Abstract:
Strain is directly estimated in ultrasound elasticity imaging without computing displacement or resorting to spectral analysis. Conventional ultrasound elasticity imaging relies on calculating displacement and strain is computed from a derivative of the displacement. However, for typical parameter values used in ultrasound elasticity imaging, the displacement can be as large as a hundred times or displacement differences. If a tiny error in the calculation of displacement occurs, this could drastically affect the calculation of strain. By directly estimating strain, image quality is enhanced and the reduction in computational effort facilitates commercialization to aid in diagnosing disease or cancerous conditions.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
       [0001]    This application claims the benefit of U.S. Provisional Application No. 60/604,886, filed Aug. 27, 2004, which is incorporated herein by reference. 
     
    
     TECHNICAL FIELD 
       [0002]    The present invention generally relates to measuring strain in biological structures. 
       BACKGROUND 
       [0003]    Medical examination of the human body by touch, called palpation, reveals external abnormalities that may instigate additional investigation. In breast exams or prostate exams, clinicians use palpation to detect abnormal stiffness, which may point to diseased or cancerous bodily structures. Scientists have measured changes in the stiffness of biological structures by measuring their elasticity moduli ex vivo. For example, scientists have found that infiltrating ductile carcinomas are much stiffer than any other breast tissues, and prostate cancers are stiffer than normal prostate tissues. Stiff tissue has a high elasticity modulus and shows less strain under applied force than soft tissue. Thus, by applying compression and estimating the strain, stiffness in bodily structures can be obtained. 
         [0004]    An early medical application of ultrasound imaging included the monitoring of blood flow by the use of the displacement technique. Strain can be computed from a derivative of the displacement. With this understanding, conventional elasticity imaging is based on estimating displacements from echoed ultrasound signals to calculate strain in bodily structures. However, conventional elasticity imaging suffers from both image quality and computational burden, which have prevented successful commercialization of the technology for clinicians to use. Because of the domination of the displacement technique, many attempts at improving the calculation of strain have focused on measuring displacement ever more accurately. 
         [0005]    In medical ultrasound, strain in an object can be estimated using ultrasound echo signals acquired before and after object compression. Strain represents the amount of deformation and is defined as: 
         [0000]    
       
         
           
             
               
                 
                   
                     Δ 
                      
                     
                         
                     
                      
                     L 
                   
                   L 
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where L is the length of an object and ΔL is the amount of change in the object length under uni-axial force. Conventional ultrasonic strain estimation methods are based on estimating displacements between ultrasound signals acquired before and after compression. Once displacements are estimated, strain is computed from a spatial derivative of the displacement as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       d 
                       2 
                     
                     - 
                     
                       d 
                       1 
                     
                   
                   L 
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where d 1  and d 2  are the displacements at two different locations and L is the distance between these two locations (defined as the strain sample length). 
         [0006]    For conventional strain estimation, displacements can be very large (e.g., more than a hundred micrometer), whereas the displacement difference converted to strain is very small (e.g., few micrometers). Consequently, if the strain were computed from large displacements, a tiny error in displacement estimation could magnify the error in the computation of the strain. Because strain constitutes such a small part of the displacement, scientists have resorted to displacement interpolation techniques to find the strain. To obtain desired results, hundreds of points may need to be interpolated, resulting in onerous computational effort, hindering production of marketable machinery. 
       SUMMARY 
       [0007]    In accordance with this invention, a computer-readable medium, system, and method for calculating strain is provided. The method form of the invention includes a computer-implemented method for calculating strain that comprises transmitting and acquiring ultrasound signals echoed from an object before the object is compressed and after the object is compressed. The method further comprises estimating an amount of deformation of the object by using the phase of temporal and spatial correlation between ultrasound baseband signals from different frames so as to determine strain without having to estimate displacement. 
         [0008]    In accordance with further aspects of this invention, a system form of the invention includes a system for calculating strain that comprises a temporal correlator for correlating groups of pixels of interest between two frames of baseband signals containing deformation information of an object. The system further comprises a spatial correlator for correlating information produced by the temporal correlator to render a complex number that contains information pertaining to an angular strain. The system yet further comprises a phase computer that calculates the phase of the complex number to obtain the angular strain. The system also comprises a gross motion vector estimator that locates correlated baseband signals from different frames and the results are provided to the temporal correlator. The system additionally comprises an angular strain converter, which takes the angular strain and converts it to strain. 
         [0009]    In accordance with further aspects of this invention, a computer-readable medium form of the invention includes a storable computer-readable medium having computer-executable instructions thereon for implementing a method for calculating strain, which comprises transmitting and acquiring ultrasound signals echoed from an object before the object is compressed and after the object is compressed. The computer-readable medium also comprises estimating an amount of deformation of the object by using the phase of temporal and spatial correlation between ultrasound baseband signals from different frames so as to determine strain without having to estimate displacement. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0010]    The foregoing aspects and many of the attendant advantages of this invention will become more readily appreciated as the same become better understood by reference to the following detailed description, when taken in conjunction with the accompanying drawings, wherein: 
           [0011]      FIG. 1  is a block diagram illustrating an exemplary system for directly calculating strain using echoed ultrasound signals from bodily structures; 
           [0012]      FIG. 2  is a block diagram illustrating an exemplary system for calculating an angular strain, which can be converted directly to strain; 
           [0013]      FIG. 3  is a pictorial diagram illustrating the calculation of gross motion vectors involving the use of different frames of baseband signals obtained at different times; and 
           [0014]      FIGS. 4A-4H  are process diagrams illustrating a method for computing strain in bodily structures, including the computation of temporal and spatial correlation. 
       
    
    
     DETAILED DESCRIPTION 
       [0015]      FIG. 1  illustrates a system  100  that directly calculates strain using baseband signals without the use of displacement. Transmitted ultrasound  102  is produced when an ultrasound transducer transmits pulses into an object  104  to be imaged. Received ultrasound  106  represents the ultrasound signals that are reflected back from boundaries of the object and/or scattered by scatterers in the object  104  to be imaged. The object  104  is compressed or expanded and its deformation is estimated by the system  100  using the received ultrasound  106 . The compression or expansion source of force can be external, such as by a mechanical device (e.g., a vibrator) or by hand; or the compression or expansion source of force can be internal, such as by naturally occurring arterial pulsation. 
         [0016]    The received ultrasound  106  is converted into radio-frequency electrical signals by the transducer that originally produced the transmitted ultrasound  102 . The radio-frequency electrical signals have a carrier frequency, which is removed by a demodulator  108 , resulting in complex baseband signals  110 . As will be appreciated by one skilled in the art, any suitable ultrasound signals can be used in various embodiments of the present invention to calculate strain without the use of displacement. Although the following discussion uses the baseband signals  110  for calculation of strain, various embodiments of the present invention are not limited to the use of baseband signals, and other signals, such as radio-frequency signals, can be used for strain estimation. 
         [0017]    The baseband signals  110  are provided to an angular strain estimator  112  which calculates the angular strain connected with the object prior to compression and after compression. Preferably, at least two pieces of ultrasound baseband data acquired at different times are used in estimating the angular strain by the angular strain estimator  112 . Also preferably, correlated ultrasound baseband signals obtained from different frames are used to estimate strain without displacement computations. The resulting strain calculation that is derived from the angular strain can be used by clinicians to determine object strain  114 , among other things. The object includes tissues, aggregates of cells, internal or external organs, and other bodily or implanted structures. 
         [0018]      FIG. 2  illustrates the angular strain estimator  112  in more detail. The baseband signals  110  are presented to a gross motion vector estimator  200  and a temporal correlator  202 . The gross motion vector estimator  200  estimates gross motion vectors to find locations of correlated signals in pre-compression frames and post-compression frames in the baseband signals  110 . A gross motion vector preferably indicates a distance in terms of the number of samples between two correlated signals from different frames. Any suitable motion estimation technique, such as block matching techniques that include cross-correlation or the sum of absolute differences, can be used by the gross motion vector estimator  200  to estimate gross motion vectors. Many other suitable motion estimation techniques can be used, including those that perform a full block matching search (but those that perform a coarse block matching search will suffice as well). As will be appreciated by one skilled in the art, the gross motion vectors roughly indicate where a portion of an object being imaged or the imaged object is located between two frames of the baseband signals and need not be exact. 
         [0019]    The temporal correlator  202  receives the gross motion vectors from the gross motion vector estimator  200  and baseband signals  110  to calculate a temporal correlation. The temporal correlator  202  then presents the temporal correlation to a spatial correlator  204 , which calculates the spatial correlation. The spatial correlation is then presented to a phase computer  206  which calculates the argument or the phase of a complex number representing the spatial correlation. Alternatively, aliasing could be used to help compute the phase instead of using the spatial correlation. The phase is then presented to an angular strain converter  208 , which converts the angular strain to the strain of the object imaged by the ultrasound signals. Alternatively, the angular strain can be directly displayed on a screen or used for further characterization of the imaged object (instead of strain estimation having to be used). Additionally, spatial filters, such as a boxcar filter or median filter, can be applied at any step to estimate strain in various embodiments of the present invention. Moreover, center frequency estimation or frequency variance compensation can be applied for more accurate conversion of angular strain to strain. 
         [0020]      FIG. 3  illustrates pictorially the calculation of a gross motion vector. A frame  302  has a nomenclature “FRAME  1 ” taken at time T 1  of the baseband signals  110 . Another frame  304  has the nomenclature “FRAME  2 ” taken at time T 2  of the baseband signals  10 . Within the frame  302  is a group  302   a  of pixels of interest that represent a portion of an object  104  or the object  104  being imaged. The centroid of the group  302   a  can be identified by indexes I, J, which define the dimensions of the frame  302 . The frame  304  occurs at a time different from the time at which the frame  302  occurs. Similar to the frame  302 , the frame  304  is indexed by indices I, J, which together define the dimensions of the frame  304 . Frames  302 ,  304  denote distinct periods of time during which ultrasound samples are obtained from the baseband signals  110 . Each frame  302 ,  304  can be displayed by taking its envelope information and rendering the envelope information on a grayscale screen. Each frame is a collection of information formed from complex numbers. 
         [0021]    A group  304   a  consists of pixels of interest that can be determined to be correlated with the group  302   a  by any suitable motion estimation technique. If the group  302   a  and the group  304   a  are correlated, a gross motion vector  308  can be defined to determine the magnitude by which the group  302   a  in the frame  302  has moved by looking at the location of the group  304   a  in the frame  304 . To provide a concrete example, suppose that the group  302   a  is located at (2, 3) and the group  304   a  is located at (2, 1). Because the groups  302   a ,  304   a  are correlated, the number 2 indicates the number of samples separating the group  302   a  and the group  304   a . Thus, the gross motion vector for the group  302   a  at (2, 3) is 2. However, the gross motion vector  308  can be of any suitable magnitude to show the extent to which the group  302   a  has moved from its original position. In those cases where the object deforms not only axially but also in other directions, a window and a search range may be defined to reflect the movement of the object in other directions. Although  FIG. 3  illustrates only axial movement by the group  302   a , one with ordinary skill in the art would appreciate that the group  302   a  consists of pixels of interest that may move not only axially but can also be moved laterally as well. 
         [0022]      FIGS. 4A-4H  illustrates a method  400  for computing the strain of bodily structures. The method  400  or a portion of it can be implemented by software executing on any programmable computer or by hardware, such as application specific integrated circuits, field programmable gate arrays, field programmable logic devices. From a start block, the method  400  proceeds to a set of method steps  402  defined between a continuation terminal (“Terminal A”) and another continuation terminal (“Terminal B”). The set of method steps  402  calculates gross motion vectors to ascertain the whereabouts of pixels of interest in two different frames of the baseband signals  110 . Artificial compression or natural compression of objects to be imaged can occur and strain can be calculated. Various embodiments of the present invention are not limited to artificial compression, such as by pressing on an object to be imaged. 
         [0023]    From Terminal A ( FIG. 4B ), the method  400  proceeds to block  408  where ultrasound signals are transmitted into an object of interest. See  FIG. 1 . Echoed radio-frequency ultrasound signals are received by the method at block  410 . Next at block  412 , the method demodulates the echoed radio-frequency ultrasound signals into baseband signals. The method  400  then performs correlation analysis between two different frames of the baseband signals by using any suitable motion estimation technique. See block  414 . At block  416 , the distance in terms of the number of samples between correlated groups of pixels from different frames is calculated. Next at block  418 , the gross motion vector (N) is formed from the location of the correlated groups of pixels and the calculated distance in samples. The method  400  then continues to Terminal B. 
         [0024]    From Terminal B ( FIG. 4A ), the method  400  proceeds to another set of method steps  404  defined between a continuation terminal (“Terminal C”) and another continuation terminal (“Terminal D”). The set of method steps  404  calculates the temporal correlation of pixels of interest in two different frames of the baseband signals  110 . 
         [0025]    From Terminal C ( FIG. 4C ), the method  400  proceeds to block  420  where a lateral window size (W TR, LAT ) for temporal correlation is defined, such as three samples. Next at block  422 , the lateral window size (W TR, LAT ) is used to calculate a range of a first summation and is indexed by a variable H. One end of the lateral range is equated to 
         [0000]    
       
         
           
             
               
                 
                   h 
                   = 
                   
                     - 
                     
                       
                         
                           W 
                           
                             TR 
                             , 
                             LAT 
                           
                         
                         - 
                         1 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    See block  424 . At block  426 , another end of the lateral range is equated to 
         [0000]    
       
         
           
             
               
                 
                   h 
                   = 
                   
                     
                       
                         
                           W 
                           
                             TR 
                             , 
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                         - 
                         1 
                       
                       2 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0026]    The method  400  continues to block  428  where an axial window size (W TR, AXI ) for temporal correlation is defined, such as 39 samples. Preferably, a window size between 0.5-2 mm is used; for a sampling frequency of 30 MHz, a 1 mm window size is approximately 39 samples. At block  430 , the axial window size (W TR, AXI ) is used to calculate a range of a second summation and is indexed by a variable K. One end of the axial range is equated to 
         [0000]    
       
         
           
             
               
                 
                   k 
                   = 
                   
                     - 
                     
                       
                         
                           W 
                           
                             TR 
                             , 
                             AXI 
                           
                         
                         - 
                         1 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    See block  432 . Next at block  434  another end of the axial range is equated to 
         [0000]    
       
         
           
             
               
                 
                   k 
                   = 
                   
                     
                       
                         W 
                         
                           TR 
                           , 
                           AXI 
                         
                       
                       - 
                       1 
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The method  400  then continues at another continuation terminal (“Terminal C 1 ”). 
         [0027]    From Terminal C 1  ( FIG. 4D ), the method continues to block  436 , at which the value (Z 1 ) of the baseband signal Z(I+H, J+K, T) is obtained, where Z( ) is the baseband signal at the time index T, I is the lateral (width) index, and J is the axial (depth) index. At block  438 , another value (Z 2 ) of the baseband signal Z(I+H, J+K+N(I, J, T), T+ΔT) is obtained, where Z( ) is the baseband signal at the time index T+ΔT and N( ) is a gross motion vector at (I, J, T). In an embodiment where the gross motion vector is not restricted to the distance of correlated signals in the axial direction only, the gross motion vector can represent three dimensional movement of an object (N 1 , N 2 , N 3 ), and in which case the value (Z 2 ) of the baseband signal Z(I+H+N 1 (I, J, T), J+K+N 2 (I, J, T), T+ΔT) is obtained. The method  400  then proceeds to block  440  where the method obtains a product by multiplying Z 1  with a conjugate of Z 2 . The product is a summand at (I, J, T). See block  442 . The summand is added to the second summation. See block  444 . The method  400  then continues to another continuation terminal (“Terminal C 2 ”). 
         [0028]    From Terminal C 2  ( FIG. 4E ), the method  400  proceeds to decision block  446  where a test is performed to determine whether the axial index K has reached its limit. If the answer is NO to the test at decision block  446 , the axial index K is incremented. See block  448 . The method  400  then proceeds to Terminal C 1  where it loops back to block  436  and repeats the above-discussed processing steps. Otherwise, if the answer to the test at decision block  446  is YES, the method  400  continues to another decision block  450  where another test is performed to determine whether the lateral index H has reached its limits. If the answer is NO to the test at decision block  450 , the lateral index H is incremented and the axial index K is reset. See block  452 . The method then continues to Terminal C 1  where it loops back to block  436  and the above-identified processing steps are repeated. Otherwise, if the answer to the test at decision block  450  is YES, the method  400  continues to Terminal D. 
         [0029]    From Terminal D ( FIG. 4A ), the method  400  proceeds to a set of method steps  406  defined between a continuation terminal (“Terminal E”) and another continuation terminal (“Terminal F”). The set of method steps  406  calculates spatial correlation using the results of the temporal correlation and directly computes strain. 
         [0030]    From Terminal E ( FIG. 4F ), the method  400  proceeds to decision block  454  where a test is performed to determine whether there is a desire for an alternate spatial correlation calculation. If the answer to the test at decision block  454  is YES, the method  400  continues to another continuation terminal (“Terminal E 1 ”). Otherwise, if the answer to the test at decision block  454  is NO, the method  400  proceeds to block  456  where a value (TR 1 ) of the temporal correlation TR(I, J+L, T) is obtained where L is the strain sample length. Next at block  458 , another value (TR 2 ) of the temporal correlation TR(I, J, T) is obtained. At block  460 , a product for the spatial correlation is obtained by multiplying TR 1  and the conjugate of TR 2 . The method  400  then continues to another continuation terminal (“Terminal E 2 ”). 
         [0031]    From Terminal E 1  ( FIG. 4G ), a subtrahend is formed from the argument of a complex number, which is the temporal correlation TR(I, J, T). See block  462 . At block  464 , a minuend is formed from the argument of another complex number, which is the temporal correlation TR(I, J+L, T). The method  400  then proceeds to block  466  where the difference between the minuend and the subtrahend is a temporary value calculated at dimension (I, J, T). If the temporary value is greater than π, the angular strain Φ S  the sum of −2π and the temporary value. See block  468 . Otherwise, at block  470 , if the temporary value is less than −π, the angular strain Φ S  is the sum of 2π and the temporary value. See block  470 . Otherwise, at block  472 , the angular strain Φ S  is the temporary value. The method then continues to another continuation terminal (“Terminal E 3 ”). 
         [0032]    From Terminal E 2  ( FIG. 4H ), the method  400  proceeds to block  474  where the angular strain is calculated by taking the argument of a complex number represented by the spatial correlation ARG(STR(I, J, T)). The method  400  continues to Terminal E 3  which then continues to block  476  where the phase corresponding to the strain sample length Φ L  is calculated by taking the product (4πF 0 L) and dividing it by C, where L is the strain sample length, F 0  is the ultrasound carrier frequency, and C is the sound velocity. At block  478 , the strain is obtained by dividing the angular strain Φ S  by the phase corresponding to the strain sample length Φ L . The method  400  then continues to Terminal F where it terminates execution. 
         [0033]    While the preferred embodiment of the invention has been illustrated and described, it will be appreciated that various changes can be made therein without departing from the spirit and scope of the invention. 
       APPENDIX 
       [0034]    As in Eq. (2), strain can be computed by taking a spatial derivative of the displacement. Utilizing the relationship between the displacement and the corresponding phase (angular displacement), φ d , strain can be computed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                         ( 
                         
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                           , 
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                           , 
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                          
                         
                             
                         
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                         π 
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                           f 
                           0 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
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                           strain 
                           = 
                           
                             
                               
                                 d 
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                                     , 
                                     
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                                       + 
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                                  
                                 
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                             L 
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                               
                           
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                                   φ 
                                   d 
                                 
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                                   d 
                                 
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                                     , 
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                               L 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               
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                                 s 
                               
                                
                               
                                 ( 
                                 
                                   i 
                                   , 
                                   j 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                             
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                               L 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
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                      
                     
                         
                     
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         [0000]    where φ s  is the phase corresponding to strain (defined as angular strain) and φ I , is the phase corresponding to the strain sample length L. The angular displacement, φ d , can be expressed as: 
         [0000]      φ d ( i,j,t )=2 πm ( i,j,t )+φ m ( i,j,t ), 
         [0000]      −π≦φ m ( i,j,t )&lt;π  (A2) 
         [0000]    where m(i,j,t) is an integer value. Then, the change in angular displacements can be derived by: 
         [0000]      φ d ( i,j +1 ,t )=2π m ( i,j+ 1 ,t )+φ m ( i,j+ 1 ,t )→φ d2 =2π m   2 +φ m2 , 
         [0000]      φ d ( i,j,t )=2π m ( i,j,t )+φ m ( i,j,t )→φ d1 =2π m   1 +φ m1 , 
         [0000]      φ d2 −φ d1 2π m   2 +φ m2 −2π m   1 −φ m1 =2π( m   2   −m   1 )+φ m2 −φ m1   (A3) 
         [0000]    Since the range of φ m2 −φ m1  is from −2π to 2π, φ m2 −φ m1  can be expressed as: 
         [0000]      φ m2 −φ m1 =2π k+φ   k , 
         [0000]        k= 0,±1, 
         [0000]      −π≦φ k &lt;π  (A4) 
         [0000]    From Eqs. (A1), (A3), and (A4), the angular strain, φ s , is 
         [0000]      φ s =φ d2 −φ d1 =2π( m   2   −m   1   +k )+φ k , 
         [0000]        k= 0,±1, 
         [0000]      −π≦φ k &lt;π  (A5) 
         [0000]    Due to the decorrelation noise for large compression levels, ultrasonic strain estimation is typically limited to small compression levels, such as up to 3%. The angular strain corresponding to the typical compression range in ultrasound elasticity imaging, such as up to 3%, is small enough to be limited within the range of [−π,π]. For example, with the strain sample length (L) of 200 μm and the ultrasound center frequency (f 0 ) of 7.5 MHz, the angular strain corresponding to up to 25% strain does not exceed the range of [−π,π]. This implies that the angular strain to be estimated, φ s , is same as φ k : 
         [0000]      −π&lt;φ s =φ d2 −φ d1 =2π( m   2   −m   1   +k )+φ k &lt;π, 
         [0000]                2π( m   2   −m   1   +k )=0 
         [0000]                φ s =φ k   (A6) 
         [0000]    φ k  can be computed from φ m1  and φ m2  based on their relationship defined in Eq. (A4). If a phase-aliasing operator, alias[ ], is defined as: 
         [0000]      φ2π h+φ   h , −π&lt;φ h &lt;π, 
         [0000]      alias[φ]=φ h   (A7) 
         [0000]    then φ k  is the result of the phase aliasing operation using φ m1  and φ m2  as: 
         [0000]      φ k =alias[φ m2 −φ m1 ]  (A8) 
         [0035]    In medical ultrasound systems, φ m  can be obtained from the phase of the temporal correlation function using correlated baseband signals (TR(i, j, t) in Eq. (A10)) as: 
         [0000]      φ m ( i,j,t )= arg ( TR ( i,j,t ))  (A9) 
       Where TR(i, j, t) is 
       [0036]    
       
         
           
             
               
                 
                   
                     TR 
                      
                     
                       ( 
                       
                         i 
                         , 
                         j 
                         , 
                         t 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         h 
                         = 
                         
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                                 W 
                                 
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                                   , 
                                   LAT 
                                 
                               
                               - 
                               1 
                             
                             2 
                           
                         
                       
                       
                         
                           
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                               , 
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                           - 
                           1 
                         
                         2 
                       
                     
                      
                     
                         
                     
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                           k 
                           = 
                           
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                                   W 
                                   
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                                     , 
                                     AXI 
                                   
                                 
                                 - 
                                 1 
                               
                               2 
                             
                           
                         
                         
                           
                             
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                                 , 
                                 AXI 
                               
                             
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                                 + 
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                                 + 
                                 
                                   n 
                                    
                                   
                                     ( 
                                     
                                       i 
                                       , 
                                       j 
                                       , 
                                       t 
                                     
                                     ) 
                                   
                                 
                               
                               , 
                               
                                 t 
                                 + 
                                 
                                   Δ 
                                    
                                   
                                       
                                   
                                    
                                   t 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     A 
                      
                     
                         
                     
                      
                     10 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    From Eqs. (A8), and (A9), the phase of the temporal and spatial correlation function (STR(i, j, t) in Eq. (A12)) is equivalent to φ k  as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           arg 
                            
                           
                             ( 
                             
                               STR 
                                
                               
                                 ( 
                                 
                                   i 
                                   , 
                                   j 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           arg 
                            
                           
                             ( 
                             
                               
                                 TR 
                                  
                                 
                                   ( 
                                   
                                     i 
                                     , 
                                     
                                       j 
                                       + 
                                       1 
                                     
                                     , 
                                     t 
                                   
                                   ) 
                                 
                               
                               × 
                               
                                 
                                   TR 
                                   * 
                                 
                                  
                                 
                                   ( 
                                   
                                     i 
                                     , 
                                     j 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           alias 
                            
                           
                             [ 
                             
                               
                                 arg 
                                  
                                 
                                   ( 
                                   
                                     TR 
                                      
                                     
                                       ( 
                                       
                                         i 
                                         , 
                                         
                                           j 
                                           + 
                                           1 
                                         
                                         , 
                                         t 
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 arg 
                                  
                                 
                                   ( 
                                   
                                     TR 
                                      
                                     
                                       ( 
                                       
                                         i 
                                         , 
                                         j 
                                         , 
                                         t 
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           alias 
                            
                           
                             [ 
                             
                               
                                 
                                   φ 
                                   m 
                                 
                                  
                                 
                                   ( 
                                   
                                     i 
                                     , 
                                     
                                       j 
                                       + 
                                       1 
                                     
                                     , 
                                     t 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   φ 
                                   m 
                                 
                                  
                                 
                                   ( 
                                   
                                     i 
                                     , 
                                     j 
                                     , 
                                     t 
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             φ 
                             k 
                           
                            
                           
                             ( 
                             
                               i 
                               , 
                               j 
                               , 
                               t 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     A 
                      
                     
                         
                     
                      
                     11 
                   
                   ) 
                 
               
             
           
         
       
     
       Where STR(i, j, t) is 
       [0037]        STR ( i,j,t )= TR ( i,j+L,t )× TR* ( i,j,t )  (A12) 
         [0000]    From Eqs. (A6) and (A11), the angular strain, φ s , is same as the phase of the temporal and spatial correlation function as: 
         [0000]      φ s ( i,j,t )=φ k ( i,j,t )= arg ( STR ( i,j,t ))  (A13)