BACKGROUND OF INVENTION 
     The invention relates generally to Magnetic Resonance Imaging (MRI) systems and more specifically to a method and apparatus for reconstructing images in parallel MRI systems. 
     Generally, MRI is a well-known imaging technique. A conventional MRI device establishes a homogenous magnetic field, for example, along an axis of a person&#39;s body that is to undergo MRI. This homogeneous magnetic field conditions the interior of the person&#39;s body for imaging by aligning the nuclear spins (in atoms and molecules forming the body tissue) along the axis of the magnetic field. If the orientation of the nuclear spins is perturbed out of alignment with the magnetic field, the nuclei attempt to realign their spins with the axis of the magnetic field. Perturbation of the orientation of nuclear spins may be caused by application of radio frequency (RF) pulses tuned to the Larmor frequency. During the realignment process, the nuclei precess about the axis of the magnetic field and emit electromagnetic signals that may be detected by one or more RF detector coils placed on or about the person. 
     The frequency of the magnetic resonance (MR) signal emitted by a given precessing nucleus depends on the strength of the magnetic field at the nucleus&#39; location. As is well known in the art, it is possible to distinguish signals originating from different locations within the person&#39;s body by applying a gradient to the magnetic field across the person&#39;s body. For the sake of convenience, direction of this field gradient may be referred to as the left-to-right direction. Signals of a particular frequency acquired during application of the field gradient may be assumed to originate at a given position within the field gradient, and hence at a given left-to-right position within the person&#39;s body. The application of such a field gradient is also referred to as frequency encoding. 
     However, the application of a field gradient does not allow for two-dimensional spatial encoding, since all nuclei at a given left-to-right position experience the same field strength, and hence emit signals of the same frequency. Accordingly, the application of a frequency-encoding gradient, by itself, does not make it possible to discern signals originating from the top versus signals originating from the bottom of the person at a given left-to-right position. Spatial encoding has been found to be possible in this second direction by application of gradients of varied strength in a perpendicular direction prior to acquisition of the signal, to thereby twist the phase of the nuclear spins by varied amounts. The application of such additional gradients is referred to as phase encoding. 
     Frequency-encoded data sensed by the RF detector coils after a phase encoding step is stored as a line of data in a data matrix known as the k-space matrix. Multiple phase encoding steps are performed in order to fill the multiple lines of the k-space matrix. An image may be generated from this matrix by performing a two-dimensional Fourier transformation of the matrix to convert this frequency information to spatial information representing the distribution of nuclear spins or density of nuclei of the image material. 
     Alternatively, spatial encoding can be performed in three dimensions by applying phase-encoding gradients in two orthogonal directions, followed by a frequency-encoding gradient in the third orthogonal direction, with signals acquired during the frequency-encoding gradient, in order to generate a three-dimensional matrix of k-space data. Three-dimensional Fourier transformation of the matrix will then convert this frequency information to spatial information representing the distribution of nuclear spins or density of nuclei within a volume of the image material. 
     Imaging time is largely a function of desired signal-to-noise ratio (SNR) and the speed with which the MRI device can fill the k-space matrix. In conventional MRI, the k-space matrix is filled one line at a time. Although many improvements have been made in this general area, the speed with which the k-space matrix may be filled is limited. To overcome these inherent limits, several techniques have been developed to effectively simultaneously acquire multiple lines of data for each application of a magnetic field gradient. These techniques, which may collectively be characterized as “parallel imaging techniques”, use spatial information from arrays of RF detector coils to substitute for the encoding which would otherwise have to be obtained in a sequential fashion using field gradients alone. The use of multiple detectors has been shown to multiply imaging speed, without increasing gradient switching rates or RF power deposition. Parallel imaging techniques fall into one of two categories. They can fill in the omitted k-space lines prior to Fourier transformation, by constructing a weighted combination of neighboring lines acquired by the different RF detector coils. Or, they can first Fourier transform the limited k-space data to produce an aliased image from each coil, and then unfold the aliased signals by a linear transformation of the superimposed pixel values. In either case, the reconstructed images tend to suffer from incomplete removal of aliasing artifacts, especially for large speedup factors. In images corrupted by aliasing, the edges of the image are wrapped into the center of the image. 
     Another problem in the reconstruction of the image is degraded signal-to-noise ratio (SNR) in some regions of the image. One reason for both of the above effects is the overlapping of the sensitivity profiles of the coils in the array. Insufficiently distinct sensitivity profiles can lead to both residual aliasing and degraded regional SNR. One way to overcome the above problems is to keep the sensitivity profile of each coil orthogonal to all of the others. 
     It is therefore desirable to reconstruct an artifact-free image, by reducing the effects of aliasing and degraded SNR. 
     SUMMARY OF INVENTION 
     Briefly, in accordance with one embodiment of the invention, a reconstruction method for use in a parallel MRI system is provided. The MRI system comprises a plurality of MR detector coils arranged in an array and each coil has a corresponding spatial sensitivity profile. The method comprises detecting a plurality of MR signals from the plurality of MR coils and processing the detected MR signals with at least one filter bank to reconstruct at least one image. 
     In an alternate embodiment, a parallel Magnetic Resonance Imaging (MRI) system is provided for reconstruction of images. The MRI system comprises an array of magnetic resonance (MR) detector coils arranged in an array for detecting a plurality of MR signals, each of the coils having a corresponding spatial sensitivity profile and a processing means for processing the plurality of MR signals with at least one filter bank to reconstruct at least one image. 
    
    
     BRIEF DESCRIPTION OF DRAWINGS 
     These and other features, aspects, and advantages of the present invention will become better understood when the following detailed description is read with reference to the accompanying drawings in which like characters represent like parts throughout the drawings, wherein: 
     FIG. 1 is a schematic block diagram of an exemplary MR imaging system suitable for use with the present invention embodiments; 
     FIG. 2 is a block diagram illustrating a method for reconstructing images by a MRI system; 
     FIG. 3 is an exemplary embodiment of a lapped transform filter bank used for reconstruction of images when the sensitivity profiles of the array elements overlap; 
     FIG. 4 is a schematic block diagram of an exemplary embodiment for a strip array; 
     FIG. 5 is a block diagram of a mathematical equivalent of the embodiment shown in FIG. 4; and, 
     FIG. 6 is an exemplary block diagram of a reconstruction for the embodiments shown in FIGS.  4  and  5 . 
    
    
     DETAILED DESCRIPTION 
     FIG. 1 is an example embodiment of a magnetic resonance (MR) imaging system  100 . MRI system  100  includes a computer  110 , which controls gradient coil power amplifiers  130  through a pulse control module  120 . The pulse control module  120  and the gradient amplifiers  130  together produce the proper gradient waveforms Gx, Cy, and Gz, for either a spin echo, a gradient recalled echo pulse sequence, a fast spin echo, or other type of pulse sequences. The gradient waveforms are connected to gradient coils  16 , which are positioned around the bore of an MR magnet assembly  130  so that gradients Gx, Gy, and Gz are impressed along their respective axes on the polarizing magnetic field B 0  from magnet assembly  140 . 
     The pulse control module  120  also controls a radio frequency synthesizer  178  that is part of RF transceiver system  170 , portions of which are enclosed by dashed line block. The pulse control module  120  also controls an RF modulator  179 , which modulates the output of the radio frequency synthesizer  178 . The resultant RF signals, amplified by RF power amplifier  180  and applied to a RF coil assembly  147  through transmit/receive switch  160 , are used to excite the nuclear spins of the imaged object (not shown). 
     In embodiments of the present invention, RF coil assembly  147  is a RF detector array consisting of multiple detector elements, such as loops or conductive strips and may also be configured as an array of multiple coils or conductive strips. Other conductive materials and structures, for example copper rods, pipes, wires or other similar line structures, may also be used as detector elements. In a further embodiment, RF coil assembly  147  is an MRI phased array. In an embodiment, the detector elements may be non-overlapping or overlapping. 
     The MR signals from the excited nuclei of the imaged object are picked up by RF coil assembly  147  and presented to preamplifier  171  through transmit/receive switch  160 , to be amplified and then processed by a quadrature phase detector  173 . The detected signals are digitized by a high speed A/D converter  175  and applied to computer  110  for processing to produce MR images of the object. Computer  110  also controls shimming coil power supplies  141  to power shimming coil assembly  143 . The manner in which computer  110  processes the detected signal to reconstruct an image is described in further detail below. 
     FIG. 2 is a block diagram illustrating a method for reconstructing images by an MRI system. FIG. 2 is shown comprising spatial filter bank  210 , decimated gradient phase encoding  220 , interpolating system  230  and synthesis filter bank  240 . Each component is described in further detail below. It is to be noted that the array of RF detector coils forms the spatial filter bank. 
     Spatial filter bank  210  is formed with the respective sensitivity profiles for spatially filtering the plurality of detected MR signals. In an embodiment, the sensitivity profiles are represented by F(z), where F(z) equals:                F        (   z   )       =     (             F   1          (   z   )                   F   2          (   z   )               ⋮               F   N          (   z   )             )             Equation                   (   1   )                                  
     where F 1 (z), . . . , F N (z) are the sensitivity profiles of coil 1 to coil N. 
     The overall detected signal S′(z) after the spatial filter bank F(z) can be                described                 as        :                       S   ′          (   z   )         =         F        (   z   )            S        (   z   )         =     (             S   1   ′          (   z   )                   S   2   ′          (   z   )               ⋮               S   N   ′          (   z   )             )               Equation                   (   2   )                                  
     where S1(z), . . . , SN(z) are the original signals, and S′1(z), . . . , S′N(z) are the signals after spatially filtered by sensitivity profiles of each coil. Desirably, the detector coils of the array are arranged to optimize the spatial encoding of the spatial filter bank. 
     Decimated gradient phase encoding  220  collects a reduced set of phase-encoded MR signals resulting in fewer lines of k-space. Interpolating system  230  interpolates the plurality of RF and gradient-encoded signals to generate a plurality of interpolated signals. Alternatively, the decimated gradient encoding comprises collecting sparse subsets of non-rectilinear trajectories in k-space, for example fewer interleaves of an interleaved spiral trajectory or fewer radial lines of a radial trajectory. The signal generated after RF and gradient encoding and interpolation is represented by:                  S   ″          (   z   )       =         1   M            ∑     m   =   0       M   -   1                         S   ′          (     zW   ″     )           =       1   M            ∑     m   =   0       M   -   1              F        (     zW   ″     )            S        (     zW   ″     )                       Equation                   (   3   )                                  
     where M is the decimation factor (which is less than or equal to the number of coils or elements in the array), and W=exp(−j2π/M). 
     The image can be reconstructed by applying a lapped transform or filter bank to reconstruct interpolated signals of Equation (3). The description is continued first with reference to the filter bank method of reconstructing images and then with reference to the lapped-transform method. 
     Synthesis filter bank  240  transforms the interpolated signal to generate a corresponding image. The signal generated by synthesis filter bank  240  is represented by:                  S   ^          (   z   )       =           G        (   z   )       T            S   ″          (   z   )         =         1   M            ∑     m   =   0       M   -   1                G        (   z   )       T          F        (     zW   ″′     )            S        (     zW   ″′     )             =       1   M          (         G   1          (   z   )              G   2          (   z   )                     …                     G   N          (   z   )         )            F   0          (   z   )            (           S        (   z   )                 S        (   zW   )               ⋮             S        (     zW     M   -   1       )             )                   Equation                   (   4   )                                  
     where G 1 (z), . . . , G N (z) are the designed synthesis filters, superscript T denotes conjugate transposition, 
     
       
         Ŝ(z)  
       
     
     represents the signals after parallel reconstruction, and the aliasing-component matrix is                  F   a          (   z   )       =     (             F   0          (   z   )               F   0          (   zW   )           …           F   0          (     zW     M   -   1       )                   F   1          (   z   )               F   1          (   zW   )           …           F   1          (     zW     M   -   1       )               ⋮       ⋮       ⋰       ⋮               F   N          (   z   )               F   N          (   zW   )           ⋯           F   N          (     zW     M   -   1       )             )             Equation                   (   5   )                                  
     In an embodiment, the image S(z)is substantially free of aliasing and amplitude distortion. To obtain Ŝ(z) free of aliasing, both spatial filter bank  210  and synthesis filter bank  240  should satisfy 
     
       
           G ( z ) T   F   a ( z )=( C ( z )0 . . . 0),  Equation (6)  
       
     
     where C(z) is an arbitrary filter function. However Equation (6) ensures only that there is no aliasing in the final signal, but does not guarantee absence of amplitude distortion in Ŝ(z). The condition for an image which is fee of both aliasing and amplitude distortion is 
     
       
           G ( z ) T   F   a ( z )=( z   −d 0 . . . 0),  Equation (7)  
       
     
     where z −d  is an arbitrary delay. 
     The above equation can be realized by derivation of a synthetic filter bank  240  based on the aliasing component matrix, 
     
       
           G ( z ) T =( z   −d 0 . . . 0) F   a   −1 ( z )  Equation (8)  
       
     
     However if there are any poles in 
     
       
         F a   −1 (z),  
       
     
     the reconstruction system may not be stable. 
     To guarantee a stable perfect reconstruction (as used herein “perfect reconstruction” refers to the condition where the signals after reconstruction have the same amplitudes and phase as the original signal with only a time delay), an intermediate filter bank V(z) (not shown) can be inserted between the decimated gradient phase-encoding block  220  and the interpolation block  230  for stabilizing the reconstruction. Thus Equation (3) becomes                  S   ″          (   z   )       =         1   M          V        (     z   M     )              ∑     m   =   0       M   -   1              S   ′          (     zW   ″′     )           =       1   M          V        (     z   M     )              ∑     m   =   0       M   -   1              F        (     zW   ″′     )            S        (     zW   ″′     )                       Equation                   (   9   )                                  
     From Equation (4), (5) and (6), the perfect reconstruction condition will be 
     
       
           G ( z ) T   V ( z   M ) F   a ( z )=( z   −s 0 . . . 0)  Equation (10)  
       
     
     Therefore the synthetic filter bank  240  is represented by: 
     
       
           G ( z ) T =( z   −d 0 . . . 0) F   a   −1 ( z ) V   −1 ( z   M )  Equation (11)  
       
     
     where V(z) can be designed to keep G(z) stable. 
     In many cases of parallel MRI, especially for arrays containing a large number of detector elements, the sensitivity profiles of the array are overlapping. In such cases G (z) can be derived much more easily from the lapped transform. The manner in which lapped transforms may be used for deriving G(z) is described below with reference to FIG.  3 . 
     The lapped transform is a filter bank in which the impulse responses of the synthesis filters are the LT basis functions, and the impulse responses of the spatial filters are the time-reversed basis functions. FIG. 3 is an exemplary embodiment of a lapped transform filter bank. 
     In FIG. 3, P is represented by              P   =     (           P   1                                                                           P   0                                                                         ⋰                                                                         P   0                                                                           P   2           )             Equation                   (   12   )                                  
     P 0  is a L×M matrix. L=DM, and D is the overlap factor. In P 1  and P 2 , L=(D−1)M. The relation between input and output signals is given by 
     
       
           {circumflex over (x)} ( k )= PP   T   x ( k )  Equation (13)  
       
     
     The superscript T denotes complex-conjugate transposition. The perfect reconstruction condition for the lapped transform from Equation (13) is: 
     
       
         PP T =I  Equation (14)  
       
     
     Equation (14) implies that the basis functions of P0 (the columns of P0) not only must be orthogonal to one another, 
     
       
         P 0 P 0   T =I  Equation (15)  
       
     
     but also must be orthogonal to the overlapping portion of the basis functions from neighboring blocks, 
     
       
         P 0 ωP 0   T =0,  Equation (16)  
       
     
     where the one-block shift matrix is              ω   =     (         0       I           0       0         )             Equation                   (   17   )                                  
     and the identity matrix I is of order (D−1)M. 
     In a further embodiment, the detector elements of the array (RF coil assembly  147  of FIG. 1) comprise linear strips. Such an embodiment is referred to hereinafter as SAFARI or Strip Array For Analytic Reconstruction of Images. As used herein, a strip array refers to a coil array in which the conductive elements are microstrips of conductive (e.g. copper) material of a selected length connected to a ground plane and arranged to form a planar or cylindrical shape. In general, there are 8 or 16 strips, but it is to be appreciated that variations known to one skilled in the art could also be employed. 
     Inherent with strip arrays, the signal from each strip introduces a different frequency modulation, as shown in FIG.  4 . The mathematical representation of FIG. 4 is illustrated in FIG.  5 . 
     From FIG. 5, it is seen that: 
     
       
           P   i ( m )= f ( mM+i )  
       
     
     
       
           q   i ( m )= g ( mM+i ).  Equation (18)  
       
     
     Where p(m), and q(m) represent mathematical equivalents for spatial filter bank f (mM+I) and synthesis filter bank g(mM+I), respectively. Here i+0, 1, . . . , M−1. 
     When a large number of strips are concentrated in a region that is relatively small compared to the sensitivity profiles, FIG. 5 can be simplified to FIG. 6, where the parallel reconstruction is simply a block-in From FIG. 5, it is seen that:verse discrete Fourier transform (DFT). 
     When the RF coil assembly is a strip array, each strip has an explicit phase relationship relative to a spatial location within the object being imaged. Referring to FIG. 4, the relationship is expressed as W k   M =exp(j2πk/M) for M=0 to (M−1), where M is the number of strips in the array. Referring to FIGS. 4-6, the explicit phase relationship in the sensitivity profiles can be used for simplifying the lapped transform reconstruction described earlier. 
     In further embodiments of the present invention, reconstruction may be improved by adjusting the sensitivity profiles of the coils of the RF detector array (RF coil assembly  147  of FIG. 1) to realize an orthogonal sensitivity profile. Orthogonality is useful for reducing the matrix condition number, thereby making the reconstruction more robust. In a first embodiment, the RF detector array is a planar strip array comprised of multiple conductive strips and coupling capacitors interconnecting the strips. By adjustment of the interconnecting capacitors and input ports for each respective strip, the desired sensitivity profile can be obtained. In a second embodiment for adjusting sensitivity profiles, post processing can be used to improve orthogonality. 
     While only certain features of the invention have been illustrated and described herein, many modifications and changes will occur to those skilled in the art. It is therefore to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the invention.
A parallel Magnetic Resonance Imaging (MRI) system is provided for reconstructing images, to enable accelerated MRI with minimal artifacts. The MRI system comprises an array of magnetic resonance (MR) coils arranged in an array for detecting a plurality of MR signals, each coil having a corresponding spatial sensitivity profile and a processing means for processing the plurality of MR signals with at least one filter bank to reconstruct at least one image.