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...ransform F shall be referred to as the reconstruction matrix. Its size is nV � n Cn K. Assembling sample and image values in vectors, image reconstruction may be rewritten in matrix notation: v � Fm. =-=[7]-=- With such linear mapping the propagation of noise from sample values into image values is conveniently described by noise matrices. The �-th diagonal entry of the image noise matrix X represents the ...

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...or a the complex image values the chosen pixel has in the intermediate images. The complex coil sensitivities at the nP superimposed positions form an nC � nP sensitivity matrix S: S �,� � s � (r �), =-=[1]-=- where the subscripts �, � count the coils and the superimposed pixels, respectively, r � denotes the position of the pixel �, and s� is the spatial sensitivity of the coil �. The sensitivity matrix i...

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...ents the superposition of pixels forming a Cartesian grid. In this example four of these pixels are in the full FOV; thus the actual degree of aliasing is four. trix U: U � (S H � �1 S) �1 S H � �1 , =-=[2]-=- where the superscript H indicates the transposed complex conjugate, and � is the n C � nC receiver noise matrix (see Appendix A), which describes the levels and correlation of noise in the receiver c...

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...mpare the noise levels as obtained with full and reduced Cartesian Fourier encoding. According to Appendix D the partial image noise matrix for the relevant unfolding step is X � 1 (S nK H� �1S) �1 . =-=[20]-=- Let � denote the index of the voxel under consideration within the set of voxels to be separated. With full Fourier encoding no aliasing occurs and the matrix S has only one column. Note that this si...

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...he n C � nC receiver noise matrix (see Appendix A), which describes the levels and correlation of noise in the receiver channels. Using the unfolding matrix, signal separation is performed by v � Ua, =-=[3]-=- where the resulting vector v has length n P and lists separated pixel values for the originally superimposed positions. By repeating this procedure for each pixel in the reduced FOV a non-aliased ful...

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... by Lagrange calculus (see Appendix C), yielding F � (E H � ˜ �1 E) �1 E H � ˜ �1 . [16]sSENSE: Sensitivity Encoding for Fast MRI 955 In this case the image noise matrix reads X � (E H � ˜ �1 E) �1 . =-=[17]-=- The reconstruction formulae [12] and [16] permit image reconstruction from data obtained with hybrid gradient and sensitivity encoding. Both are numerically challenging as they imply the inversion of...

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... compare the noise levels as obtained with full and reduced Cartesian Fourier encoding. According to Appendix D the partial image noise matrix for the relevant unfolding step is X 5 1 nK (SH C21S)21. =-=[20]-=- Let r denote the index of the voxel under consideration within the set of voxels to be separated. With full Fourier encoding no aliasing occurs and the matrix S has only one column. Note that this si...

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... voxel function fulfil the orthonormality relations of its ideal counterpart: � i�*(r)f��(r)dr ���,�� VOI ��, ��. [14] Using Eqs. [10] and [11], Eq. [14] may be rewritten in matrix form: FE � Id nv , =-=[15]-=- where Id nv denotes nV � nV identity. By this condition the reconstruction matrix F is generally not yet entirely determined. It leaves nCn K - n V degrees of freedom per voxel, which may be utilized...

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...ne. In Appendix B it is shown that it yields F � E H C �1 , [12] where C denotes the correlation matrix of the encoding functions. The image noise matrix [8] is then given by X � E H C �1 � ˜ C �1 E. =-=[13]-=- The second approach, dubbed the weak one, uses a different concept of similarity between real voxel functions and ideal shapes. It requires that each voxel function fulfil the orthonormality relation...

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...al set of ideal voxel shapes, e.g., box functions. The relation between ideal voxel shapes and encoding functions is described by the n Cn K � n V encoding matrix E (�,�),� � � VOI i �*(r)e �,�(r)dr. =-=[11]-=- There are many possible ways of approximating ideal voxels. Here we discuss two concepts. The first approach is to choose those voxel functions that exhibit the least square deviation from the ideal....

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... object be within the volume of interest (VOI). Then a sample value m obtained from the �-th coil at the �-th position in k-space is given by m �,� � � VOI where r denotes 3D position, c(r)e �,�(r)dr =-=[4]-=- e �,�(r) � e ik�r s �(r) [5] is the net encoding function composed of harmonic modulation and the complex spatial sensitivity s � of coil �, and c results from tissue and sequence parameters. The eff...

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...of interest (VOI). Then a sample value m obtained from the �-th coil at the �-th position in k-space is given by m �,� � � VOI where r denotes 3D position, c(r)e �,�(r)dr [4] e �,�(r) � e ik�r s �(r) =-=[5]-=- is the net encoding function composed of harmonic modulation and the complex spatial sensitivity s � of coil �, and c results from tissue and sequence parameters. The effects of non-uniform k-space w...

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...ept of similarity between real voxel functions and ideal shapes. It requires that each voxel function fulfil the orthonormality relations of its ideal counterpart: � i�*(r)f��(r)dr ���,�� VOI ��, ��. =-=[14]-=- Using Eqs. [10] and [11], Eq. [14] may be rewritten in matrix form: FE � Id nv , [15] where Id nv denotes nV � nV identity. By this condition the reconstruction matrix F is generally not yet entirely...

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... Derivation of DFT Based Reconstruction in the Cartesian Case Here the practical procedure described at the beginning of the Theory and Methods section is derived from the weak reconstruction formula =-=[16]-=-, assuming Dirac distributions as ideal voxel functions. Consider sampling and voxel positions forming regular grids in 2D k-space and image domain, respectively. The grid constants shall be chosen su...

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...st square deviation from the ideal. This criterion entirely determines reconstruction; the approach is therefore referred to as the strong one. In Appendix B it is shown that it yields F � E H C �1 , =-=[12]-=- where C denotes the correlation matrix of the encoding functions. The image noise matrix [8] is then given by X � E H C �1 � ˜ C �1 E. [13] The second approach, dubbed the weak one, uses a different ...

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...g it is clear that image reconstruction must essentially be linear as well. That is, each of n V image values is to be calculated as a linear combination of sample values: v � � � �,� F �,(�,�)m �,�, =-=[6]-=- where � counts the voxels to be resolved. The transform F shall be referred to as the reconstruction matrix. Its size is nV � n Cn K. Assembling sample and image values in vectors, image reconstructi...

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...� � VOI c(r) (� �,� F�,(�,�)e�,�(r)) dr. [9] The term in brackets describes the spatial weighting of signal in � �. It is therefore called the corresponding voxel function: f�(r) � � F�,(�,�)e�,�(r). =-=[10]-=- �,� Hence, the matrix F has to be chosen such that the resulting voxel functions approximate the desired voxel shapes. Let i �(r) denote an orthonormal set of ideal voxel shapes, e.g., box functions....

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...h image value selectively reflect signal from the voxel it represents. To trace the origin of signal in image values, insert Eq. [4] into Eq. [6], to find � � � � VOI c(r) (� �,� F�,(�,�)e�,�(r)) dr. =-=[9]-=- The term in brackets describes the spatial weighting of signal in � �. It is therefore called the corresponding voxel function: f�(r) � � F�,(�,�)e�,�(r). [10] �,� Hence, the matrix F has to be chose...

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...k approach is less robust in terms of ensuring voxel quality. The limitations of weak reconstruction may be understood by considering Dirac distributions as ideal voxel functions: i �(r) ��(r � r �), =-=[18]-=- where r � denotes the center of the �-th voxel. The encoding matrix then reduces to E (�,�),� � e �,�(r �). [19] In this case the weak criterion [15] may be restated as follows: each voxel function m...

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...stood by considering Dirac distributions as ideal voxel functions: i �(r) ��(r � r �), [18] where r � denotes the center of the �-th voxel. The encoding matrix then reduces to E (�,�),� � e �,�(r �). =-=[19]-=- In this case the weak criterion [15] may be restated as follows: each voxel function must be equal to one in the center of the voxel it belongs to and equal to zero in all other voxels’ centers. A vo...

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... require the calculation of the matrix C and poses the smaller inversion problem when nV + nCnK. Furthermore, it yields optimized SNR. On the other hand, the strong approach is always applicable, whereas the second algorithm works only if condition [15] can be fulfilled. In particular, for weak reconstruction the rank of the matrix E must be equal to nV, thus nV , nCnK must hold. Moreover, the weak approach is less robust in terms of ensuring voxel quality. The limitations of weak reconstruction may be understood by considering Dirac distributions as ideal voxel functions: i# (r) $ )(r & r#), [18] where r# denotes the center of the #-th voxel. The encoding matrix then reduces to E(",'),# $ e",' (r#). [19] In this case the weak criterion [15] may be restated as follows: each voxel function must be equal to one in the center of the voxel it belongs to and equal to zero in all other voxels’ centers. A voxel function with this property will be acceptable only as long as it is well behaved between voxel centers. In this view, the criterion becomes unreliable when there are solutions that vary considerably within voxels and at the same time yield favorably low noise. The Dirac choice in the ...

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...it yields optimized SNR. On the other hand, the strong approach is always applicable, whereas the second algorithm works only if condition [15] can be fulfilled. In particular, for weak reconstruction the rank of the matrix E must be equal to nV, thus nV , nCnK must hold. Moreover, the weak approach is less robust in terms of ensuring voxel quality. The limitations of weak reconstruction may be understood by considering Dirac distributions as ideal voxel functions: i# (r) $ )(r & r#), [18] where r# denotes the center of the #-th voxel. The encoding matrix then reduces to E(",'),# $ e",' (r#). [19] In this case the weak criterion [15] may be restated as follows: each voxel function must be equal to one in the center of the voxel it belongs to and equal to zero in all other voxels’ centers. A voxel function with this property will be acceptable only as long as it is well behaved between voxel centers. In this view, the criterion becomes unreliable when there are solutions that vary considerably within voxels and at the same time yield favorably low noise. The Dirac choice in the weak approach also is of great practical significance. It is with this choice that reconstruction in the Carte...

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...nsequence, unlike standard Fourier images the noise level in a SENSE image varies from pixel to pixel and there is noise correlation between pixels. For similar reasons the noise level does not have the common square-root dependence on the number of samples taken. In the case of Cartesian sampling with reconstruction as initially explained, this can be made yet clearer. For one particular voxel we compare the noise levels as obtained with full and reduced Cartesian Fourier encoding. According to Appendix D the partial image noise matrix for the relevant unfolding step is X $ 1 nK (SH %&1S)&1. [20] Let # denote the index of the voxel under consideration within the set of voxels to be separated. With full Fourier encoding no aliasing occurs and the matrix S has only one column. Note that this single column is identical to the #-th column of S in the case of reduced Fourier encoding. Thus, the ratio of the noise levels obtained in that voxel is given by #X #,#red #X #,#full $ #R #[(SH %&1S)&1]#,# (SH %&1S)#,# , [21] where S corresponds to reduced Fourier encoding and R denotes the factor by which the number of samples is reduced with respect to full Fourier encoding: R $ nK full nK red . ...