Multiecho reconstruction for simultaneous water-fat decomposition and T2* estimation


  • Presented in conference at the 14th Annual Scientific Meeting of ISMRM, Seattle, WA, USA, 2006. 15th Annual Scientific Meeting of ISMRM, Berlin, Germany, 2007.



To describe and demonstrate the feasibility of a novel multiecho reconstruction technique that achieves simultaneous water-fat decomposition and T2* estimation. The method removes interference of water-fat separation with iron-induced T2* effects and therefore has potential for the simultaneous characterization of hepatic steatosis (fatty infiltration) and iron overload.

Materials and Methods

The algorithm called “T2*-IDEAL” is based on the IDEAL water-fat decomposition method. A novel “complex field map” construct is used to estimate both R2* (1/T2*) and local B0 field inhomogeneities using an iterative least-squares estimation method. Water and fat are then decomposed from source images that are corrected for both T2* and B0 field inhomogeneity.


It was found that a six-echo multiecho acquisition using the shortest possible echo times achieves an excellent balance of short scan and reliable R2* measurement. Phantom experiments demonstrate the feasibility with high accuracy in R2* measurement. Promising preliminary in vivo results are also shown.


The T2*-IDEAL technique has potential applications in imaging of diffuse liver disease for evaluation of both hepatic steatosis and iron overload in a single breath-hold. J. Magn. Reson. Imaging 2007;26:1153–1161. © 2007 Wiley-Liss, Inc.

MRI IS INCREASINGLY IMPORTANT for the characterization of diffuse liver diseases. Hepatic steatosis, or fatty infiltration of the liver, is the primary feature and the earliest manifestation of nonalcoholic fatty liver disease (NAFLD), now recognized as one of the most common chronic liver diseases (1). MRI is emerging as a valuable noninvasive tool for early detection and monitoring of steatosis (2, 3). Hepatic iron overload is also a common chronic liver disease. The excessive concentration of iron in the liver tissue produces significant signal dephasing and T2* shortening, reducing T2* in liver to as short as 2 or 3 msec in extreme cases (4). MRI has been shown to have excellent sensitivity to the presence of iron with T2*-weighted sequences (4–6). It has been demonstrated that R2* (=1/T2*) is strongly correlated with liver iron and the relationships between MRI-measured R2* and biopsy-obtained HIC (hepatic iron concentration) have been derived in various studies (7–9). Furthermore, the co-occurrence of hepatic steatosis and iron overload is increasingly recognized as being common. Although the exact interaction between the two conditions remains unclear (10) and the hemochromatosis gene mutation is evidently ethnic-specific (11), various studies have found an increased prevalence of the hemochromatosis gene mutation or signs of hepatic iron overload in patients with steatosis (9%–31%) (12, 13), particularly in Anglo-Celtic subjects (22% and higher) (11, 14). It is therefore very important to consider the possible presence of iron when attempting to quantify steatosis and vice versa.

Conventional methods of measuring liver fat content or iron deposition using MRI both rely on signal intensity changes among multiple echoes. The amount of fat in liver can be determined from water and fat images decomposed from multipoint chemical shift-based techniques, known as the “Dixon” methods (15–17). However, it has been assumed that T2* decay results in negligible signal loss among the echoes. In the presence of iron deposition, however, T2* shortening may be significant on the time scale of Dixon echo shifts, severely compromising the estimation of fat. It has been shown that a rapid T2* decay may lead to substantial errors in hepatic fat estimates, and a T2* map must be collected separately for correction (2). Bydder et al (18) showed that the 2-point Dixon can underestimate the fat-signal fraction when the T2* is short. Westphalen et al (19) recently reported that no correlation was found between the radiologist reading and biopsy when quantifying liver fat for iron deposition patients. Likewise, any method that attempts to measure T2* may be confounded by the possible presence of steatosis. If fat coexists with water in a voxel, T2* relaxometry may be disturbed by the chemical shift of fat, due to constructive and destructive interference of fat and water signals, potentially causing errors in the estimate of T2*. A recently developed multipoint iterative reconstruction algorithm, known as IDEAL (Iterative Decomposition of water and fat with Echo Asymmetry and Least-squares estimation), allows water-fat decomposition with flexible echo sampling times and more than three echoes (20, 21). IDEAL has great potential for quantitative evaluation of fatty infiltration in the liver. However, T2* cannot be estimated and may interfere with the fat quantification.

A promising technique that models the effects from both fat and T2* is described by Bydder et al (18); however, it is challenging to distinguish water and fat without further assumption. In this work we propose a new reconstruction algorithm, based on the IDEAL method, to achieve simultaneous estimation of tissue water content, fat content, and T2*. A novel construct of a “complex field map” is used to include the effects of T2*, which is assumed to be equal for water and fat. The IDEAL algorithm is then modified to estimate the “complex field map,” thus achieving estimation of T2* even in the presence of fat. Like the conventional IDEAL algorithm, the new technique (called “T2*-IDEAL”) permits reconstruction from flexible echo times and thus data can be acquired as rapidly as possible. As a result, it is possible to achieve both water-fat decomposition and T2* estimation in a single breath-hold.


T2*-IDEAL Reconstruction with Weighted Least-Squares Inversion

In the presence of iron overload, we make the assumption that the shortened T2* effect is dominated by the presence of iron and, as a result, the water and fat components that coexist in the same voxel have similar values of T2*. The signals (Si) of a voxel at the echo times (ti, i=1, 2, 3… k, k = number of echoes acquired) can be represented as:

equation image(1)

where w and f denote the water and the fat components in this voxel, respectively. Δf is the chemical shift of fat with respect to water. ψ represents the B0 field inhomogeneity (in Hz), or field map, at this voxel. ni is the noise in the signal. We have used R2* for convenience. Furthermore, a “complex field map” is introduced:

equation image(2)

With this “complex field map,” equation image, Eq. (1) has the same form as the signal model used in the IDEAL algorithm (20). Therefore, water, fat, and equation image can be calculated in a similar way as the IDEAL algorithm, the details of which are described in the Appendix. The converged value of equation image is then decomposed with the real and imaginary parts assigned to the field map and the R2* map estimates. The source signals are demodulated by equation image, thereby correcting for both B0 field inhomogeneity and T2* decay simultaneously, as denoted in Eq.(3):

equation image(3)

Considering all echoes, Eq.(3) can be formulated in a matrix form:

equation image(4)

Note that with the T2* correction the variance of the noise (n') is no longer equal for all echoes:

equation image(5)

Equation (5) suggests that the source signals after correction for field map and T2* (s') have less noise at earlier echoes, which is an intuitive result as signals decay away exponentially. To account for the different noise variance, we propose to obtain water and fat components from a weighted least-squares inversion, shown in the following equation:

equation image(6)

where the weights are given by W =diag(emath imagemath image, emath imagemath image, … emath imagemath image). The value of R2* is obtained from the iterative estimation of equation image as described earlier.

Noise Performance

The T2*-IDEAL method essentially includes a process of fitting the signals to an exponential decay, where typically 5–20 echoes have been used in the context of R2* mapping in liver (4, 22). Acquiring more than three echoes improves T2* estimation and correction but results in increased scan time through an increased TR. For abdominal imaging the breath-hold time becomes the dominant factor when determining the number of echoes. We found 6-echo acquisitions achieve a good balance of short scan time and improved T2* estimation. With increasing parallel imaging accelerations or reductions in spatial resolution, longer echo trains can be used.

The SNR performance of a water-fat decomposition method can be measured by the effective NSA (number of signal averages), defined in the following equation with the water image as an example:

equation image(7)

The upper bound of the effective NSA can be predicted by the Cramer-Rao bound (CRB) (23). The CRB of the T2*-IDEAL signal model (Eq. [1]) is formulated based on previous work (23). In general, it varies with the water-fat ratio in the voxel, the echo times used, the number of echoes, and the actual T2* value. With a fixed number of echoes (=6), the NSAeff of water was studied to guide the distribution of the echo times. Figure 1 shows the water NSAeff maps predicted from the CRB. To be more general, the plots were made based on the water-fat phase shifts, defined in the following:

equation image(8)

Θ1 and ΔΘ are the corresponding water-fat phase shifts of TE1 and ΔTE. Plots for two representative T2* values, 40 msec (Fig. 1a) and 20 msec (Fig. 1b) are shown. The NSAeff values from T2*-IDEAL reconstruction is reduced from IDEAL due to the additional degree of freedom (T2*). The correction for the T2* decay, which exponentially magnifies the relative noise in the source images, also contributes to the apparent decrease in NSAeff. Specifically, the correction for faster T2* decay leads to more noise magnification, which explains the larger NSAeff values in Fig. 1a than the ones in Fig. 1b. Both NSAeff maps show a similar pattern of higher NSAeff values with a shorter Θ1, which suggests that a short TE1 is favored in order to improve the noise performance. With a fixed Θ1, NSAeff is less sensitive to the change of ΔΘ, unless ΔΘ is close to 0 or 2π, when water and fat are close to in-phase. As a result, the minimum echo times (including minimum TE1 and ΔTE) can be used, provided that the corresponding ΔΘ is not close to 0 or 2π. This strategy allows acquiring as many echoes as possible and achieving the best (or near best) possible noise performance without increasing the scan time.

Figure 1.

Noise performance of the T2*-IDEAL model: water NSAeff maps predicted from the Cramer-Rao Bound for equally spaced six echoes with assumed T2* of 40 msec (a) and 20 msec (b). A short TE1 is favored in order to maximize the noise performance, while the NSAeff is less sensitive to the change of echo spacing except that a value close to 0 or 2π should be avoided.


Phantom Experiments

Phantom studies were first performed to validate the T2*-IDEAL algorithm. Acetone was used as an alternative to fat because it is soluble in water. The chemical shift of acetone was measured by spectroscopy and determined to be 2.4 ppm, or −155 Hz at 1.5T. Equal volumes of water and acetone were mixed in six 50-mL tubes, which were then doped with increasing concentrations of Feridex IV (Berlex Laboratories, Wayne, NJ), a superparamagnetic iron oxide (SPIO) contrast agent that shortens T2*. The composition of the phantom tubes is illustrated in Fig. 2a. A 2D spoiled gradient echo (SPGR) IDEAL sequence was used for all phantom scans. Three echoes were acquired to simulate the most challenging situation for the T2*-IDEAL method. Echo times ti = [3.8, 8.2, 12.5] msec were selected to achieve optimal SNR performance for IDEAL reconstruction (21, 23). The data were processed with both the T2*-IDEAL algorithm and IDEAL. In addition, the phantom was imaged at three echo times chosen when water and acetone are in-phase, ie: Θi = [2π, 4π, 6π], ti = [6.5, 13.1, 19.6] msec at 1.5T. TR was 500 msec and the flip angle was 60° for both scans. The R2* map was measured independently by fitting the in-phase signals to an exponential decay. Comparison between the R2* maps obtained from the two approaches was made.

Figure 2.

Results from water-acetone phantom scans. a: The composition of the phantom. The center tube has only water. The other six tubes contain the same volume of water and acetone, which were doped with increasing concentrations of an SPIO contrast agent that shortens T2*. b,c: Water and acetone images from 3-point IDEAL reconstruction. d,e: Water and acetone images from T2*-IDEAL reconstruction with the same data, with water and acetone images corrected for the T2* decay. f,g: R2* maps obtained from T2*-IDEAL reconstruction and an alternative approach of using three in-phase signals. h: R2* values estimated from T2*-IDEAL and in-phase signals are plotted versus iron concentrations. The two approaches are in close agreement. The linear behavior is evident. Error bars indicate standard deviation.

In Vivo Scans

Abdominal imaging was performed to demonstrate the feasibility of the T2*-IDEAL technique in vivo with 5 healthy volunteers and 12 liver patients with known or suspected chronic liver disease on 1.5T scanners (Signa HDx, GE Healthcare, Waukesha, WI). All scanning was performed after Institutional Review Board approval and informed consent was obtained. A multiecho 3D SPGR sequence with flyback gradients (24) was used.

A variety of imaging parameters were used to evaluate the flexibility of the sequence and the reconstruction technique. A minimum of six echoes was acquired. Unlike 3-point IDEAL, it is no longer necessary to choose an echo spacing of ΔΘ = 2π/3, allowing increased flexibility in the choice of other imaging parameters, such as spatial resolution, field of view (FOV), and bandwidth. In practice, the first TE (TE1) ranged from 1.6–2.8 msec, and the echo spacing (ΔTE) ranged from 1.6–3.2 msec. Imaging matrices ranging from 192 × 128 to 256 × 128 and slice thickness of 8 mm were used with bandwidths ranging from ±100 kHz to ±167 kHz. Scans with higher resolutions (384 × 256) were performed on healthy volunteers. These imaging parameters lead to a breath-hold time between 22 and 33 seconds. An efficient autocalibrating parallel imaging technique (25) with reduction factor of 2 was used to reduce acquisition time for high-resolution scans. All data were reconstructed with both IDEAL and T2*-IDEAL methods. Synthesized in-phase, out-of-phase, and fat-signal fraction (fat/(water+fat)) images were created using the two methods and compared. R2* maps were also obtained from T2*-IDEAL processing. For one patient scan with fatty infiltration of liver, 12 echoes were acquired. Fat-signal fraction images were calculated from the IDEAL and T2*-IDEAL methods using an incremental number of echoes, starting from the first three echoes up to all 12 echoes.


Phantom Experiments

Figure 2 shows the decomposed water and acetone images from the 3-point phantom experiments. At higher SPIO concentrations with shorter T2*, the signal decay was propagated into the decomposed water (Fig. 2b) and acetone (Fig. 2c) images for IDEAL. In contrast, T2*-IDEAL (Fig. 2d,e) compensates for the signal loss from the T2* decay. For the outer six tubes with both water and acetone, the shorter T2* phantom tubes have higher signal intensity in the water and acetone images due to the known T1 shortening effect of SPIOs. Furthermore, the T2*-IDEAL method provided accurate estimates of the R2* values (Fig. 2f), demonstrating close agreement with the in-phase R2* measurements (Fig. 2g). Figure 2h plots R2* calculated from the T2*-IDEAL method and the in-phase signals against the iron concentration. Mean values in each tube were used and error bars indicate standard deviation. A linear relation is evident, in close agreement with previous measurements of R2* (4, 9). Close agreement between the two approaches suggests high confidence in the R2* estimated using the T2*-IDEAL method even with only three echoes. Unlike the in-phase approach to measure R2*, the T2*-IDEAL method also allows a shorter TR and separation of water and acetone signals.

In Vivo Results

Figure 3 shows images from a patient with known genetic hemochromatosis and the resulting severe hepatic iron overload. In this acquisition, six echoes ranging from 2.0–10.8 msec were used. The R2* map (Fig. 3c) demonstrates a markedly elevated R2* (shortened T2*) in the liver. The mean R2* value measured in the indicated region of interest (ROI) is 187 s−1, translating to T2* = 5.3 msec. Note that R2* in the spleen appears normal, typical for hemochromatosis patients where iron overload primarily affects the liver and pancreas. The T2*-IDEAL also maintains a uniform water-fat decomposition (Fig. 3a,b). However, the T2*-IDEAL water and fat images appear noisy due to the intrinsic low SNR of the source images with a significantly shortened T2*. Due to the interference from the T2* decay, IDEAL reconstruction results in a biased, 11% fat-signal fraction (Fig. 3e) in liver. This bias is not present from T2*-IDEAL reconstruction (Fig. 3d). Compared to Fig. 3a, a noisier water image was obtained from T2*-IDEAL reconstruction with nonweighted least-squares inversion (Fig. 3f). As expected, the water image from IDEAL shows low signal intensity (Fig. 3g) due to the uncorrected T2* decay. Finally, Fig. 3h shows the acquired and fitted signal curves of a typical pixel in liver. The acquired signals clearly suggest the presence of a T2* decay. While the fitted signals from T2*-IDEAL could mostly follow the acquired signals, the IDEAL fitted signals resulted in significant residue errors. The fitted signal variation at different echoes from IDEAL was caused by the water-fat constructive and destructive effects, suggesting a nonnegligible and biased estimation of fat content in this pixel.

Figure 3.

Results from a hepatic iron overload patient. a–c: Water, fat and R2* map obtained from T2*-IDEAL reconstruction. An elevated R2* level can be seen in the liver, which indicates the presence of a high iron concentration. d,e: Fat-signal fraction images from T2*-IDEAL and IDEAL, respectively, showing a fat bias in IDEAL. f: The water image from T2*-IDEAL reconstruction without applying the weighted least-squares inversion. g: The water image from IDEAL, where the liver appears dark due to the T2* decay. h: The acquired signals and the fitted signals at the six echoes from T2*-IDEAL and IDEAL of a typical pixel in liver. Six echoes were collected, TE1 = 2.0 msec, ΔTE = 1.8 msec. Other imaging parameters include: 224 × 128 matrix, FOV = 35 × 21 cm, BW = ±143 kHz, TR = 15.5 msec, flip angle = 10°; 28 locations were covered in two 22-second breath-holds with no parallel imaging acceleration.

Results from a patient with mild iron deposition are presented in Fig. 4. The mean R2* value measured in the indicated ROI is 60 s−1 (Fig. 4c), corresponding to T2* = 17 msec. The decomposed water and fat images (Fig. 4a,b) show only a moderate noise magnification with the correction for a mild T2* decay.

Figure 4.

Results from a liver patient with mild iron overload. a–c: Water, fat, and R2* images from T2*-IDEAL reconstruction, respectively. The mean T2* value measured in the indicated liver ROI is 17 msec, suggesting the presence of mild iron. Six echoes were collected, TE1 = 2.0 msec, ΔTE = 1.6 msec. Other imaging parameters include: 192 × 128 matrix, 35 × 28 cm FOV, BW = ±125 kHz, TR = 12.5 msec, flip angle = 15°; 24 locations were covered in a 29-second breath-hold without the use of parallel imaging.

In vivo scans were also performed on healthy volunteers and a representative case is shown in Fig. 5. A higher imaging matrix (384 × 256) was used. As a result, the minimum possible echo spacing was extended to 2.7 msec (ΔΘ = 1.1π). The strategy of using this minimum echo spacing in combination with a parallel imaging reduction factor of 2 (25) ensures a total scan time (33 seconds) achievable with a single breath-hold for normal volunteers. The measured T2* values ranged from 21–45 msec in our asymptomatic volunteers, consistent with previously reported T2* values in normal liver (7, 22).

Figure 5.

Results from a healthy volunteer with a high-resolution scan. a–c: Water, fat and R2* images from T2*-IDEAL reconstruction. d,e: Water and fat images from IDEAL reconstruction. The mean T2* value in the indicated ROI is 21 msec, in the range of T2* values of a normal liver. The IDEAL reconstruction provides high SNR water and fat images. Six echoes were collected, TE1 = 1.6 msec, ΔTE = 2.7 msec. Other imaging parameters include: 384 × 256 matrix, 33 × 26 cm FOV, BW = ±167 kHz, TR = 20 msec, flip angle = 15°; 10 locations were covered in a 33-second breath-hold accelerated by an efficient autocalibrating parallel imaging technique (R = 2).

Finally, results from a patient with known hepatic steatosis are shown in Fig. 6. Twelve echoes were collected for this acquisition, although the images (Fig. 6a–d) shown are from T2*-IDEAL reconstruction using the first six echoes. The decomposed water and fat images (Fig. 6a,b) clearly show an abnormal amount of fat content in liver. The fat-signal fraction image (Fig. 6c) confirms this observation, with measured mean fat-signal fraction of 31% in the ROI. The R2* map suggested no shortened T2* in liver (T2* ≈25 msec) for this patient. Furthermore, images were reconstructed using both IDEAL and T2*-IDEAL with an increasing number of echoes, starting from the first three echoes up to all 12 echoes. The corresponding fat-signal fraction images were obtained. Mean fat-signal fraction values in the ROI (Fig. 6c) are plotted against the number of echoes for both IDEAL and T2*-IDEAL methods. As can be seen, the fat-signal fraction values calculated from IDEAL appear inconsistent due to a confounding effect from the T2* decay. In contrast, T2*-IDEAL provides stable estimation, so long as four or more echoes are used. This study demonstrates the importance of T2* correction when quantifying hepatic steatosis, even in the presence of a normal T2* decay.

Figure 6.

Results from a patient with hepatic steatosis. Twelve echoes were collected. a–d: Water, fat, fat-signal fraction, and R2* images from 6-point T2*-IDEAL reconstruction, using the first six of the 12 echoes. 31% fat-signal fraction is measured in the indicated liver ROI. e: Fat-signal fraction values measured from IDEAL and T2*-IDEAL using an increasing number of echoes. The measurements from IDEAL vary with the number of echoes used, due to the interference from the uncorrected T2* decay. Such variation is not observed from the T2*-IDEAL measurements. Therefore, it is important to correct for T2* when quantifying fat content even in the presence of a normal T2* decay. Imaging parameters include: TE1 = 2.8 msec, ΔTE = 3.2 msec, 224 × 128 matrix, 36 × 25 cm FOV, BW = ±100 kHz, TR = 39 msec, flip angle = 10°; eight locations were covered in a 29-second breath-hold with no parallel imaging acceleration.


We have presented a novel multiecho reconstruction method capable of decoupling and estimating the fat content and the T2* decay. This method shows great potential for simultaneous assessment of fatty infiltration and iron overload in patients with chronic liver disease. To remove the possible confounding effects of fat and iron, conventional methods require careful design of the imaging parameters (eg, acquire in-phase images only for R2* mapping), often leading to limitations in sequence flexibility or multiple breath-holds. The T2*-IDEAL technique introduced in this work accounts for the effects of both fat and shortened T2*, and therefore is promising for reliable and rapid measurement of both fat content and the presence of iron.

The multiecho sequence allows the collection of more than three echoes with a relatively small scan time penalty. Depending on the specific imaging parameters, it may still be difficult to cover the entire liver in a single breath-hold. Some of our patient scans shown may have insufficient coverage due to long scan time, which can be resolved by using parallel imaging acceleration in one or more directions.

A major assumption of the T2*-IDEAL technique is that water and fat that coexist in a voxel will have the same T2* relaxation. With severe iron overload T2* decay is dominated by the presence of iron and this may be a reasonable assumption. With mild or no iron present, the T2* values of the water and fat within a voxel may be different. However, a previous study has shown that fat-signal fraction calculated from a single-valued T2* correction is within 10% of the true percentage when the T2* value is 35 msec for fat and 25 msec for water (2). We performed a simulation with T2*-water = 49 msec and T2*-fat = 71 msec (26). Using the T2*-IDEAL reconstruction, a true fat-signal fraction of 30% resulted in measured fat-signal fraction of 30.8% and estimated T2* of 50 msec. If T2-water and T2-fat are known, or its difference is known, they can be easily incorporated in the model, and the T2*-IDEAL can be used to estimate the part of the T2* decay that is caused by the nontissue-specific factors, such as susceptibility and iron. Future work will also explore the possibility of extending the algorithm to estimate multicomponent T2* decays, similar to the technique described previously (18). In addition, the formulation shown in Eq.(1) does not take into account the spectral complexity of the fat (27). The resultant error in R2* estimation can be reduced by increasing the echo train duration or completely removed by incorporating a measured fat spectrum in the model.

The accuracy of the T2*-IDEAL technique, including the T2* estimation and fat quantification, requires clinical validation. Ultimately, co-registration of biopsy with the results from T2*-IDEAL reconstruction must be performed in order to obtain a meaningful correlation.

In conclusion, we have described an algorithm that can achieve chemical species separation and T2* quantification simultaneously. As a result, the scope of applications for water-fat decomposition methods such as IDEAL is greatly extended. Even though its clinical utility requires further validation, T2*-IDEAL is a promising technique for fast and reliable assessment of patients with chronic liver diseases with concomitant hepatic steatosis and iron overload.


Modified IDEAL Algorithm to Calculate the Complex Field Map equation image

With the signals collected at all echoes, Eq.(1) can be formatted in a matrix form:

equation image(A.1)


equation image
equation image

The vector s denotes the acquired signals. The matrix A is considered known. The matrix P(equation image) is a function of the complex field map and represents the field map and R2* modulation on the signals. The noise term in Eq.(1) has been dropped for convenience. At each pixel the following algorithm is used to estimate the complex field map equation image.

  • 1Starting from the initial guess of the complex field map equation image = ψ0. An initial guess of 0 is used for R2* at all pixels. equation image represents the current estimate of the “complex field map.”
  • 2With the estimated equation image, the corresponding complex water and fat can be determined from a least-squares inversion:
    equation image(A.2)
    where AT represents the complex conjugate transpose of the A matrix. Here, we have used the fact that P(− equation image) · P(equation image)=P(equation image) · P(−equation image)=I.
  • 3Equation (1) can be approximated by Taylor expansion as in the following, with the second and higher-order terms neglected.
    equation image(A.3)
    Considering all echoes, Eq. (A.3) can be formulated in a matrix form:
    equation image(A.4)
    equation image
    Therefore, error terms can be obtained by another least-squares inversion:
    equation image(A.5)
    where B(w̃,f̃) has been simplified as B.
  • 4Update the estimated complex field map:
    equation image(A.6)
  • 5With the new equation image, repeat steps 2–4 until the following convergence criterion is achieved or a predefined maximum number of iterations (30) is met:
    equation image(A.7)
    where ε denotes a small number. In practice, ε = 1 can be used.