ⁡
                
                  (
                  η
                  )
                
              
            
            )
          
        
        =
        
          
            
              1
              2
            
            ⁢
            
              
                
                  
                    [
                    
                      
                        
                          R
                          ⁡
                          
                            (
                            η
                            )
                          
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              φ
                              ⁡
                              
                                (
                                η
                                )
                              
                            
                            )
                          
                        
                      
                      -
                      
                        x
                        i
                      
                    
                    ]
                  
                  2
                
                +
                
                  
                    [
                    
                      
                        
                          R
                          ⁡
                          
                            (
                            η
                            )
                          
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              φ
                              ⁡
                              
                                (
                                η
                                )
                              
                            
                            )
                          
                        
                      
                      -
                      
                        y
                        i
                      
                    
                    ]
                  
                  2
                
              
            
          
          +
          
            
              1
              2
            
            ⁢
            
              
                
                  
                    [
                    
                      
                        
                          R
                          ⁡
                          
                            (
                            η
                            )
                          
                        
                        ⁢
                        
                          cos
                          ⁡
                          
                            (
                            
                              φ
                              ⁡
                              
                                (
                                η
                                )
                              
                            
                            )
                          
                        
                      
                      -
                      
                        x
                        f
                      
                    
                    ]
                  
                  2
                
                +
                
                  
                    [
                    
                      
                        
                          R
                          ⁡
                          
                            (
                            η
                            )
                          
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              φ
                              ⁡
                              
                                (
                                η
                                )
                              
                            
                            )
                          
                        
                      
                      -
                      
                        y
                        f
                      
                    
                    ]
                  
                  2
                
              
            
          
          +
          γ
          -
          
            R
            ⁡
            
              (
              η
              )
            
          
        
      
    
    
      
        (
        56
        )
      
    
  




This equation replaces Eq. (1) for the near-field problem. The remainder of the analysis is straightforward, wherein the integrations/summations over θ are replaced by the corresponding integrations/summations over the scalar parameter η. It can also be verified by a Taylor series expansion that the near-field equation (56) reduces to the far-field equation (1) as the radius R(η) becomes much larger than the size of the scene of interest. Equation (56) can be further extended to apply for a non-planar collection surface by defining the sensor location in terms of the spherical coordinates (R(η), φ(η), Ω(η)) parameterized by the scalar parameter η via




  
    
      
        
          s
          ⁡
          
            (
            
              
                R
                ⁡
                
                  (
                  η
                  )
                
              
              ,
              
                φ
                ⁡
                
                  (
                  η
                  )
                
              
              ,
              
                ω
                ⁡
                
                  (
                  η
                  )
                
              
            
            )
          
        
        =
        
          
            
              1
              2
            
            ⁢
            
              
                
                  
                    
                      
                        
                          [
                          
                            
                              
                                R
                                ⁡
                                
                                  (
                                  η
                                  )
                                
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    φ
                                    ⁡
                                    
                                      (
                                      η
                                      )
                                    
                                  
                                  )
                                
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    ω
                                    ⁡
                                    
                                      (
                                      η
                                      )
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              x
                              i
                            
                          
                          ]
                        
                        2
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          [
                          
                            
                              R
                              ⁢
                              
                                (
                                η
                                )
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    φ
                                    ⁡
                                    
                                      (
                                      η
                                      )
                                    
                                  
                                  )
                                
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    ω
                                    ⁡
                                    
                                      (
                                      η
                                      )
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              y
                              i
                            
                          
                          ]
                        
                        2
                      
                      +
                      
                        
                          [
                          
                            
                              
                                R
                                ⁡
                                
                                  (
                                  η
                                  )
                                
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    ω
                                    ⁡
                                    
                                      (
                                      η
                                      )
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              z
                              i
                            
                          
                          ]
                        
                        2
                      
                    
                  
                
              
            
          
          +
          
            
              1
              2
            
            ⁢
            
              
                
                  
                    
                      
                        
                          [
                          
                            
                              
                                R
                                ⁡
                                
                                  (
                                  η
                                  )
                                
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    φ
                                    ⁡
                                    
                                      (
                                      η
                                      )
                                    
                                  
                                  )
                                
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    ω
                                    ⁡
                                    
                                      (
                                      η
                                      )
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              x
                              f
                            
                          
                          ]
                        
                        2
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          [
                          
                            
                              R
                              ⁢
                              
                                (
                                η
                                )
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    φ
                                    ⁡
                                    
                                      (
                                      η
                                      )
                                    
                                  
                                  )
                                
                              
                              ⁢
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    ω
                                    ⁡
                                    
                                      (
                                      η
                                      )
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              y
                              f
                            
                          
                          ]
                        
                        2
                      
                      +
                      
                        
                          [
                          
                            
                              
                                R
                                ⁡
                                
                                  (
                                  η
                                  )
                                
                              
                              ⁢
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    ω
                                    ⁡
                                    
                                      (
                                      η
                                      )
                                    
                                  
                                  )
                                
                              
                            
                            -
                            
                              z
                              f
                            
                          
                          ]
                        
                        2
                      
                    
                  
                
              
            
          
          +
          γ
          -
          
            R
            ⁡
            
              (
              η
              )
            
          
        
      
    
    
      
        (
        57
        )
      
    
  




This type of modeling also permits targets to have different heights z.

Further, the location of the transmitter and receiver can be different. The required modifications for such a bistatic collection are well-known to those skilled in the art. There are two primary modifications required. First, the measurement angle θ for a two-dimensional monostatic collection (wherein the transmitter and receiver are co-located) in the far-field is replaced by the bisector angle between the angle of the transmitter and that of the receiver. Second, the measured spatial frequencies for this problem are scaled by the factor cos(β/2), with β equal to the angle between that of the transmitter and that of the receiver. Further extension of the bistatic problem to the near-field case and to a non-planar collection surface, as presented above for the monostatic case, can be accomplished by one who is skilled in the art.
In a further embodiment of the present invention, an implementation of the IRAMS processing is based upon the cross-range drift in SAR magnitude imagery of the multipath responses with respect to the mean aspect angle at which the phase history data are measured. In particular, the ideal response in SAR magnitude imagery of each direct-return scattering event does not drift with regard to aspect angle. In contrast, the SAR magnitude response of each multipath event does drift in the cross-range direction with respect to aspect angle, and the amount of drift is directly proportional to the value of the multipath delay. The ideal responses of these multipath cross-range drifts can be used within a deconvolution framework in order to estimate the multipath reflectivity based upon the separate co-registered magnitude SAR images formed at different aspect angles.
The analytic expression for the cross-range drift for a particular multipath response characterized by the round-trip delay {tilde over (γ)}=2γ is found by simply computing the maximum of the magnitude response in conventional SAR imagery as a function of the mean aspect angle of the phase history measurements. A derivation is given below.
Suppose that it is desired to form a conventional SAR image (i.e., assuming only direct scattering) at a nominal aspect angle of θ=ω. To perform this operation, the range-angle measurement function of Eq. (3) is linearized in terms of the angle θ about the mean aspect angle ω, retaining only the constant and linear terms in the local aperture angle defined by

φ≡θ−ω.  (58)

The distinction between φ and ω is that the SAR image at some mean aspect angle ω is computed using phase history data over some span of local aperture angles φ. The resulting linearization of Eq. (3) above yields to following first-order Taylor series:

s(φ,ω,x,y,γ)=[x cos(θ)+y sin(θ)+γ]+[−x sin(θ)+y cos(θ)]φ+O(φ2)  (59)

with the notation O(φ2) denoting second and higher order terms in φ. Conventional direct-path image formation at each aspect angle ω tacitly assumes that the range-angle function has the form of this equation with γ=0, i.e.,

{tilde over (s)}(φ,ω,x,y)=[x cos(θ)+y sin(θ)]+[−x sin(θ)+y cos(θ)]φ+O(φ2).  (60)

It is useful to examine the variation with respect to aspect angle ω of the point response in the conventional direct-scattering SAR complex imagery due to a single multipath scattering event. In particular, the response due to the multipath reflectivity

ƒ(x,y,γ,ω)=α(ω)δ(x−x0,y−y0,γ−γ0).  (61)

is examined, with α(ω) a complex-valued, aspect-dependent scatterer response function and δ(•) the standard delta function defined by

δ(y)=∫ƒ(x)δ(x−y)dx.  (62)

It is assumed that this response function is approximately constant over the relatively narrow span of aperture angles φ and can vary significantly only over the larger span of aspect angles ω. The contribution to the phase history due to this multipath scattering event is obtained by using the following expression:




  
    
      
        
          
            G
            ⁡
            
              (
              
                ξ
                ,
                
                  φ
                  ;
                  ω
                
              
              )
            
          
          =
          
            
              ∫
              
                -
                ∞
              
              ∞
            
            ⁢
            
              
                d
                x
              
              ⁢
              
                
                  ∫
                  
                    -
                    ∞
                  
                  ∞
                
                ⁢
                
                  
                    d
                    y
                  
                  ⁢
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      ∞
                    
                    ⁢
                    
                      
                        d
                        γ
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        f
                        ⁡
                        
                          (
                          
                            x
                            ,
                            y
                            ,
                            γ
                            ,
                            ω
                          
                          )
                        
                      
                      ⁢
                      
                        exp
                        ⁡
                        
                          (
                          
                            
                              -
                              j
                            
                            ⁢
                            
                              
                            
                            ⁢
                            2
                            ⁢
                            π
                            ⁢
                            
                              
                            
                            ⁢
                            
                              s
                              ⁡
                              
                                (
                                
                                  φ
                                  ,
                                  ω
                                  ,
                                  x
                                  ,
                                  y
                                  ,
                                  γ
                                
                                )
                              
                            
                            ⁢
                            ξ
                          
                          )
                        
                      
                    
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        63
        )
      
    
  




with s(θ,x,y,γ) given through first order in φ by Eq. (59). Thus, the contribution to the phase history due to this multipath event becomes

G(ξ,φ;ω)=α(ω)exp(−j2πs(φ,ω,x0,y0,γ0)ξ).  (64)

The conventional SAR image can be computed using the back-projection function




  
    
      
        
          
            b
            ⁡
            
              (
              
                x
                ,
                
                  y
                  ;
                  ω
                
              
              )
            
          
          =
          
            
              ∑
              ξ
            
            ⁢
            
              
                ∑
                φ
              
              ⁢
              
                
                  G
                  ⁡
                  
                    (
                    
                      ξ
                      ,
                      
                        φ
                        ;
                        ω
                      
                    
                    )
                  
                
                ⁢
                
                  exp
                  ⁡
                  
                    (
                    
                      
                        -
                        j
                      
                      ⁢
                      
                        
                      
                      ⁢
                      2
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          s
                          ~
                        
                        ⁡
                        
                          (
                          
                            φ
                            ,
                            ω
                            ,
                            
                              x
                              0
                            
                            ,
                            
                              y
                              0
                            
                          
                          )
                        
                      
                      ⁢
                      ξ
                    
                    )
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        65
        )
      
    
  




with the direct-path range-angle function {tilde over (s)}(φ,ω,x,y) given by Eq. (60) through first order in φ. Thus, the SAR image for this multipath event alone is computed by inserting Eq. (64) into Eq. (65), giving





  
    
      
        
          
            b
            ⁡
            
              (
              
                x
                ,
                
                  y
                  ;
                  ω
                
              
              )
            
          
          =
          
            
              ∑
              ξ
            
            ⁢
            
              
                ∑
                φ
              
              ⁢
              
                
                  G
                  ⁡
                  
                    (
                    
                      ξ
                      ,
                      
                        φ
                        ;
                        ω
                      
                    
                    )
                  
                
                ⁢
                
                  exp
                  ⁡
                  
                    (
                    
                      
                        -
                        j
                      
                      ⁢
                      
                        
                      
                      ⁢
                      2
                      ⁢
                      
                        π
                        [
                        
                          
                        
                        ⁢
                        
                          
                            
                              s
                              ~
                            
                            ⁡
                            
                              (
                              
                                φ
                                ,
                                ω
                                ,
                                
                                  x
                                  0
                                
                                ,
                                
                                  y
                                  0
                                
                              
                              )
                            
                          
                          -
                          
                            s
                            ⁡
                            
                              (
                              
                                φ
                                ,
                                ω
                                ,
                                
                                  x
                                  0
                                
                                ,
                                
                                  y
                                  0
                                
                                ,
                                
                                  γ
                                  0
                                
                              
                              )
                            
                          
                        
                        ]
                      
                      ⁢
                      ξ
                    
                    )
                  
                
              
            
          
        
        ,
      
    
    
      
        (
        66
        )
      
    
  




with s(φ,ω,x0,y0,γ0) and {tilde over (s)}(φ,ω,x,y) given by Eqs. (59) and (60), respectively.

The location in the conventional direct-path SAR image of the multipath ghost point response corresponds to a vanishing exponential in Eq. (66) for all ξ and φ. The resulting direct-path SAR image coordinates (x,y) of the multipath scattering event characterized by the parameters (x0,y0,γ0) are given by the solution of the linear equations

(x−x0)cos(ω)+(y−y0)sin(ω)=γ0,  (67)

−(x−x0)sin(ω)+(y−y0)cos(ω)=0.  (68)

The solution to these equations yields the SAR image coordinates (x,y) with respect to the aspect angle ω corresponding to the multipath point response (x0,y0,γ0):

x(ω)=x0+γ0cos(ω),  (69)

y(ω)≅y0+γ0sin(ω).  (70)

Therefore, the conventional SAR image response of a particular multipath contribution characterized by (x0,y0,γ0) merely rotates on a circle with radius γ0 about the true scattering event position at (x0,y0). For small values of the aspect angle ω such that second and higher order terms in ω can be ignored, only the variation in the cross-range coordinate is significant, i.e.,

x(ω)≅x0+γ0,  (71)

y(ω)≅y0+γ0ω.  (72)

This resulting cross-range drift γ0ω, which is directly proportional to the total multipath delay γ0 through first order in the aspect angle ω, forms the basis for the IRAMS processing described below.

Given the drift of the multipath ghost responses with respect to aspect angle, in order to account for the drift during image analysis, a method for estimating the various contributions to the multipath reflectivity ƒ(x,y,γ,ω) is utilized. In particular, the effects of direct scattering must be separated from that due to multipath. The overall processing is initiated based upon some number N of SAR magnitude images formed at N uniformly incremented mean aspect angles. Of course, the individual sets of phase history data for each these initial SAR images can overlap in order for the processing to be practical.
The processing begins by separating the N initial SAR images into N/2 pairs of adjacent images. Then, within each pair of images, it is necessary to estimate the contributions of two different functions. The first such function p(y) corresponds to direct scattering events that exhibit little or no cross-range drift in progressing from the first image of the pair to the second. The other essential function q(y) for this image pair contains the contributions due to multipath scattering events characterized by a particular delay γ, such that there is a cross-range drift in progressing from the first image to the second. The dependence of the along-range coordinate x is suppressed in these two functions and the remainder of the analysis, since Eqs. (71) and (72) showed that only the cross-range coordinate y of a multipath scattering event exhibits a significant variation with respect to the mean aspect angle ω. Thus, the two adjacent SAR magnitude images can be delineated by b1(y) and b2(y).
The mathematical relation describing the dependence of the two adjacent SAR magnitude images upon the direct and multipath functions can be expressed in the form

b1(y)=p(y)+q(y+1/2),  (73)

b2(y)=p(y)+q(y−1/2).  (74)

The offsets of −1/2 and 1/2 in the arguments of q in Eqs. (73) and (74) correspond to a shift to the right of one unit of length (e.g., one pixel) in the b2 SAR magnitude image with respect to that of b1. Half-integral shifts are used in Eqs. (73) and (74) in order to make the drift effects symmetric relative to that of the non-drift effects. The Fourier transform of Eqs. (73) and (74) with respect to the cross-range coordinate yields

B1(η)=P(η)+Q(η)exp(jπη),  (75)

B2(η)=P(η)+Q(η)exp(−jπη).  (76)

in terms of the cross-range spatial frequency η. For values of η that are not equal to an integer, this system yields the following estimates for the Fourier transforms of the desired functions p(y) and q(y):

{circumflex over (P)}(η)=H0(η)−cos(πη){circumflex over (Q)}(η),  (77)





  
    
      
        
          
            
              Q
              ^
            
            ⁡
            
              (
              η
              )
            
          
          =
          
            -
            
              
                j
                ⁢
                
                  
                
                ⁢
                
                  
                    H
                    1
                  
                  ⁡
                  
                    (
                    η
                    )
                  
                
              
              
                sin
                ⁡
                
                  (
                  
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    η
                  
                  )
                
              
            
          
        
        ,
      
    
    
      
        (
        78
        )
      
    
  




in terms of the Fourier-transformed average and difference images defined by

H0(η)≡[B2(η)+B1(η)]/2,  (79)

H1(η)≡[B2(η)−B1(η)]/2.  (80)

The inversion given in Eqs. (77) and (78) is not well-defined for η equal to zero or unity, but the solution exists for the open set between these two values, i.e., 0<|η|<1. However this problem becomes ill-conditioned for some non-zero span of low frequencies in the neighborhood of η=0 for the realistic case of noisy estimates of the different images b1(y) and b2(y) resulting from noisy phase history measurements. A practical means for ameliorating these inversion issues is to apply an appropriate weighting to H1(η) in Eq. (78) prior to applying the inverse Fourier transform to obtain the desired function q(y). Empirical analysis has revealed that the application of a Hanning window over the span of 0<|η|<1 seems to give a reasonably good estimate of both the direct and multipath scattering effects, especially for the realistic case in which multipath results from spatially localized interactions between scattering centers. Results can be further improved by preconditioning the input images b1(y) and b2(y) at each stage in the pyramid synthesis to ensure that only effects consistent with the direct p(y) and multipath q(y) effects remain.
By way of particular example, as shown in the example of FIG. 10, N=8, resulting in 4 pairs including original images b1(y) through b8(y). Then, within each pair of images, it is necessary to estimate the contributions of two different functions. The first such function p(y) corresponds to direct scattering events that exhibit little or no cross-range drift in progressing from the first image of the pair to the second. The other essential function q(y) for this image pair contains the contributions due to multipath scattering events characterized by a particular delay, such that there is a cross-range drift in progressing from the first image to the second.
An estimation of separate pairs of the direct p(y) and multipath q(y) functions is applied to each of the N/2 pairs of adjacent images. The variation in the values of these functions with respect to aspect angle occurs since the response of each scatterer is not necessarily uniform with respect to changes in the relative aspect angle (ω). The q(y) functions contain the highest resolvable multipath delay, i.e., those satisfying |ω1−ω2|γmax=1, with ω1 and ω2 the mean aspect angles corresponding to the two images corresponding to a pair of interest. However, the p(y) functions contain not only the effects of direct scattering events, but also those with lower values of multipath delay γ<γmax. Thus, it is possible to further this analysis in a multi-delay-resolution approach by considering the N/2 distinct p(y) functions as the input images into the process. If we delineate these images via p1, with the subscript 1 delineating the first level in the multi-delay-resolution approach, then the application of the inversion process yields N/4 pairs of multipath q2(y) and direct p2(y) functions of the second level. The functions q2(y) contain the multipath effects corresponding to the delay at one-half of the maximum unambiguous value, i.e., γ<γmax/2. This processing is repeated at each level until the final pair of pM−1(y) images, with M=log2(N), are inverted to obtain the multipath qM(y) and direct pM(y) functions of the final level. The direct pM(y) image contains suppresses the effects of multipath point scattering, and the remaining set of q1(y) functions contain the effects of multipath at different levels of delay resolution and at different relative aspect angles ω.
FIG. 10 shows the various direct-path and multi-path images produced in this multi-delay-resolution IRAMS architecture. The power of this new embodiment of IRAMS is that it produces a good separation of single scatterer and multiple scatterer data for the case of anisotropic scattering events, i.e., those in which the scattering intensity depends upon the aspect angle ω. For example, suppose that there exist multiple scattering data in the original images b1(y) and b2(y) in FIG. 10, but the intensity of these multiple scattering data is weak or non-existent at the different aspect angles ω corresponding to the original images b5(y) and b6(y). Then, the multi-delay-resolution embodiment of IRAMS shown in FIG. 10 will estimate the multiple scattering data of the original images b1(y) and b2(y) into the multiple scattering function q1,1(y) while placing the residual data in the single scattering function p1,1(y). In contrast, the multiple scattering function q1,3(y) will contain no multiple scattering data, since there was no multiple scattering data in its original images b5(y) and b6(y). Thus, this IRAMS embodiment provides the additional information of the particular aspect angles at which multiple scattering effects are most important, thereby improving both the estimates of the multiple scattering data and the single scattering data.
The IRAMS processing described in the embodiment above is validated using simulated phase history data that contain the contributions of multipath scattering between point locations. The “truth” plot of FIG. 11 shows an example containing five actual scattering centers 20 and the simulated multipath ghosts 25 arising from the various pairings of the actual scattering centers 20.
The true values of the scattering centers and their corresponding multipath couplings were used in Eq. (63) to simulate the phase history measurements G(ξ,φ;ω) for a signal-to-noise ratio (SNR) of 60 dB. In this example, a total of N=32, resulting in 16 distinct co-registered SAR image pairs (i.e., 32 images, b1 through b32) were formed at different aspect angles ω using overlapping sets of phase history data. FIG. 12 shows one of the original images, which is the result of conventional SAR image formation applied to the simulated phase history measurements containing multipath scattering effects. Although the point locations of the various direct scattering centers of FIG. 11 are well-reproduced, FIG. 12 contains many additional spurious scattering centers that arise from the multipath contributions to the simulated phase history data.
The multi-delay-resolution IRAMS processing described above is applied to the original 16 pairs of co-registered images. Referring to the IRAMS processing tree of FIG. 10, the processing of the 16 pairs of co-registered images would follow a tree that at its base has 4 times as many starting images (i.e., 32 images, b1 through b32) and thus would reach Level 5 at the final processed image. These conventional SAR images exhibit the predicted cross-range drift of multipath scattering ghosts, as shown in FIGS. 13a and 13b and as predicted by the theory presented above for differing aspect angles, ω, i.e., −2.8571 degrees and 2.8571 degrees, respectively in exemplary FIGS. 13a and 13b. In contrast, the locations of the true point scattering events do not exhibit such a drift.
The result of the two direct images at the fourth level is shown in FIGS. 14 and 15. A slight shift between the two images of the multipath ghosts is evident, whereas the direct scatterers do not drift. The application of the final inversion of Eqs. (77) and (78) yields the direct and multipath path images of FIGS. 16 and 17, respectively. Clearly, direct path image of FIG. 16 contains primarily the effects of the actual scattering centers while suppressing the multipath ghost effects, thereby yielding a more literal image. Likewise, the multipath image of FIG. 17 contains primarily the multipath ghosts while suppressing the direct scattering centers, thus potentially facilitating target recognition capabilities.
FIG. 18 summarizes all of the images corresponding to the last three levels of the multi-delay-resolution IRAMS processing tree in the manner of FIG. 10. FIGS. 14 and 15 correspond to the two direct-path images at level four in FIG. 18, and FIGS. 16 and 17 give the direct-path and multipath images, respectively, at level five in FIG. 18. One possible means of delineating the effects of multipath from that of direct-path scattering can be obtained by taking the difference between the final direct-path image at the highest level and the integrated sum of all of the multipath images in the multi-resolution tree. The result of applying this discrimination technique is shown in FIG. 19. Comparison with the truth image of FIG. 11 reveals that the direct-path scattering effects correctly correspond to positive values of the discriminator, whereas the integrated multipath scattering events are given by negative values.
In a further embodiment of the present invention, the analysis methodology described above can also be implemented to analyze images of an object-of-interest for the presence of different materials. Conventional SAR image formation assumes that all scattering events which affect the measured phase history data are due to non-delayed reflection effects. That is, the theory of conventional SAR does not account for the delayed components of radar signature, which are indicative of the particular material properties. However, the physical reality of radar scattering dictates that the material relaxation time can be significantly larger than the temporal resolution cell size. Thus, there can be significant interference between the immediate reflection at one spatial position and the delayed material reflection component at a “closer-range” spatial position. Recent advances in the understanding of the interactions of radio-frequency (RF) waveforms with different materials have yielded the potential of performing remote identification of different materials within synthetic aperture radar (SAR) imagery. By way of example, referring to FIG. 20a the locations of the five actual scattering centers 20 are shown along with the projected locations of their corresponding delayed material responses 30. FIG. 20b illustrate the resulting SAR image, wherein the simulated phase history data includes the effects of simulated delayed echoes 30 from each of the scattering centers 20 of FIG. 20a. As shown in FIG. 21, various immediate-path and delayed-path images are produced in this multi-delay-resolution IRAMS architecture.
The analysis methodology described above is utilized as part of an overall methodology that utilizes advanced signal and image processing algorithms in order to obtain estimates of the material precursor signatures in the midst of a myriad of interfering effects, e.g., relative stability with regard to the narrow span of aspect angles typical of SAR data collections; relatively weak precursor signals overwhelmed by the side-lobes of primary radar returns from neighboring objects and clutter; and interference from multipath interactions. These material precursor signatures are input into a classification algorithm in order to provide a map of the material compositions contained within each image pixel, in a fashion similar to that of hyper-spectral imaging for material identification.
Consequently, this additional processing of the SAR data yields a more literal image containing the estimated direct-path scattering contributions with the MBGs suppressed, thus resulting in improved target recognition capabilities.
The specific examples described above are not intended to be limiting. Alteration of the processes described above is contemplated by one skilled in the art for application of the IRAMS process to various types of reflection tomographic data, including radar, sonar, and the like.

## Claims

1. A process for estimating the contributions of single path and multipath scattering events to reflectance data in order to reconstruct an image comprising:
separating the single path events from the multipath scattering events including:
identifying N images formed at N uniformly incremented mean aspect angles;
separating the N images into N/2 pairs of adjacent first and second images,


wherein the adjacent images are separated by a time delay;
estimating the cross-range drift of direct path and multipath scattering events in progressing from the first to the second images; and
reconstructing an image utilizing the estimation of cross-range drift from the reflectance data.

2. The process according to claim 1, wherein the direct path scattering events exhibit negligible cross-range drift between adjacent first and second images.

3. The process according to claim 1, wherein the multipath scattering events exhibit measurable cross-range drift adjacent first and second images.

4. The process according to claim 1, wherein the direct path scattering events are represented by single bounce reflectance data and the multipath scattering events are represented by multiple bounce reflectance data.

5. The process according to claim 1, wherein the reflectance data is synthetic aperture radar data.

6. The process according to claim 1, wherein the reflectance data represents two-dimensions of the image.

7. The process according to claim 1, wherein the image is reconstructed in three dimensions.

8. The process according to claim 1, further including collecting the reflectance data using a co-located transmitter and receiver.

9. The process according to claim 8, wherein the reflectance data is comprised of waveforms when it is collected by the co-located transmitter and receiver.

10. The process according to claim 9, wherein the waveforms are collected at a far-field location, such that the wavefronts impinging on a region to be imaged and collected at the receiver are approximately linear.

11. The process according to claim 9, wherein the waveforms are collected at a near-field location, such that the wavefronts impinging on a region to be imaged and collected at the receiver are curved.

12. The process according to claim 1, wherein the reflectance data is collected using non co-located transmitters and receivers.

13. The process according to claim 1, wherein the reflectance data exhibits anisotropic scattering characteristics.