Abstract:
Methods and systems for the efficient determination of the direction of multiple signal sources in both near and far field using hierarchical combinations of sets of raw input signals to convert spatial input to angular output. Each increasing hierarchical combination increases angular resolution, improving image quality with low computational expenditure.

Description:
BACKGROUND 
       [0001]    Imaging technologies based on frequencies include technologies such as ultrasound, radar, sonar and radio astronomy. Ultrasound, radar and sonar technologies generally send a signal and determine a position and/or shape of an object by processing the reflections of the signals and use that reflection to create an image whereas radio astronomy processes signals assuming they are coming from far away. 
         [0002]    While standard transducers comprise either a single active element that both generates and receives high frequency sound waves, or two paired elements, one for transmitting and one for receiving; phased arrays comprise transducer assemblies that can each be pulsed separately. The transducer assemblies may be arranged in a strip (linear array), a ring (annular array), a circular matrix (circular array), curved, or a more complex shape. A phased array system varies the time between a series of outgoing pulses in such a way that the individual wave fronts generated by each element in the array combine with each other to add or cancel energy in predictable ways that effectively steer and shape the sound beam or radio signal. 
         [0003]    Beamforming is a signal processing technique used to precisely align the phases of an incoming signal from different parts of an array to form a well understood beam in a specific direction and focus depth. The signals from each of the elements are delayed such that when they are summed they all have the same delay corresponding to a specific direction. However, each independent beam requires an independent analog path or delay and sum calculation causing increasing complications and computational complexity as the size of the array and the corresponding number of independent beams increases. 
         [0004]    There is therefore a need for alternative methods of signal processing to align the phases of an incoming signal to efficiently create images from phased arrays in imaging technology. 
       BRIEF SUMMARY 
       [0005]    Described herein is a method of beamform signal processing to enhance image quality and angular resolution from signals received from a phased array in near and far field imaging data and to transform raw transducer data into a form that can be displayed on a screen with a high degree of image resolution. The methods and systems described herein may take a single or multi-dimensional image generated from a transducer array and generate multi-dimensional images one dimension more than the dimension from which the image was generated, e.g. 2D images may be generated from 1D images, 3D images may be generated from 2D images, et cetera. The signal processing systems and methods described herein may be applied to radar, sonar, ultrasound and radio astronomy. 
         [0006]    In the methods and systems and methods described here, translations are made from spatial input data to angular output. Two or more raw input signals are combined to form a similar number of output signals in a series of subarrays across an array of input sensors/transducers less than a wavelength apart. In some embodiments, such transducers may be about 0.7 wavelengths apart or any fraction&gt;1 wavelength. 
         [0007]    In some embodiments, a first group of raw input signals from a first set of neighboring transducers in a phased array is combined to form a first set of output angular signals and a second group of raw input signals from a second set of neighboring sensors in the phased array is combined to form a second set of output angular signals. Two or more raw input signals may be combined to form a set of output angular signals until the spatial input form the entire sensor array has been converted to angular information of resolution at the Abbe limit. Each set of output angular signals of input is then combined with adjacent angular beams from neighboring subarrays to refine the angular resolution with the amount of angular resolution increasing with each subsequent pairing such that a fourth set in the hierarchy would have twice as much angular resolution as the second set in the hierarchy. The output angular signals may be combined in the same or different ways to refine angular resolution and improve image quality data. For example, inputs can be combined in any desired adjacent fashion including, but not limited to, up, down, azimuthal angle θ, and polar angle φ. In some embodiments, additional refinements to the data may be added such that some or all of the data including, but not limited to the raw data, may be apodized. 
         [0008]    The sample frequency used to generate the raw input signals in the methods and compositions described herein is at least two times per wavelength or period (T). In some embodiments, the sampling frequency may be about 10 times per wavelength, or any fraction in between including, but not limited to, 3, 4, 5, 6, 6.25, 7, 7.5 8, 9, and the like. The same or different sample frequencies may be used to generate each group of raw input frequencies, creating a frequency agnostic system. In some embodiments, the beam directions may be determined in parallel. 
         [0009]    In some embodiments, a method of generating a multi-dimensional image from a phase array with one less dimension than the multi-dimensional image may include receiving a signal of amplitude (A) from an array of sensors over time (t); apodizing received raw signals A; pairing each signal A received by a sensor with an adjacent signal above, below and across forming a first set of paired signals B; compensating for the time delay due to the location difference of each sensor receiving the signal; pairing each pair in the first set of paired signals B in accordance with their spatial placement and angle in volume in the phase array to form a second set of paired signals C; and/or pairing each pair in the second set of paired signals C in accordance with their spatial placement and angle in volume in the phase array to form a third set of paired signals D. Such third sets of paired signals D and/or subsequently grouped signals may be used to produce the multi-dimensional image. 
         [0010]    In some embodiments, a beamforming apparatus may include a phased array and a processor configured to translate spatial input data from beam signals of angles between +π/4 radians to −π/4 radians to angular output by apodizing each set of signals, interpolating the angular information, and forming an image from the interpolated angular information. The phased array may include transducers separated by a distance L=λ/1.44 configured to receive signals of a frequency (f) with a period (T) and a time delay of +t/4 to −t/4 between signals received by adjacent transducers. In some embodiments, raw input data is paired according to space and time to create a first set of paired received signals. The first set of received signals is grouped according to space and time to create a second set of paired signals. In further embodiments, each of the second set of paired signals is grouped according to space, angle and time to create a third set of paired signals. In additional embodiments, each of the third set of paired signals is paired according to space, angle and time to create a fourth set of paired signals. Such grouping may continue hierarchically to combine angles of interest with adjacent angular beams from neighboring sub-arrays to refine angular resolution until all of the spatial input has been converted to angular information of resolution at the Abbe limit and used to generate images from phased arrays. 
         [0011]    To the accomplishment of the foregoing and related ends, certain illustrative aspects of the system are described herein in connection with the following description and the attached drawings. The features, functions and advantages that have been discussed can be achieved independently in various embodiments of the present disclosure or may be combined in yet other embodiments, further details of which can be seen with reference to the following description and drawings. This summary is provided to introduce a selection of concepts in a simplified form that are elaborated upon in the Detailed Description. This summary is not intended to identify key features or essential features of any subject matter described herein. 
     
    
     
       BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS 
         [0012]    To easily identify the discussion of any particular element or act, the most significant digit or digits in a reference number refer to the figure number in which that element is first introduced. 
           [0013]      FIG. 1  illustrates an embodiment of processing of raw input received from a phased array with a beamforming apparatus as described herein. 
           [0014]      FIG. 2  illustrates a routine for interpreting beam direction imaging data in accordance with one embodiment. 
           [0015]      FIG. 3  illustrates a routine for increasing the angular resolution of raw input. 
           [0016]      FIG. 4  illustrates a method of generating a multi-dimensional image from a phase array with one less dimension than the multi-dimensional image. 
           [0017]      FIG. 5  illustrates the focal point for a single point in the near field. 
           [0018]      FIG. 6  illustrates an hierarchical pairing of received signals A, with increased angular resolution with each subsequent pairing B and C; a 2D transducer array generating 3D images. 
           [0019]      FIG. 7  illustrates a an alternative means of combining spatial input to obtain angular information where each A has location at a single direction and each C has the center location but diverse directions. 
           [0020]      FIG. 8  is a graph showing the increased efficiency (logarithmic) of an embodiment described herein (fbf) in comparison to the current standard (delay_sum). 
           [0021]      FIG. 9  is a chart illustrating a graph showing the increased efficiency of an embodiment described herein (fbf) in comparison to the current standard (delay_sum). 
           [0022]      FIG. 10  illustrates a system  1000  in accordance with one embodiment. 
       
    
    
     DETAILED DESCRIPTION 
       [0023]    “Abbe diffraction limit” in this context refers to that light with wavelength λ, traveling in a medium with refractive index n and converging to a spot with angle \theta will make a spot with radius 
         [0000]    
       
         
           
             d 
             = 
             
               
                 λ 
                 
                   2 
                    
                   n 
                    
                   
                       
                   
                    
                   sin 
                    
                   
                       
                   
                    
                   θ 
                 
               
               . 
             
           
         
       
     
         [0024]    “Anistrophic” in this context refers to exhibiting properties with different values when measured in different directions. 
         [0025]    “Apodization” in this context refers to the process of altering a signal (such as one emitted by an electrical sensor) to make its mathematical description smoother and more easily manipulatable. 
         [0026]    “Far Field” in this context refers to the region from the antenna where the radiation pattern does not change shape with distance. It satisfies the following three equations: R&gt;(2D̂2)/λ, R&gt;&gt;D, and R&gt;&gt;λ. 
         [0027]    “FLOPS” in this context refers to floating-point operations per second where “floating-point” is a method of encoding real numbers within the limits of finite precision available on computers. 
         [0028]    “Fresnel zone” in this context refers to is a series of concentric ellipsoidal regions of alternating double strength and half strength volumes of a wave&#39;s propagation, caused by a wave following multiple paths as it passes by an object and is partially refracted by it, resulting in constructive and destructive interference as the different length paths go in and out of phase. 
         [0029]    “Near Field” in this context refers to the area in the immediate vicinity of an antenna having a boundary of R&lt;0.62√(D̂3/λ) where D is the maximum linear dimension of an antenna and λ is the wavelength. 
         [0030]    “Nyquist limit” in this context refers to the minimum sampling rate required to avoid aliasing. Specifically, there must be at least two samples per period of the wave being observed. 
         [0031]    “Piezoelectric” in this context refers to the ability of certain materials to generate an AC (alternating current) voltage when subjected to mechanical stress or vibration, or to vibrate when subjected to an AC voltage, or both. 
         [0032]    “Secant of an angle” in this context refers to the length of the hypotenuse divided by the length of the adjacent side. The abbreviation is sec. sec θ=hypotenuse/adjacent. It is equal to 1/cosine. 
         [0033]    “Voxel” in this context refers to each of an array of elements of volume that constitute a notional three-dimensional space, especially each of an array of discrete elements into which a representation of a three-dimensional object is divided. 
       Description 
       [0034]    Described herein are methods and systems for translating spatial input data to angular output for a phased array in order to generate images with high resolution. Phased arrays may be in any shape desired including, but not limited to, linear, curved, annular, rectangular, circular, matrix, and the like. The methods and systems described herein decrease computational costs, allowing for increasingly complex phased arrays to be used in imaging technologies with near or far field focus including, but not limited to, ultrasound, radar, sonar and radio astronomy. In some embodiments, the system can produce images of high resolution and frame rate as well as handle high numbers of array elements. 
         [0035]    During pulse propagation in a phased array, echoes are scattered by acoustic impedance perturbations and detected by the array elements. The data is sampled at a rate of about four to about ten of the fundamental frequency (f) though any fraction of that frequency can be used including but not limited to, 5, 6, 7, 8, and 9 or any fraction thereof, and digitally integrated through beamforming, allowing reception in a predefined direction corresponding to the transmission angle, and optimal focus at each depth. 
         [0036]    In some embodiments, raw input signals received by a transducer in a phased array are combined hierarchically using spacial, angular and time coordinates using Cartesian coordinates, i.e. the placement of the input signal in space, the angle in volume, i.e. the direction facing into the void, and time delay, i.e. the difference in time for a beam to return based on the position of a transducer. The number of signals combined at each level in the hierarchy may vary from about 2 to about the square root of the total sensor/transducer count. In some embodiments, the translation of the input data from a plurality of raw input signals may take place using parallel processing. 
         [0037]    As shown in the phased array with beamforming apparatus in  FIG. 1 , a CPU  102  sends instructions to a transmitter beamformer  104  that a signal should be sent from the transmitter  106 . The transmitter  106  emits or instructs a transducer (not shown) to emit a series of pulsed signals  110  the series of the pulsed signals  110  are captured by a receiver  108  which may be the same or a different transducer than the transmitter  106 . In some embodiments the transmitter  106  and the receiver  108  are the transducers or sensors. They may be the same or different transducers located in a phased array or in a probe. The echo received by the receiver  108  is then sent to the CPU  102  for receive beamformer  114 . The CPU  102  may process the received data in real-time during a scanning/signal capture event or after a delay. Processed frames of data are presented on a display  116 . In some embodiments, the CPU  102 , transmitter beamformer  104 , transmitter  106 , and receiver  108 , may all be contained in a probe. In other embodiments, they may be in one or more different wireless and/or wired devices. 
         [0038]    Receive beamformer  114  may proceed as shown in  FIG. 2 , where a first plurality of raw input signals from a first set of neighboring transducers in a phased array is acquired  202 . Once the signals are acquired, the first plurality of raw input signals is combined to form a first set of output angular signals  204 . A second plurality of raw input signals from a second set of neighboring sensors in the phased array is acquired  206  and the second plurality of raw input signals from the second set of neighboring sensors in the phased array is combined to form a second set of output angular signals  208 . Angles of interest from the first set of output angular signals are combined with the second set of output angular signals to refine angular resolution  210 , continuing until the spatial input from the entire sensor array has been converted to angular information of resolution at the Abbe limit at which point Routine  200  ends  212 . The angles of interest may be combined in a plurality of independent ways to refine angular resolution including up, down, at an azimuthal angle θ and polar angle φ and the like. 
         [0039]    In additional embodiments, receive beamformer  114  may proceed as shown in  FIG. 3 , where Routine  300  receives raw signals A of amplitude (A) from array of sensors over time (t) at  302 . The raw signals A are apodized at  304 . Each signal A received by a sensor is paired with an adjacent signal above, below and across forming a first set of paired signals B at  306  and compensating for the time delay due to the location difference of each sensor receiving a raw signal at  308 . Each pair in the first set of paired signals B are grouped in accordance with their special placement and angle in volume in the phase array to form a second set of paired signals Cat  310 . Each of the second set of signals C is grouped according to their spatial placement and angle in volume in the phase array to form a third set of paired signals D at  312 . Angles of interest from the first set of output angular signals are combined with the second set of output angular signals to refine angular resolution, continuing until the spatial input from the entire sensor array has been converted to angular information of resolution at the Abbe limit at which point Routine  300  ends  314 . The angles of interest may be combined in a plurality of independent ways to refine angular resolution including up, down, at an azimuthal angle θ and polar angle φ and the like. 
         [0040]    Signal pairing or grouping occurs in a hierarchical manner using spacial and angular coordinates from the phased array as well as time displacement based on the location of the transducers in relation to one another. As shown in  FIG. 4 , routine  400  receives a signal of amplitude (A) from an array of sensors over time (t) at  402 . Routine  400  then pairs each signal A received by a sensor with an adjacent signal above, below and across forming a first set of paired signals B. Time delays in the signals due to difference in location of the sensors receiving the signal is compensated for at  406 . Routine  400  then pairs each pair in the first set of paired signals B in accordance with their spacial placement and angle in volume in the phase array to form a second set of paired signals C at  408 . Paired signals C are then paired in accordance with their spacial placement and angle in volume in the phase array to form a third set of paired signals D at  410 . The pairing continues until the spacial input from the entire sensor array has been converted to angular information of resolution at the Abbe limit. In some embodiments, the raw input and the paired signals may be apodized. The angular information is then interpolated to form an image that may be displayed. 
         [0041]    In a phased array, sensors/transducers are separated by a constant distance L. As shown in  FIG. 5 , L separating exemplary sensors or transducers is a distance of λ/1.4 where λ is the wavelength though any distance less than a wavelength λ may be used and the beam angle is +π/4 radians to −η/4 radians. In the near field, each signal approaches a focal point at a different angle arriving at a different time point (t). The first layer of signals, α, β, χ, and δ have no focal depth. The second layer of signal output, ε, ι and γ is centered on pairs of transducers and the third layer of signal output δ is centered in four transducers. Exemplary distance/delay can be summarized as follows:
       α: depth   β: depth/cos(arctan(lambda/1.4*depth))   β: sqrt(deptĥ2+L̂2/2)   χ: sqrt(deptĥ2+2*L̂2)   δ: sqrt(deptĥ2+2*L̂2)   ε: sqrt(deptĥ2+L̂2/8)   ι: sqrt(deptĥ2+12.5*L̂2)   γ: sqrt(deptĥ2+4.5*L̂2)       
 
         [0050]    As shown in  FIG. 6  and  FIG. 7 , in an array subtile, raw input signals A are paired hierarchically to form a first set of paired received signals B pairs which are then paired to form a second set of signals C. 
         [0051]    As shown in  FIG. 6 , raw input A 11  and A 12  are combined with a time compensation t according to Aij[t]=raw inputs amplitudes to yield an equation Bijkm(t)=Yi[2j−1]*Ai[2j−1](t+{2m−3})+Yi[2j]*Ai[2j](t−{2m−3}) such that: 
         [0000]        A 12( t+ 1)+ A 11( t− 1) to yield  B  in an upwards direction ( B up) 
         [0000]        A 12( t− 1)+ A 11( t+ 1) to yield  B  in a downwards direction ( B down) 
         [0000]    where Xij, Yij are constant tables derived from the windowing function for the full aperture such that X and Y are apodizing functions such as, but not limited to, exp−(x̂2+ŷ2), where x and y are distances from the center of the sensor array; i and j represent coordinates in space (Cartesian) and k and m represent angular location where k corresponds to azimuthal angle θ and m represents polar angle φ. Inputs can be combined in any desired adjacent fashion including up, down, azimuthal angle θ and polar angle φ. 
         [0052]    This first set of paired received signals B is then combined as shown in  FIG. 6  with a time compensation to form a second plurality of paired signals of C pairs according to the formula: 
         [0000]        Cijkm ( t )= X[ 2 i−m]j*Bi[ 2 j−m ]( t+{ 2 m− 3})+ X[ 2 i]j*Bi[ 2 j ]( t−{ 2 m− 3}) 
         [0000]    where Xij, Yij are constant tables derived from the windowing function for the full aperture such that X and Y are apodizing functions, i and j represent coordinates in space (Cartesian) and where k corresponds to azimuthal angle θ and m represents polar angle φ. Inputs can be combined in any desired adjacent fashion including up, down, azimuthal angle θ and polar angle φ. 
         [0053]    The first set of paired signals B is combined to form: 
         [0000]        C ,up,left,( t )= B 11up( t− 1)+ B 12up( t+ 1) 
         [0000]        C ,up,right,( t )= B 11up( t )+ B 12up( t −1)
 
         [0000]        C ,down,left,( t ) =B 11down( t− 1) +B 12down( t+ 1) 
         [0000]        C ,down,right,( t )= B 11down( t )+ B (12)down( t− 1) 
         [0000]    where the first two indices in the array, in this instance, “11”, identify the location of the subarray in the sensor array, e.g. the first index counts from left to right and the second index counts from the top to the bottom of the array. In addition to the location of the subarray in the sensor array, the second set of paired signals such as adds the polar angles of a beam associated with the subarray. As the Aij array in most embodiments has no angle associated it with it, just two indices are listed. The pairing process continues until the spatial input from the entire sensor array has been converted to angular information of resolution at the Abbe limit. While these exemplary calculations demonstrate pairwise combinations of signals, the number of signals combined at each hierarchy in each subarray could vary from two to the square root of the total sensor count. For example, a third set of third set of paired signals D could be calculated according to the formula: 
         [0000]        Dijkm ( t )= Dijkm ( t )= Yi[ 2 j−m]*Ci[ 2 j−m]k[m/ 2]( t+{ 2 m− 3})+ Yi[ 2 j]Ci[ 2 j]k[m/ 2]( t−{ 2 m− 3}). 
         [0000]    with D having twice as much angular resolution as when the raw input data was paired to form the first set of paired signals B, where Xij, Yij are constant tables derived from the windowing function for the full aperture such that X and Y are apodizing functions, i and j represent coordinates in space (Cartesian) and where k corresponds to azimuthal angle θ and m represents polar angle φ. Inputs can be combined in any desired adjacent fashion including up, down, azimuthal angle θ and polar angle φ. 
         [0054]    In an alternative embodiment as shown in  FIG. 7 , each A may have a location in but a single direction and each C has the center location but diverse directions such that: 
         [0000]        Aij ( t )=raw inputs amplitudes 
         [0000]        Bijkm ( t )= Ai[ 2 j−m ]( t+{ 2 m− 4})+ Ai[ 2 j ]( t−{ 2 m− 4}) 
         [0000]        Cijkm ( t )= B[ 2 i−m]j[k/ 2] m ( t+{ 2 k− 4})+ B[ 2 i]j[k/ 2] m ( t−{ 2 k− 4}) 
         [0000]    where: 
         [0055]    i: space index left/right 
         [0056]    j: space index up/down 
         [0057]    k: angle index left/right 
         [0058]    m: angle index up/down 
         [0000]    such that: 
         [0000]        A 12( t+ 1)+ A 11( t− 1) to yield  B  in an upwards direction ( B up) 
         [0000]        A 12( t )+ A 11( t ) to yield  B  at an azimuthal angle ( B θ)
 
         [0000]        A 12( t− 1)+ A 11( t+ 1) to yield  B  in a downwards direction ( B down) 
         [0000]      and 
         [0000]        C ,up,left,( t )= B 11up( t− 1)+ B 12up( t+ 1) 
         [0000]        C ,up,θ,( t )= B 11up( t )+ B 12up
 
         [0000]        C ,up,right,( t )= B 11up( t )+ B 12up( t− 1) 
         [0000]        C ,down,left,( t ) =B 11down( t− 1) +B 12down( t+ 1) 
         [0000]        C ,down,θ,( t )= B 11down( t )+ B 12down
 
         [0000]        C ,down,right,( t )= B 11down( t )+ B (12)down( t− 1) 
         [0000]    with the pairing process continuing until the spatial input from the entire sensor array has been converted to angular information of resolution at the Abbe limit. While these exemplary calculations demonstrate pairwise combinations of signals, the number of signals combined at each hierarchy in each subarray could vary from two to the square root of the total sensor count. 
         [0059]    The direction of a signal in a phased array may be detected by selectively delaying the signals received from each sensor and running cross-correlations of the received return signals. While this is feasible for phased arrays with small number of sensors, the computational cost for a larger array is the square of the number of sensors in the array. By hierarchically ranking pairs of sensors in space, angle, and accounting for time, the computational cost is the number of sensors times the log of the number of sensors rather than the square, decreasing the computational cost considerably. As shown in Table 1, even 512×512 inputs and outputs calculated according to the methods described herein can be calculated four orders of magnitude more efficiently than traditional methods as shown in lines 1-5 (delay and sum (4.12E+04) to 1.61E+02) of Table 1. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Computational Cost of Beamforming 
               
             
          
           
               
                   
                 input 
                 output 
                   
                 time 
                 make 1 
                 sum over 
                 sum layer; 
                 GFLOPS at 
               
               
                 case 
                 nxm 
                 nxm 
                 window 
                 interpolation 
                 beam 
                 beams 
                 FLOPS/frame 
                 100 fps 
               
               
                   
               
             
          
           
               
                 delay 
                 262144 
                 262144 
                 262144 
                 786432 
                 524288 
                 4.12E+11 
                 4.12E+11 
                 4.12E+04 
               
               
                 and sum 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 one of 
                 512 
                 512 
                 512 
                 1536 
                 1024 
                 1572864 
                 8.05E+08 
                   
               
               
                 two 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 two of 
                 512 
                 262144 
                 512 
                 1536 
                 1024 
                 805306368 
                 8.05E+08 
                   
               
               
                 two 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                   
                   
                   
                   
                   
                   
                   
                 1.61E+09 
                 1.61E+02 
               
               
                 one of 
                 32 
                 32 
                 32 
                 96 
                 64 
                 6144 
                 1.01E+08 
                   
               
               
                 four 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 two of 
                 16 
                 512 
                 16 
                 48 
                 32 
                 49152 
                 2.52E+07 
                   
               
               
                 four 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 three of 
                 32 
                 16384 
                 32 
                 96 
                 64 
                 3145728 
                 1.01E+08 
                   
               
               
                 four 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 four of 
                 16 
                 262144 
                 16 
                 48 
                 32 
                 25165824 
                 2.52E+07 
                   
               
               
                 four 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                   
                   
                   
                   
                   
                   
                   
                 2.52E+08 
                 2.52E+01 
               
               
                 one of 
                 4 
                 4 
                 4 
                 12 
                 8 
                 96 
                 6.29E+05 
                   
               
               
                 nine 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 two of 
                 4 
                 16 
                 4 
                 12 
                 8 
                 384 
                 6.29E+06 
                   
               
               
                 nine 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 three of 
                 4 
                 64 
                 4 
                 12 
                 8 
                 1536 
                 6.29E+06 
                   
               
               
                 nine 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 four of 
                 4 
                 256 
                 4 
                 12 
                 8 
                 6144 
                 6.29E+06 
                   
               
               
                 nine 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 five of 
                 4 
                 1024 
                 4 
                 12 
                 8 
                 24576 
                 6.29E+06 
                   
               
               
                 nine 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 six of 
                 4 
                 4096 
                 4 
                 12 
                 8 
                 98304 
                 2.52E+07 
                   
               
               
                 nine 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 seven of 
                 4 
                 16384 
                 4 
                 12 
                 8 
                 393216 
                 6.29E+06 
                   
               
               
                 nine 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 eight of 
                 4 
                 65536 
                 4 
                 12 
                 8 
                 1572864 
                 6.29E+06 
                   
               
               
                 nine 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 nine of 
                 4 
                 262144 
                 4 
                 12 
                 8 
                 6291456 
                 6.29E+06 
                   
               
               
                 nine 
                   
                   
                   
                   
                   
                   
                   
                   
               
             
          
           
               
                 max depth for 
                   
                   
                   
                   
                   
                 7.55E+07 
                 7.55E+00 
               
               
                 512 × 512 
               
               
                   
               
             
          
         
       
     
         [0060]    This is additionally shown in  FIG. 8  and  FIG. 9 , as the number of sensors in the array increase, the processing delay ( FIG. 8 ) and runtime of traditional methods increases significantly in proportion to the methods described herein ( FIG. 9 ). 
         [0061]      FIG. 10  illustrates several components of an exemplary system  1000  in accordance with one embodiment. In various embodiments, system  1000  may include a desktop PC, server, workstation, mobile phone, laptop, tablet, set-top box, appliance, or other computing device that is capable of performing operations such as those described herein. In some embodiments, system  1000  may include many more components than those shown in  FIG. 10 . However, it is not necessary that all of these generally conventional components be shown in order to disclose an illustrative embodiment. Collectively, the various tangible components or a subset of the tangible components may be referred to herein as “logic” configured or adapted in a particular way, for example as logic configured or adapted with particular software or firmware. 
         [0062]    In various embodiments, system  1000  may comprise one or more physical and/or logical devices that collectively provide the functionalities described herein. In some embodiments, system  1000  may comprise one or more replicated and/or distributed physical or logical devices. 
         [0063]    In some embodiments, system  1000  may comprise one or more computing resources provisioned from a “cloud computing” provider, for example, Amazon Elastic Compute Cloud (“Amazon EC2”), provided by Amazon.com, Inc. of Seattle, Wash.; Sun Cloud Compute Utility, provided by Sun Microsystems, Inc. of Santa Clara, Calif.; Windows Azure, provided by Microsoft Corporation of Redmond, Wash., and the like. 
         [0064]    System  1000  includes a bus  1002  interconnecting several components including a network interface  1008 , a display  1006 , a central processing unit  1010 , and a memory  1004 . 
         [0065]    Memory  1004  generally comprises a random access memory (“RAM”) and permanent non-transitory mass storage device, such as a hard disk drive or solid-state drive. Memory  1004  stores an operating system  1012  as well as routine  200 , routine  300  and routine  400 . 
         [0066]    These and other software components may be loaded into memory  1004  of system  1000  using a drive mechanism (not shown) associated with a non-transitory computer-readable medium  1016 , such as a floppy disc, tape, DVD/CD-ROM drive, memory card, or the like. 
         [0067]    Memory  1004  also includes database  1014 . In some embodiments, system  1000  may communicate with database  1014  via network interface  1008 , a storage area network (“SAN”), a high-speed serial bus, and/or via the other suitable communication technology. 
         [0068]    In some embodiments, database  1014  may comprise one or more storage resources provisioned from a “cloud storage” provider, for example, Amazon Simple Storage Service (“Amazon S3”), provided by Amazon.com, Inc. of Seattle, Wash., Google Cloud Storage, provided by Google, Inc. of Mountain View, Calif., and the like. 
         [0069]    References to “one embodiment” or “an embodiment” do not necessarily refer to the same embodiment, although they may. Unless the context clearly requires otherwise, throughout the description and the claims, the words “comprise,” “comprising,” and the like are to be construed in an inclusive sense as opposed to an exclusive or exhaustive sense; that is to say, in the sense of “including, but not limited to.” Words using the singular or plural number also include the plural or singular number respectively, unless expressly limited to a single one or multiple ones. Additionally, the words “herein,” “above,” “below” and words of similar import, when used in this application, refer to this application as a whole and not to any particular portions of this application. When the claims use the word “or” in reference to a list of two or more items, that word covers all of the following interpretations of the word: any of the items in the list, all of the items in the list and any combination of the items in the list, unless expressly limited to one or the other. “Logic” refers to machine memory circuits, non-transitory machine readable media, and/or circuitry which by way of its material and/or material-energy configuration comprises control and/or procedural signals, and/or settings and values (such as resistance, impedance, capacitance, inductance, current/voltage ratings, etc.), that may be applied to influence the operation of a device. Magnetic media, electronic circuits, electrical and optical memory (both volatile and nonvolatile), and firmware are examples of logic. Logic specifically excludes pure signals or software per se (however does not exclude machine memories comprising software and thereby forming configurations of matter). Those skilled in the art will appreciate that logic may be distributed throughout one or more devices, and/or may be comprised of combinations memory, media, processing circuits and controllers, other circuits, and so on. Therefore, in the interest of clarity and correctness logic may not always be distinctly illustrated in drawings of devices and systems, although it is inherently present therein. The techniques and procedures described herein may be implemented via logic distributed in one or more computing devices. The particular distribution and choice of logic will vary according to implementation. 
         [0070]    Those having skill in the art will appreciate that there are various logic implementations by which processes and/or systems described herein can be effected (e.g., hardware, software, and/or firmware), and that the preferred vehicle will vary with the context in which the processes are deployed. “Software” refers to logic that may be readily readapted to different purposes (e.g. read/write volatile or nonvolatile memory or media). “Firmware” refers to logic embodied as read-only memories and/or media. Hardware refers to logic embodied as analog and/or digital circuits. If an implementer determines that speed and accuracy are paramount, the implementer may opt for a hardware and/or firmware vehicle; alternatively, if flexibility is paramount, the implementer may opt for a solely software implementation; or, yet again alternatively, the implementer may opt for some combination of hardware, software, and/or firmware. Hence, there are several possible vehicles by which the processes described herein may be effected, none of which is inherently superior to the other in that any vehicle to be utilized is a choice dependent upon the context in which the vehicle will be deployed and the specific concerns (e.g., speed, flexibility, or predictability) of the implementer, any of which may vary. 
         [0071]    Those skilled in the art will recognize that optical aspects of implementations may involve optically-oriented hardware, software, and or firmware. The foregoing detailed description has set forth various embodiments of the devices and/or processes via the use of block diagrams, flowcharts, and/or examples. Insofar as such block diagrams, flowcharts, and/or examples contain one or more functions and/or operations, it will be understood as notorious by those within the art that each function and/or operation within such block diagrams, flowcharts, or examples can be implemented, individually and/or collectively, by a wide range of hardware, software, firmware, or virtually any combination thereof. Several portions of the subject matter described herein may be implemented via Application Specific Integrated Circuits (ASICs), Field Programmable Gate Arrays (FPGAs), digital signal processors (DSPs), or other integrated formats. However, those skilled in the art will recognize that some aspects of the embodiments disclosed herein, in whole or in part, can be equivalently implemented in standard integrated circuits, as one or more computer programs running on one or more computers (e.g., as one or more programs running on one or more computer systems), as one or more programs running on one or more processors (e.g., as one or more programs running on one or more microprocessors), as firmware, or as virtually any combination thereof, and that designing the circuitry and/or writing the code for the software and/or firmware would be well within the skill of one of skill in the art in light of this disclosure. In addition, those skilled in the art will appreciate that the mechanisms of the subject matter described herein are capable of being distributed as a program product in a variety of forms, and that an illustrative embodiment of the subject matter described herein applies equally regardless of the particular type of signal bearing media used to actually carry out the distribution. Examples of a signal bearing media include, but are not limited to, the following: recordable type media such as floppy disks, hard disk drives, CD ROMs, digital tape, flash drives, SD cards, solid state fixed or removable storage, and computer memory. 
         [0072]    In a general sense, those skilled in the art will recognize that the various aspects described herein which can be implemented, individually and/or collectively, by a wide range of hardware, software, firmware, or any combination thereof can be viewed as being composed of various types of “circuitry.” Consequently, as used herein “circuitry” includes, but is not limited to, electrical circuitry having at least one discrete electrical circuit, electrical circuitry having at least one integrated circuit, electrical circuitry having at least one application specific integrated circuit, circuitry forming a general purpose computing device configured by a computer program (e.g., a general purpose computer configured by a computer program which at least partially carries out processes and/or devices described herein, or a microprocessor configured by a computer program which at least partially carries out processes and/or devices described herein), circuitry forming a memory device (e.g., forms of random access memory), and/or circuitry forming a communications device (e.g., a modem, communications switch, or optical-electrical equipment). 
         [0073]    Those skilled in the art will recognize that it is common within the art to describe devices and/or processes in the fashion set forth herein, and thereafter use standard engineering practices to integrate such described devices and/or processes into larger systems. That is, at least a portion of the devices and/or processes described herein can be integrated into a network processing system via a reasonable amount of experimentation. 
         [0074]    The foregoing described aspects depict different components contained within, or connected with different other components. It is to be understood that such depicted architectures are merely exemplary, and that in fact many other architectures can be implemented which achieve the same functionality. In a conceptual sense, any arrangement of components to achieve the same functionality is effectively “associated” such that the desired functionality is achieved. Hence, any two components herein combined to achieve a particular functionality can be seen as “associated with” each other such that the desired functionality is achieved, irrespective of architectures or intermedial components. Likewise, any two components so associated can also be viewed as being “operably connected,” or “operably coupled,” to each other to achieve the desired functionality. 
         [0075]    Embodiments of methods and systems for interpreting beam direction imaging data have been described. The following claims are directed to said embodiments, but do not preempt application performance testing in the abstract. Those having skill in the art will recognize numerous other approaches to interpreting beam direction imaging data possible and/or utilized commercially, precluding any possibility of preemption in the abstract. However, the claimed system improves, in one or more specific ways, the operation of a machine system for interpreting beam direction imaging data, and thus distinguishes from other approaches to the same problem/process in how its physical arrangement of a machine system determines the system&#39;s operation and ultimate effects on the material environment. The terms used in the appended claims are defined herein in the glossary section, with the proviso that the claim terms may be used in a different manner if so defined by express recitation.