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 Ω_{C} is a p×p Fourier matrix, it is a matrix with (j,k)^{th} element [Ω_{C}]_{jk}= κ (ω^{jk}) 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 p

_{y}×p

_{x}(p

_{y} rows and p

_{x} 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 R

_{C} that has a real part R

_{R} as in an imaginary part R

_{I} that is the zero matrix so that R

_{C}=R

_{R}+iR

_{I}. The encoded data, or Fourier transform of this image, can be found as in

Eq. (1) by pre-multiplying the p

_{y}×p

_{x} dimensional complex-valued matrix R

_{C} by a standard complex-valued forward Fourier matrix

_{yC}=

_{yR}+i

_{yI}, that is of dimensions p

_{y}×p

_{y}, and post-multiplying R

_{C} by the transpose of another standard forward Fourier matrix

, where T denotes matrix transposition, that is of dimensions p

_{x}×p

_{x}. The result of the pre- and post-multiplications is a complex-valued array of spatial frequency (

*k*-space) measurements, S

_{C}, with real part S

_{R} and imaginary part S

_{I} as also shown in

Eq. (1).

This mathematical procedure is graphically illustrated in using the aforementioned 8×8 image. In the 8×8 image, R

_{C}, is utilized to mimic an image from a magnetic resonance echo planar imaging experiment. R

_{C} is displayed with real part, R

_{R}, in and imaginary part, R

_{I}, in . The spatial frequency (

*k*-space) values, S

_{C}=(S

_{R}+iS

_{I}), 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, S

_{C}, for the complex-valued image R

_{C} are presented as an image with real part, S

_{R}, in and imaginary part, S

_{I}, in . Note that, as mentioned earlier, the image does not have to be square.

However, as previously described, in MRI encoded (*k*-space) measurements, S_{C}, 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 p

_{y}×p

_{x} dimensional complex-valued spatial frequency matrix, S

_{C}, by a complex-valued inverse Fourier matrix, Ω

_{y}, that is of dimensions p

_{y}×p

_{y}, and post-multiplying S

_{C} by the transpose of another Fourier matrix,

_{x}^{T}, that is of dimensions p

_{x}×p

_{x}, where T denotes matrix transposition. The result of the pre- and post-multiplications is a complex-valued array of image measurements R

_{C}, with real part R

_{R} and imaginary part R

_{I} as also shown in

Eq. (2).

The complex-valued image R_{C} can be recovered as seen in . The process of recovering the original complex-valued image R_{C} is to pre-multiply the complex-valued spatial frequency (*k*-space) values S_{C} by the complex-valued inverse Fourier matrix Ω_{Cy}, () then post-multiply the result by the transpose of the symmetric inverse Fourier matrix Ω_{Cx}, (). The recovered complex-valued image, R_{C}, is presented with real part, R_{R}, in and imaginary part, R_{I}, in .

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 2p

_{x}p

_{y} dimensional vector of complex-valued spatial frequencies from an image where the first p

_{x}p

_{y} elements are the rows of the real part of the spatial frequency matrix, S

_{R}, shown in , and the second p

_{x}p

_{y} elements are the rows of the imaginary part of the spatial frequency matrix, S

_{I}, shown in . The real-valued vector of spatial frequencies is thus formed as s=vec(S

_{R}^{T},S

_{I}^{T}), where (S

_{R}^{T},S

_{I}^{T}) is a p

_{x}×2p

_{y} matrix formed by joining the transpose of the real and imaginary parts of S

_{C} 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 .

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

and

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 Ω

_{Cy}, with real and imaginary parts Ω

_{yR} and Ω

_{yI} given in , along with the complex-valued Fourier matrix Ω

_{Cx}, with real and imaginary parts Ω

_{xR} and Ω

_{xI} 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 2p_{x}p_{y}×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 p

_{x}. 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 .

In the above described procedure, measurement noise was not considered. Redefine S_{C} to be the p_{y}×p_{x} dimensional complex-valued spatial frequency measurement of a slice with noise that consists of a p_{y}×p_{x} dimensional matrix of true underlying noiseless complex-valued spatial frequencies, S_{0C}, and a p_{y}×p_{x} dimensional matrix of complex-valued measurement error, E_{C}. 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 S_{0R}, S_{0I}, E_{R}, and E_{I} are real and imaginary matrix valued parts of the true spatial frequencies and measurement noise, respectively. Let Ω_{Cx} and Ω_{Cy} be p_{x}×p_{x} and p_{y}×p_{y} complex-valued Fourier matrices as described above. Then, the p_{y}×p_{x} complex-valued inverse Fourier transformation reconstructed image, R_{C}, of S_{C} can be written as

where R

_{C} has a true mean R

_{0C} and measurement error N

_{C}. Note that the complex-valued matrices for reconstruction, Ω

_{x} and Ω

_{y} 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=s_{0}+ε where this 2p_{x}p_{y} 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 2p_{x}p_{y}×1, are s_{0} and Γ, then the mean and covariance of the reconstructed voxel measurements, r, are Ωs_{0} and ΩΓΩ^{T}.