Homodyne reconstruction and IDEAL water–fat decomposition



Multipoint water–fat separation methods have received renewed interest because they provide uniform separation of water and fat despite the presence of B0 and B1 field inhomogeneities. Unfortunately, full-resolution reconstruction of partial k-space acquisitions has been incompatible with these methods. Conventional homodyne reconstruction and related algorithms are commonly used to reconstruct partial k-space data sets by exploiting the Hermitian symmetry of k-space in order to maximize the spatial resolution of the image. In doing so, however, all phase information of the image is lost. The phase information of complex source images used in a water–fat separation acquisition is necessary to decompose water from fat. In this work, homodyne imaging is combined with the IDEAL (iterative decomposition of water and fat with echo asymmetry and least squares estimation) method to reconstruct full resolution water and fat images free of blurring. This method is extended to multicoil steady-state free precession and fast spin-echo applications and examples are shown. Magn Reson Med, 2005. © 2005 Wiley-Liss, Inc.

Reliable separation of water from fat using chemical-shift-based approaches has shown renewed interest in recent years (1–6), as it provides uniform separation of water from fat despite the presence of B0 and B1 field inhomogeneities. These methods typically acquire three images, each with slightly different echo times (TE), and analytical (2, 3) or least squares (7) methods are then used to decompose these “source” images into separate water and fat images. Extension to multicoil applications has also been described recently (5, 7).

Partial k-space acquisitions in the readout direction are important for applications that must reduce the minimum TE, first moment phase shifts from motion or flow in the readout direction, and the minimum TR. Short TRs are essential for good image quality for SSFP water–fat separation applications to prevent banding artifacts while maintaining high spatial resolution in the readout direction (7–10). Partial readout acquisitions would also be important for rapid SPGR water–fat separation methods that require short TE and TR.

The threefold increase in image acquisition time for water–fat separation methods is often problematic for many applications, particularly when oversampling strategies in the phase encoding direction (“no phase wrap”) are used to prevent aliasing. Although water–fat separation methods are highly SNR efficient (2, 7), there is a threefold increase in scan time compared with a conventional fat-saturated exam. In order to prevent aliasing in the phase encoding direction, additional interleaved lines of k-space can be acquired to increase the field of view in the phase encoding direction. Doubling the number of acquired lines of k-space, for example, would result in a sixfold increase in the minimum scan time of a conventional fat-saturated exam. This lengthy scan time is unacceptable for many clinical settings. Reductions in scan time through partial ky acquisitions would be very helpful toward addressing this problem.

Partial k-space acquisitions have seen very limited use with water–fat separation methods. Homodyne reconstruction (11) and other related methods (12–15) are commonly used to reconstruct partial k-space acquisitions, exploiting the Hermitian symmetry of k-space in order to maximize spatial resolution. Unfortunately, homodyne methods demodulate all phase information from complex images. The phase of the source images acquired at the different echo times contains the information required to decompose water from fat. For this reason, the unsampled portions of k-space matrices acquired with partial acquisitions are filled with zeroes and although the essential phase information is preserved, the resulting images will experience moderate blurring.

Initial attempts to apply homodyne reconstruction to water–fat separation methods have been described by Ma et al. for echo sampling schemes that acquired echoes at echo times with phase differences between water and fat of 0, π/2, and π (16). In this work, conventional homodyne reconstruction was applied to the images acquired with water–fat phase shifts of 0 and π when signal from water and fat are exactly in phase (0) or exactly out of phase (π). At these particular phase values, no phase information has been introduced from chemical shift. In this approach the middle source image (π/2) was reconstructed with simple zero-filling, however, and some blurring of the middle source image would be expected.

In this work, we describe the combination of homodyne reconstruction with an iterative least squares water–fat decomposition method (IDEAL) (7, 17, 18). Using this combination, resolution of calculated water and fat images can be maximized for partial k-space acquisitions. This method is extended to multicoil applications and applied to partial kx (frequency encoding) acquisitions for IDEAL-SSFP water–fat separation and partial ky (phase encoding) acquisitions for IDEAL-FSE water–fat separation. Examples in a water–oil resolution phantom and in normal volunteers are shown.


Signal Behavior

The signal from a pixel at position r containing magnetization from water (W) and fat (F) acquired at discrete echo times tn (n = 1,…, N), in the presence of field inhomogeneity, ψ(r) (Hz), can be written

equation image(1)

where the relative chemical shift of fat relative to water is Δffw, approximately −210 Hz at 1.5 T and −420 Hz at 3 T, cn = emath image and dn = emath image. It is important to emphasize that cn is a known complex coefficient, independent of position, r, while dn is unknown and dependent on position. It is also important to note that W(r) and F(r) are complex, with independent phase, such that W(r) = |W(r)|emath image and F(r) = |F(r)|emath image. If the field map, ψ(r), is known, then the exponential dn = emath image can be demodulated from Eq. [1] and water and fat are easily decomposed in the least-squares sense (7). If dn is unknown, an iterative method can be used to calculate the field map and water and fat are subsequently decomposed in the least-squares sense (7).

Undersampled Data and Filters

If a fully sampled data set is represented by Eq. [1], then a partial k-space acquisition could be represented by

equation image(2)

where F{ } represents the Fourier transform operator, and

equation image(3)

and kmin is distance from the center to the edge of the undersampled half of k-space, and kmax is the distance to the edge of the fully sampled half of k-space. sn(r) will be blurred in comparison to the fully sampled image, sn(r), because g(r) has a finite-width imaginary component (11).

As part of the homodyne reconstruction below, one of two filters is typically used (11). The first is a ramp transition filter,

equation image(4)

and the second choice is the step weighting function

equation image(5)

Both of these functions can be used for homodyne reconstruction, and each has its advantages/disadvantages (11, 12).

Finally, we define a low-pass filter,

equation image(6)

which will also be used in the homodyne reconstruction described below.

Water–Fat Separation and Homodyne Reconstruction

The first step is to filter the sampled k-space data with the low pass filter GL(k), and perform the Fourier transform to obtain images that have low resolution in the undersampled direction,

equation image(7)

Assuming that the field map, ψ(r), is smoothly varying, a good estimate of the field map (equation image(r), d̂n = eiequation imagemath image) can be made using the iterative approach (7). In addition, low-resolution estimates of the complex water and fat images can also be made, and from these, estimates of the phase maps of the water and fat images, i.e.,

equation image(8)

If equation imageW(r) ≈ ϕW(r) and equation imageF(r) ≈ ϕF(r), then these terms can be used below to demodulate the phase maps of the final water and fat images, as is done with conventional homodyne reconstruction (11).

Next, the sampled data are filtered with either choice of GR(k) (ramp transition function or step function), such that

equation image(9)

If it can be assumed that the field map is smoothly varying such that dn(r) varies only slightly over the width of gR(r), similar to the assumptions made by Noll et al. for phase maps (11), which allows dn(r) to be brought through the convolution, i.e.,

equation image(10)

Assuming d̂n(r) is approximately equal to dn(r), low-resolution estimates of d̂n(r) obtained above from the low-pass filtered images are now used to demodulate dn(r) from Eq. [10], and estimates of filtered water and fat images are made with the least-squares method (7), such that

equation image(11a)


equation image(11b)

where it has been assumed that the ϕW(r) and ϕF(r)vary only slightly over the width of gR(r), as is assumed with conventional homodyne reconstruction, so that the phasor terms can be brought through the convolution (11).

The phase of the water and fat images is then demodulated from Eqs. [11a] and [11b] using the low-resolution estimates of the phase of the water and fat images (equation imageW(r), equation imageF(r)). Finally, the water and fat images are calculated from the real part of the demodulated water and fat images,

equation image(12a)


equation image(12b)

in the same manner as conventional homodyne reconstruction.

Application with Multicoil Acquisitions

The combination of homodyne imaging with water–fat separation for multicoil applications is extended from previous work combining the iterative least squares water–fat decomposition with multicoil acquisitions (7). Using a similar approach as this work, the low-resolution field map is calculated for each of the P coils. A combined field map is then calculated for each pixel as the sum of the P low-resolution field maps, each weighted by the square of the signal from the source images,

equation image(13)

where sp is the local signal, determined from the average magnitude of the three source images at that pixel (7). The combined field map is then demodulated from the source images for each coil and water and fat calculated in the least-squares sense, in the same manner described previously for multicoil acquisitions (7). The phase of the low-resolution complex water and fat images calculated from each coil (equation imagemath image(r), equation imagemath image(r)) will be demodulated later from the high-resolution images calculated in the next step. The ramp transition or step function filter (GR(k)) is applied to each k-space data set for each coil, and the 2D Fourier transform is performed. The combined field map is then demodulated from the filtered source images (Eq. [10]), and P water and P fat images are calculated using the least-squares decomposition (7). Next, the low-resolution phase terms, equation imagemath image(r) and equation imagemath image(r), are demodulated from the calculated complex water and fat images and the real part of these images taken. The phase of the water and fat images must be demodulated separately for each coil, because there may be constant phase differences between the images acquired with the different coils. Finally, the P water images are combined using a standard multicoil reconstruction method (square-root of the sum of the squares) described by Roemer et al. (19). The same calculations are performed for the fat images, and the algorithm is summarized below.

  • 1For each coil:
    • aApply low-pass filter GL(k) to asymmetric k-space data;
    • bPerform 2D Fourier transform;
    • cCalculate low-resolution field maps, equation imagep(r), as well as low-resolution phase maps of water (equation imagemath image(r)) and fat (equation imagemath image(r)) images with iterative least-squares method.
  • 2Calculate combined field map, equation imagec, using Eq. [13].
  • 3For each coil:
    • aFilter the asymmetric k-space data with ramp transition or step function filter (GR(k));
    • bPerform 2D Fourier transform;
    • cDemodulate dn(r) terms from source images using combined low-resolution field map, (equation imagec(r), d̂math image = eiequation imagemath image);
    • dCalculate filtered P water and P fat images with least-squares decomposition (Eqs. [11a, 11b]);
    • eDemodulate phase terms (equation imagemath image(r), equation imagemath image(r)) from P water and P fat images;
    • fKeep real part of each water and fat image.
  • 4Calculate combined water (and fat) image using standard multicoil reconstruction (square root of the sum of the squares).


All scanning was performed at 1.5 T (Signa TwinSpeed, GE Healthcare, Milwaukee, WI, USA) and 3.0 T (vH/i, GE Healthcare). All human studies were approved by our institutional review board and informed consent was obtained for all human studies.

Imaging of a water–oil resolution phantom was performed at 1.5 T with a conventional transmit/receive head coil and three-point fast spin-echo (FSE) sequence modified to allow arbitrary echo shifts, necessary to decompose water from fat (7, 17, 18). Echoes were positioned asymmetrically with respect to the spin-echo (−0.40 ms, 1.19 ms, 2.78 ms), based on a chemical shift of −210 Hz at 1.5 T, to maximize the SNR performance of the water–fat decomposition (7, 17, 18). The resolution phantom was a Plexiglas cylinder divided into two equal halves, one filled with 7 mM CuSO4 in distilled water and the other filled with peanut oil. Within each half, two Plexiglas bars containing multiple holes of varying sizes were placed centrally in order to create detailed structure within images.

A helpful measure of sampling asymmetry in partial k-space acquisitions is the “echo fraction,” which is defined as the quotient of the actual number points acquired and the fully encoded matrix size. For example, with a 256 matrix dimension, a readout with an echo fraction of 0.625 implies that 160 points are acquired; 32 points for kx < 0, and the remaining 128 for kx > 0. For partial ky and kz acquisitions, the echo fraction is proportional to the scan time and directly reflects scan time reductions through partial k-space acquisitions.

Imaging in the knee of a normal volunteer was performed at 1.5 T with the same FSE sequence, using a conventional transmit/receive knee coil. Imaging parameters included FOV = 16 cm, slice/gap = 3.0/0.5, BW = ±20 kHz, TR/TE = 5000/48, 256 × 256 full resolution imaging matrix, and one average. Total scan time for 22 slices was 5:35 min.

Noncontrast angiographic imaging of the arteries of the lower leg of a normal volunteer was performed at 3.0 T using a transmit-receive quadrature extremity coil (MRI Devices, Waukesha, WI, USA) and a 3D-SSFP pulse sequence modified to acquire images at different echo shifts, necessary to decompose water from fat. This approach is based on the work of Brittain et al. (6). Imaging parameters included BW = ±100 kHz, TR = 4.7 ms, TE = 1.1/1.7/2.3 ms, FOVx = 24 cm, FOVy = 19.2 cm, FOVz = 9.6 cm with 256 × 204 × 96 matrix size for 0.9 × 0.9 × 1.0 mm3 resolution. The fractional readout acquired 160 points (echo fraction = 0.625).

Finally, cardiac CINE SSFP imaging acquired at 1.5 T using a retrospectively gated CINE SSFP sequence, and a four-element phased array torso coil was performed in a normal volunteer (20). Imaging parameters included FOV = 32 cm, slice = 8 mm, 224 × 128 (134 point fractional readout, echo fraction = 0.60), and BW = ±125 kHz. TE increment was 0.9 ms and TR was 4.9 ms.

Decomposition of water and fat images for full resolution and zero-filled data matrices was performed with the iterative least squares water–fat decomposition method (7), and reconstruction with the homodyne approach used the approach described above. In order to prevent ambiguities between assignment of water and fat within a calculated image, a “robust” reconstruction approach was combined with both the conventional iterative method and the proposed homodyne method (21).


Figure 1 shows the IDEAL water–fat decomposition of a full resolution acquisition in the water–oil resolution phantom, after artificial asymmetric filtering of ky lines (echo fraction = 0.625, 160 points) to simulate a partial ky acquisition. Reconstruction with simple zero-filling (Fig. 1a–c) and the proposed homodyne method (Fig. 1d–f) are shown, revealing improved apparent spatial resolution and absence of ringing in the zero-filled images. Figure 2 shows cropped water images from the same acquisition, including a full-resolution reconstruction (Fig. 2a) for comparison. In addition, an image reconstructed by filtering the raw k-space matrix with the ramp filter (Eq. [4]), but no additional processing is shown in Fig. 2d. Although there is considerable improvement in blurring and ringing from the zero-filled reconstruction shown in Fig. 2b, there is still some blurring evident in the up–down direction. The image in Fig. 2e is the difference between the full-resolution reconstruction (Fig. 2a) and the homodyne reconstructed image (Fig. 2c) and is displayed with window and level settings adjusted to accentuate the subtle differences between the two images. Areas of greatest disparity occur, as expected, at edges where spatial frequencies are greatest and may break the assumption of smoothly varying phase and field map. In addition, difference images between the homodyne reconstructed image (Fig. 2c) and the zero-filled image (Fig. 2b), as well as the difference between the homodyne reconstructed image (Fig. 2c) and the ramp-filtered image (Fig. 2d), are shown in Fig. 2f and g, respectively, revealing relatively large differences between these images. Overall, there is obvious improvement in the apparent resolution of the homodyne reconstructed image in comparison to the zero-filled image and ramp-filtered image, while the homodyne reconstructed image and full-resolution image have a very similar appearance.

Figure 1.

Recombined (a, d), calculated water (W) (b, e), and calculated fat (F) (c, f) IDEAL images of a water–oil resolution phantom, demonstrating the distribution of water and oil within the phantom. The image plane is parallel to the floor, along the long axis of the cylinder. The k-space data have been artificially filtered in the ky direction (vertical) to simulate asymmetric sampling in this direction, with echo fraction of 0.625. Images in the top row (a–c) are reconstructed with zero-filling and the bottom row (d–f) are reconstructed with the proposed homodyne method. The zero-filled images (top row) appear slightly blurred in the phase encoding direction, in comparison to the homodyne reconstructed images (bottom row).

Figure 2.

Calculated IDEAL water phantom images reconstructed with (a) full resolution acquisition (256 × 256). After artificial partial ky asymmetric under sampling (phase encoding direction = vertical), reconstruction with (b) zero-filling, (c) the proposed homodyne method, and (d) ramp filter only without homodyne reconstruction. The difference between full resolution reconstruction and homodyne reconstructed image is shown in (e) and is displayed with window/level set to 10% of the window level of the images in (a–c). (f) Difference image between the zero-filled image (b) and the homodyne reconstructed image (c). (g) Difference image between homodyne reconstructed image (c) and ramp filtered image (d). The latter two difference images (f, g) have window/level set to half that of images in (a–d).

Figure 3 shows calculated IDEAL water images obtained from a T2W FSE acquisition obtained in the sagittal plane from a normal volunteer. Figure 3a shows a full resolution image, while Fig. 3b and c contains water images reconstructed after asymmetric removal of multiple ky lines (echo fraction = 0.625), using zero-filling (Fig. 3b) and homodyne (Fig. 3c). Figure 3d contains the difference image between the full-resolution and homodyne reconstructed images. Improvement in the apparent spatial resolution of the homodyne reconstructed image is seen in comparison with the zero-filled image and has very similar appearance to the full-resolution image.

Figure 3.

Calculated IDEAL water images of a sagittal T2W FSE image from the knee of a normal volunteer, reconstructed with (a) full resolution acquisition (256 × 256). After artificial partial ky asymmetric undersampling (160 points), reconstruction with (b) simple zero-filling and (c) the proposed homodyne method. The phase encoding direction is in the left–right direction. The difference between full-resolution reconstruction and homodyne reconstructed image is shown in (d) and is displayed with very low window/level to show the subtle differences between the images. The difference between homodyne reconstruction and zero-filling reconstruction is shown in (e) and is displayed at the same window/level as the images in (a–c).

Figure 4 shows calculated water, fat, and recombined short axis CINE images from one phase (of 20) multicoil IDEAL-SSFP cardiac acquisition in a normal volunteer. The readout of this acquisition was a fractional echo with 154 of 256 points (echo fraction = 0.6), acquired asymmetrically in the kx direction (left–right). The top row is reconstructed with simple zero-filling, while the bottom row is reconstructed with the homodyne method. Subtle but definite improvement in the sharpness of the homodyne reconstructed images can be seen. This was most noticeable when the images are played as a movie.

Figure 4.

Calculated end-diastolic cardiac CINE IDEAL-SSFP recombined (a, d), water (b, e), and fat (c, f) images from a healthy volunteer acquired with a four-coil phased array. Images are reconstructed with simple zero-filling (a–c), and homodyne reconstruction (d–f). 224 × 128 SSFP images were acquired with partial kx acquisition (134 points, readout oriented up–down in image). Homodyne reconstructed images have less apparent blur than zero-filled images, most notable along the endocardial border, trabeculae, and papillary muscles (long, thin arrows). Vessels of the liver and spleen also appear sharper (short, fat arrows).

Finally, Fig. 5 contains maximum intensity projection images of 3D noncontrast enhanced angiographic IDEAL-SSFP images of the left popliteal trifurcaction of a healthy volunteer, acquired with partial kx readout (superior–inferor, 160 points). Reconstruction was performed with zero-filling (Fig. 4a) and the homodyne method (Fig. 4b). A fat-saturated SSFP angiographic image is shown for comparison (Fig. 4c) (6, 22) and has similar sharpness as the homodyne reconstructed image.

Figure 5.

Maximum intensity projection images of 3D non-contrast-enhanced angiographic IDEAL-SSFP water images of the left popliteal trifuraction of a healthy volunteer, acquired with partial kx readout (160 points). The readout direction is vertically oriented. Reconstruction was performed with (a) zero-filling and (b) homodyne. Fat-saturated acquisition is shown (c) for comparison, and a close-up of each image is shown in (df). Full matrix size of these acquisitions is 256 × 204 × 96 with spatial resolution of 0.9 × 0.9 × 1.0 mm3. Arrows depict small, horizontal vessels that appear sharper in the homodyne reconstructed image (b, e) relative to the reconstruction using zero-filling (a, d).


In this work we have described a novel combination of conventional homodyne reconstruction with a multicoil iterative least squares water–fat separation algorithm, facilitating partial k-space acquisitions with maximized spatial resolution. Both phantom and in vivo images, acquired with either conventional or phased array coils, demonstrate dramatically improved resolution over simple zero-filling of unsampled areas of k-space and demonstrate very similar image quality to full k-space acquisitions. Comparison of homodyne reconstructed phantom images with phantom images reconstructed with a ramp filter also showed improved spatial resolution.

The ability to reconstruct full resolution images from partial ky data sets will allow for substantial decreases in overall scan time. Typical echo fractions are 0.55–0.65, providing a 40% decrease in the minimum scan time with minimal compromise to image quality. Partial kx acquisitions are beneficial for short TR sequences such as SSFP and SPGR and could also be used to minimize first moment phase shifts. Maintaining a short TR is particularly important for SSFP to reduce banding artifacts caused by local field inhomogeneities (9).

As Noll et al. described in their original description of homodyne imaging, signal-to-noise ratio (SNR) will be expected to decrease as the fraction of unsampled k-space increases, although in the low SNR regime, homodyne reconstruction of fully sampled data sets may actually improve SNR (11). A full SNR analysis of this method in the context of water–fat separation was not performed for the current study.

The effects of decreasing the acquired echo fraction on image quality were also not investigated in this work. The distribution of spatial frequencies of phase shifts within an image is highly dependent on the acquisition method as well as the object itself. It should be expected that the proposed method would have similar sensitivity to aggressively small echo fractions as conventional homodyne reconstruction methods. With very low echo fractions, however, images reconstructed with zero-filling will have considerable degradation of image resolution while those reconstructed with the proposed homodyne method should maintain high resolution, so long as the assumptions that spatial frequency distributions of the field map and constant phase shifts are not grossly violated.

Conventional homodyne reconstruction algorithms assume that phase shifts within an image are smoothly varying in space. These phase shifts are a combination of those caused by field inhomogeneities (dn(r)), and constant phase shifts (ϕW(r), ϕF(r)), and are the same phase shifts seen with water–fat separation methods. It should be reasonable to expect that these assumptions will be equally valid for conventional homodyne as for homodyne water–fat separation algorithms. In fact, phase shifts caused by chemical shift between water and fat may have very rapid spatial variation and could be problematic for partial k-space acquisitions using conventional homodyne reconstruction, and the proposed IDEAL-homodyne method may actually have decreased sensitivity to aggressively small echo fractions. For example, consider a gradient echo image acquired with TE > 0 such that the phase between water and fat is nonzero (and not equal to 180°). In this case, the spatial frequencies of phase shifts from water–fat chemical shift are coupled with the spatial frequencies of the internal water–fat structures within the object, which may have high spatial frequencies, particularly at tissue interfaces. However, with the IDEAL homodyne method described above, the homodyne component of the algorithm is applied after water–fat separation has occurred. Further work must be performed to understand fully the relationship of echo fraction on image quality and water–fat decomposition.

Homodyne reconstruction methods can also be used to reduce scan time with 3D acquisitions using partial kz (depth encoding) acquisitions (23). Combination of homodyne reconstruction methods with water–fat separation methods using partial kz acquisitions should be a straightfoward extension of the algorithm described above.

This algorithm could also be applied to methods like those described by Ma et al., where prefocusing gradients are used to shift the center of partial echo readouts, shortening the time between refocusing pulses for FSE acquisitions (16). This method improves sequence efficiency and may reduce blurring artifacts in the phase encoding direction, caused by T2 decay, by reducing the spacing between refocusing pulses (24). A similar approach could be used with the IDEAL-SSFP water–fat separation method to shift the center of the echo by adjusting the readout prephaser. The proposed homodyne method would then be necessary to reconstruct the images shifted by means of prephasers rather than a bulk shift of the entire readout gradient. This would help shorten TR, which would reduce potential banding artifacts, and improve sequence efficiency.

Finally, partial k-space acquisitions could potentially be combined with parallel imaging methods that are already used with water–fat separation methods (25–27). This would facilitate even further reductions in minimum scan times.


This work has demonstrated the successful implementation of partial k-space reconstruction techniques in combination with multicoil water–fat separation methods. This approach will help improve the spatial resolution of images generated by water–fat decomposition methods that have previously relied on simple zero-filling techniques. This will facilitate the use of partial readout methods used to obtain short TR and TE, as well as partial ky and kz methods used to reduce total scan time.


The authors thank Dwight Nishimura and John Pauly for helpful discussions, John Lorbiecki for construction of the water–oil resolution phantom, and Ann Shimakawa for her assistance.