Previous work has included the development of a real-valued representation of the standard complex-valued Fourier transform [14
]. In this section we review the representation and offer a graphical example to illustrate the method. Magnetic resonance images are almost exclusively Fourier encoded. That is, one ideally measures the Fourier transform of an image and reconstruct the image via an inverse Fourier transform. The Fourier transform and inverse Fourier transforms are complex-valued procedures that results in complex-valued arrays.
Standard, complex-valued Fourier matrices are defined as follows. If ΩC is a p×p Fourier matrix, it is a matrix with (j,k)th element [ΩC]jk= κ (ωjk) where κ=1 and ω=exp[-i 2 π (j-1)(k-1)/p] for the forward transformation while κ=1/p and ω =exp[+i 2π (j-1)(k-1)/p] for the inverse transformation, where j,k=1,…,p.
Consider the Fourier transform of an image that has dimensions py
rows and px
columns). Often the image is square, although this is not necessary. More specifically, consider an 8×8, ideal, noiseless, gray scale image as presented in . Since the Fourier transform and inverse Fourier transform procedures operate on, and produce complex-valued arrays, the real-valued image in can be represented as a complex-valued image RC
that has a real part RR
as in an imaginary part RI
that is the zero matrix so that RC
. The encoded data, or Fourier transform of this image, can be found as in Eq. (1)
by pre-multiplying the py
dimensional complex-valued matrix RC
by a standard complex-valued forward Fourier matrix yC
, that is of dimensions py
, and post-multiplying RC
by the transpose of another standard forward Fourier matrix
, where T denotes matrix transposition, that is of dimensions px
. The result of the pre- and post-multiplications is a complex-valued array of spatial frequency (k
-space) measurements, SC
, with real part SR
and imaginary part SI
as also shown in Eq. (1)
This mathematical procedure is graphically illustrated in using the aforementioned 8×8 image. In the 8×8 image, RC
, is utilized to mimic an image from a magnetic resonance echo planar imaging experiment. RC
is displayed with real part, RR
, in and imaginary part, RI
, in . The spatial frequency (k
-space) values, SC
), associated with this complex-valued image, are found by pre-multiplying the complex-valued image by the complex-valued forward Fourier matrix yC
() and then post-multiplying the result by the transpose of the symmetric forward Fourier matrix (). The spatial frequency (k
-space) values, SC
, for the complex-valued image RC
are presented as an image with real part, SR
, in and imaginary part, SI
, in . Note that, as mentioned earlier, the image does not have to be square.
However, as previously described, in MRI encoded (k-space) measurements, SC, are made and reconstructed (transformed) into an image. The inverse Fourier procedure is performed. This reconstruction procedure, or inverse Fourier transform, of the spatial frequency (k-space) measurements can be found as
by pre-multiplying the py
dimensional complex-valued spatial frequency matrix, SC
, by a complex-valued inverse Fourier matrix, Ωy
, that is of dimensions py
, and post-multiplying SC
by the transpose of another Fourier matrix, xT
, that is of dimensions px
, where T denotes matrix transposition. The result of the pre- and post-multiplications is a complex-valued array of image measurements RC
, with real part RR
and imaginary part RI
as also shown in Eq. (2)
The complex-valued image RC can be recovered as seen in . The process of recovering the original complex-valued image RC is to pre-multiply the complex-valued spatial frequency (k-space) values SC by the complex-valued inverse Fourier matrix ΩCy, () then post-multiply the result by the transpose of the symmetric inverse Fourier matrix ΩCx, (). The recovered complex-valued image, RC, is presented with real part, RR, in and imaginary part, RI, in .
Fig. 3 Complex-valued 2D inverse Fourier transform.
- Inverse matrix ΩyR
- Inverse matrix ΩyI
- Spatial frequencies SR
- Spatial frequencies SI
- Inverse matrix ΩxR
- Inverse matrix ΩxI
- Image real RR
- Image imaginary RI
This complex-valued inverse Fourier transformation image reconstruction process can be equivalently described as a linear transformation with a real-valued representation [14
]. Such a transformation is often called an isomorphism in mathematics. Define a real-valued vector, s, to be a 2px
dimensional vector of complex-valued spatial frequencies from an image where the first px
elements are the rows of the real part of the spatial frequency matrix, SR
, shown in , and the second px
elements are the rows of the imaginary part of the spatial frequency matrix, SI
, shown in . The real-valued vector of spatial frequencies is thus formed as s=vec(SRT
), where (SRT
) is a px
matrix formed by joining the transpose of the real and imaginary parts of SC
as seen in , and vec(·) denotes the vectorization operator that stacks the columns, shown in , of its matrix argument. This yields us a real-valued vector representation of the matrix of spatial frequency (k
-space) values that is given in .
Fig. 4 Matrix to vector spatial frequency (k-space) values.
- Spatial frequencies ST=(SRT,SIT)
- Partitioned spatial frequencies ST
Fig. 5 Isomorphism for complex-valued 2D inverse Fourier Transform.
- Reconstruction matrix Ω
- Frequency vector s
- Image vector r
Further define a matrix Ω that is another representation of the complex-valued inverse Fourier transformation matrices as described in Eq. (3)
where the matrix elements of Ω are
denotes the Kronecker product that multiplies every element of its first matrix argument by its entire second matrix argument. Utilizing the complex-valued Fourier matrix ΩCy
, with real and imaginary parts ΩyR
given in , along with the complex-valued Fourier matrix ΩCx
, with real and imaginary parts ΩxR
given in , the resulting Ω matrix is presented in .
The real-valued vector representation s of the spatial frequency (k
-space) values in is then pre-multiplied by the (inverse Fourier) reconstruction matrix Ω as in Eq. (3)
where the real-valued representation, r, of the complex-valued image has a dimension of 2pxpy×1, true mean and no measurement error.
This is pictorially represented in . is the spatial frequency vector s and is the inverse Fourier transformation matrix Ω as described in Eq. (3)
. This matrix multiplication produces a vector representation, r, of the image voxel measurements given in as described in Eq. (3)
. The vector of voxel measurements, r, is partitioned into column blocks of length px
. These blocks are then arranged as in and formed into a single matrix image as in where the first (last) eight columns are the transpose of the real (imaginary) part of the image. As can be seen, the same resultant complex-valued image is reconstructed with the complex-valued inverse Fourier transformation procedure described in Eq. (2)
and presented in .
Fig. 6 Vector to matrix image values.
- Partitioned images RT
- Combined image RT=(RRT, RIT)
In the above described procedure, measurement noise was not considered. Redefine SC to be the py×px dimensional complex-valued spatial frequency measurement of a slice with noise that consists of a py×px dimensional matrix of true underlying noiseless complex-valued spatial frequencies, S0C, and a py×px dimensional matrix of complex-valued measurement error, EC. This partitioning of the measured spatial frequencies in terms of true noiseless spatial frequencies plus measurement error can be represented as
where i is the imaginary unit while S0R, S0I, ER, and EI are real and imaginary matrix valued parts of the true spatial frequencies and measurement noise, respectively. Let ΩCx and ΩCy be px×px and py×py complex-valued Fourier matrices as described above. Then, the py×px complex-valued inverse Fourier transformation reconstructed image, RC, of SC can be written as
has a true mean R0C
and measurement error NC
. Note that the complex-valued matrices for reconstruction, Ωx
in Eq. (5)
, need not be exactly Fourier matrices but may be Fourier matrices that include adjustments for independently measured magnetic field inhomogeneities or reconstruction matrices for other encoding procedures.
The real-valued inverse Fourier transformation method for image reconstruction can also be directly applied to noisy measurements. We can represent the noisy complex-valued spatial frequency matrix as s=s0+ε where this 2pxpy dimensional vectors includes the reals of the rows stacked upon the imaginaries of the rows of the corresponding matrix. This implies that if the mean and covariance of the spatial frequency measurement vector, s, that is of dimension 2pxpy×1, are s0 and Γ, then the mean and covariance of the reconstructed voxel measurements, r, are Ωs0 and ΩΓΩT.