Combination of complex-based and magnitude-based multiecho water-fat separation for accurate quantification of fat-fraction

Authors

  • Huanzhou Yu,

    Corresponding author
    1. Applied Science Laboratory, GE Healthcare, Menlo Park, California, USA
    • 333 Ravenswood Ave. Bldg 307, Menlo Park, California 94025
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  • Ann Shimakawa,

    1. Applied Science Laboratory, GE Healthcare, Menlo Park, California, USA
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  • Catherine D. G. Hines,

    1. Department of Radiology, University of Wisconsin, Madison, Wisconsin, USA
    2. Department of Biomedical Engineering, University of Wisconsin, Madison, Wisconsin, USA
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  • Charles A. McKenzie,

    1. Department of Medical Biophysics, The University of Western Ontario, London, Ontario, Canada
    2. Department of Physics, The University of Western Ontario, London, Ontario, Canada
    3. Department of Biomedical Engineering, The University of Western Ontario, London, Ontario, Canada
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  • Gavin Hamilton,

    1. Department of Radiology, University of California, San Diego, California, USA
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  • Claude B. Sirlin,

    1. Department of Radiology, University of California, San Diego, California, USA
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  • Jean H. Brittain,

    1. Applied Science Laboratory, GE Healthcare, Madison, Wisconsin, USA
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  • Scott B. Reeder

    1. Department of Radiology, University of Wisconsin, Madison, Wisconsin, USA
    2. Department of Biomedical Engineering, University of Wisconsin, Madison, Wisconsin, USA
    3. Department of Medical Physics, University of Wisconsin, Madison, Wisconsin, USA
    4. Department of Medicine, University of Wisconsin, Madison, Wisconsin, USA
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Abstract

Multipoint water–fat separation techniques rely on different water–fat phase shifts generated at multiple echo times to decompose water and fat. Therefore, these methods require complex source images and allow unambiguous separation of water and fat signals. However, complex-based water–fat separation methods are sensitive to phase errors in the source images, which may lead to clinically important errors. An alternative approach to quantify fat is through “magnitude-based” methods that acquire multiecho magnitude images. Magnitude-based methods are insensitive to phase errors, but cannot estimate fat-fraction greater than 50%. In this work, we introduce a water–fat separation approach that combines the strengths of both complex and magnitude reconstruction algorithms. A magnitude-based reconstruction is applied after complex-based water–fat separation to removes the effect of phase errors. The results from the two reconstructions are then combined. We demonstrate that using this hybrid method, 0–100% fat-fraction can be estimated with improved accuracy at low fat-fractions. Magn Reson Med, 2011. © 2011 Wiley-Liss, Inc.

The applications of multiecho chemical shift based water–fat separation techniques are continuously expanding, both for robust fat suppression in routine clinical scanning (1, 2) and for quantitative fat imaging such as evaluation of hepatic steatosis (3–5) and vertebral bone marrow (6). These methods rely on the different water–fat phase shifts generated at multiple echo times to estimate the amplitude of static (polarizing) field (B0) field map and water and fat content on a voxel by voxel basis. Therefore, these methods require complex source images. In addition to the phase information contained within the acquired signals, robust water–fat separation methods often use the a priori information that magnetic field inhomogeneities are smooth, allowing accurate and complete separation of water and fat (7–11). However, complex-based water–fat separation methods are sensitive to unexpected phase errors in the source images, such as those from eddy currents. Although the impact of these phase errors is small and acceptable for most qualitative applications, as we show below, such phase errors can be clinically meaningful for quantitative applications.

Alternatively, water–fat separation can be achieved using only the magnitude of the complex source signals (12, 13), including the original 2-point “Dixon” method (14). These “magnitude-based” methods are insensitive to phase errors in the source images, because the phase information is discarded. However, because of the lack of phase information, these methods alone are unable to determine fat fraction [i.e., fat/(water + fat)] over a 50% range (13). Additional information can be used to help the identification of water and fat, such as using low resolution spectroscopy (15) and difference in T1 (3, 16) or Tmath image relaxation parameters (17), which require either additional data or a more complex model.

A clinically useful fat quantification technique must be both accurate and robust. The complex-based and magnitude-based methods have complementary strengths and weaknesses. In this work, we introduce a new water–fat separation approach that combines the strengths of both complex and magnitude reconstruction methods. A complex-based approach is used to separate the fat and water signals to achieve a dynamic range of 0–100% fat-fraction, followed by a magnitude-based reconstruction to fine tune the quantification and remove phase errors. The results from the two reconstructions are then combined to provide consistent noise performance for all water–fat combinations. Using this “hybrid” approach, reliable and accurate fat-fractions can be estimated, particularly at low fat-fractions.

THEORY

Figure 1 demonstrates the impact of eddy-current induced phase errors on quantitative water–fat separation applications. Figure 1a illustrates water and fat images reconstructed from a six-point multiecho abdominal scan and Tmath image-Iterative Decomposition of Water and Fat with Echo Asymmetric and Least-squares Estimation (IDEAL) processing (18), a complex-based water–fat separation method based on IDEAL algorithm (19) that corrects for both Tmath image decay and the multiple spectral peaks of fat (20). Water and fat images are further used to produce fat-fraction maps with noise-bias correction (21) to directly visualize the fat percentage in the liver. The scan was performed in a healthy volunteer. An MR spectroscopic scan (not shown) confirmed that there was no fat present in the volunteer's liver. However, the fat-fraction map (Fig. 1a) obtained from Tmath image-IDEAL showed 8% fat in liver, which can be considered abnormal according to the 5.56% cut-off found by a 2349-patient study (22). This “fatty liver” artifact can be clinically significant as it is very important to identify low fat-fraction accurately for early diagnosis and treatment of fatty liver disease. Figure 1b shows the plot of the signal phase for mean signals measured from a region of interest (ROI) (10 × 10 voxels) placed in liver together with the read-out gradient waveforms used in this scan. All six echoes are collected in one repetition (pulse repetition time) with a 3D multiecho SPoiled Gradient Recalled acquisition (SPGR) sequence and fly-back gradients. For a voxel containing only water, the phase of the signals should evolve linearly at a rate proportional to the B0 field inhomogeneity. In Fig. 1b, however, the phase of the first echo (arrow) is slightly deviated from the linear curve that the last 5 echoes follow. This is due to the fact that the prewinder gradient preceding the first read-out gradient is different from the fly-back gradient preceding the read-out gradient of the other echoes. This difference causes inconsistent eddy current-generated phase modulation and results in imperfect water–fat separation that primarily varies in the read-out direction (right-left in this example).

Figure 1.

Reconstruction results (water, fat, and fat-fraction images) of a six-echo abdominal scan at 1.5 T using various reconstruction approaches: complex-based Tmath image-IDEAL algorithm (a), the magnitude-based algorithm (c), and the proposed complex-based and magnitude-based hybrid algorithm (d). Read-out direction is right-left in this scan (horizontal). An artifactual 8% of fat is measured with Tmath image-IDEAL, due to the phase error primarily modulated on the first echo using the multiecho “fly-back” sequence (b). Both the magnitude-based and the hybrid reconstructions reduce the apparent fat-fraction measurement to 3%. The improvement in fat-fraction is clinically relevant considering the value dropped from the abnormal range to the normal range. Imaging parameters include: 256 × 160 × 16 matrix size, field of view = 35 × 25 cm, 6 echoes, TE1 = 1.2 ms, ΔTE = 2 ms, and 25-s breath-hold.

MATERIALS AND METHODS

The flow diagram of the proposed hybrid approach is illustrated in Fig. 2. We elaborate each component in the following.

Figure 2.

Flow diagram of the proposed hybrid water-fat separation algorithm, based on a two-step approach. The first step applies complex-based Tmath image-IDEAL to reconstruct water, fat, and Rmath image maps from the multiecho complex source signals. The water and fat images are used as an initial guess for the following second step, where water–fat separation is performed using only the magnitude signals. Rmath image values from the first step are used to correct the source images, thus simplifying the fitting algorithm in the 2nd step. This two-step processing results in another set of water and fat estimates. The water and fat images from the two steps are further combined, weighted by the results from the first step, forming the final estimates of water, fat, and fat-fraction images. For simplicity, we refer to the two-step approach without weighted combination as the “magnitude-based” water–fat separation algorithm, whereas the final estimates with the weighted combination are from the “hybrid” method.

Complex-Based Water-Fat Separation Method

There are many multiecho water–fat separation methods that are based on complex source signals. In our approach, a Tmath image corrected iterative water–fat separation method (Tmath image-IDEAL) (18, 19) that uses multipeak spectral modeling of fat (20) and noise bias correction (21) is used. T1 related bias is avoided by using a low flip angle (21). Six echoes are collected with a multiecho 3D-SPGR sequence using fly-back gradients (Fig. 1b). The complex source data are then processed using the Tmath image-IDEAL algorithm to produce water (Wc), fat (Fc), and Rmath image maps. A region growing algorithm is applied to provide robust field map estimation and thus to resolve water–fat ambiguity (9).

Magnitude-Based Water-Fat Separation Method

The magnitude-based methods alone are unable to use the phase information to uniquely label water and fat signals. Therefore, these methods alone do not provide separated water and fat images. Figure 2 illustrates a two-step approach, in which the magnitude-based reconstruction is applied following the complex-based Tmath image-IDEAL. The results from the first step (Tmath image-IDEAL) are used as the initial guess for the magnitude-based reconstruction. The water and fat estimated from step 1 should be very close to the true water and fat quantities even in the presence of the phase errors. Therefore, they serve as an excellent initial guess for the estimation in the step 2, ensuring fast convergence and correct identification of water and fat. The second step, magnitude reconstruction, can be viewed as fine tuning the solution from the complex-based reconstruction to provide a more accurate estimate of fat-fraction. This 2-step approach that eventually leads to the results of (Wm, Fm) is referred to as the “magnitude-based” water–fat separation method.

The magnitude-based reconstruction aims to solve the following equation:

equation image(1)

where W and F are water and fat contents, i is the echo index (i = 1, 2… n, where n is the number of echoes). ci is the fat signal modulation term. For single peak modeling of the fat spectrum, ci can be described as:

equation image

where Δf is the chemical shift between water and fat. For convenience, we have used ai =|ci| and bi = Re{ci}.

Note that for the described modeling of the fat signal, ai = 1. Consequently, W and F are exchangeable in Eq. 1 without altering the value of the equation. This is the inherent water–fat ambiguity, which cannot be resolved without other a priori information.

When a more accurate fat spectrum is modeled with multiple (P) spectral peaks (13, 20), represented by relative amplitudes αp and frequencies Δfp, we have:

equation image

In this multipeak model, ai =|ci| is in general no longer equal to 1. Although exchanging water and fat in Eq. 1 now changes the value of the equation, the water–fat swapped solution is reflected as a local minimum when solving Eq. 1, resulting in a similar water–fat ambiguity challenge as the single peak model.

Because of the nonlinear and nonconvex nature of Eq. 1, it is often solved with iterative nonlinear data fitting algorithms. To simplify the nonlinear fitting in the magnitude-based reconstruction, the source signals are corrected for Tmath image decay using the Tmath image estimate from the complex-based Tmath image-IDEAL before the magnitude-based reconstruction, as shown in Fig. 2. This is based on the consideration that the algorithm largely relies on the magnitude changes between the echoes to estimate Tmath image, so the error in estimated Tmath image is small. In addition, Tmath image is often over-estimated using the magnitude signals if the noise bias is not taken into account (23). With Tmath image estimation removed in the magnitude-based algorithm, Eq. 1 can be simplified as:

equation image(2)

A simple iterative Gauss-Newton nonlinear curve fitting (24) is then used to solve Eq. 2, as described in the Appendix.

Noise Performance Simulations

Simulations were used to study the noise performance and accuracy of the complex-based and magnitude-based reconstructions. Source signals were generated with a typical protocol [6 echoes, echo time (TE)1 = 1.2 ms, ΔTE = 2 ms, Rmath image = 50 s−1] and sweeping fat-fraction from 0 to 100%. Gaussian distributed noise was added to the 6-echo source images. To simulate the effect of the eddy currents, a phase error (0.1π) was modulated on the first echo. This process was repeated 1000 times. Both complex-based water–fat separation (i.e., Tmath image-IDEAL) and the magnitude-based reconstruction described above were performed. The ratio of the noise variance in the source images (σmath image) and the noise variance in the resulting fat-fraction images (σmath image) was calculated and is shown in Fig. 3a. Larger ratios indicate improved noise behavior in the separation algorithm. The magnitude-based reconstruction, implemented as the two-step approach in Fig. 2, demonstrates poor noise performance near fat-fractions close to 50%, whereas the complex-based reconstruction results in relatively better noise behavior throughout all fat-fractions, in agreement with an earlier study (18). Figure 3b shows the bias (i.e., absolute error) of the fat-fraction measured from the two reconstructions. The complex-based Tmath image-IDEAL is sensitive to phase errors on the first echo near 0% or 100% fat-fraction, as expected. The magnitude-based reconstruction is generally insensitive to the phase error except near 50% fat due to its poor noise performance in that range.

Figure 3.

Noise performance (a) and errors (b) of simulations comparing the complex (blue circle), magnitude (red star), and proposed hybrid method (black dot). σs and σff denote the noise standard deviation measured in the source images and the fat-fraction images, respectively. The hybrid approach applies a Fermi weighting function (c) that combines the water and fat estimated with the magnitude-based and complex-based reconstructions.

Complex-Based and Magnitude-Based “Hybrid” Approach

As shown in Figure 3a,b, the complex-based reconstruction provides relatively superior noise performance; however, the accuracy is compromised by the bias created by the presence of phase errors, particularly at low- and high- fat-fraction. The magnitude-based reconstruction on the other hand has poor noise performance around 50%; however, it is in general more accurate in the presence of phase errors. To combine the strengths of these two approaches, we introduce a “hybrid” algorithm, in which the final water and fat images (W and F) are obtained from a weighted combination of the water and fat images from the two reconstruction steps, as illustrated in Fig. 2 (Wc and Fc from complex-based reconstruction, Wm and Fm from magnitude-based reconstruction):

equation image(3)

The weight λ is determined from the fat-fraction calculated from the first step of complex-based Tmath image-IDEAL. In our approach, a Fermi function (25) is used (Fig. 3c). When fat-fraction resulting from the complex-based reconstruction is close to 0 or 100%, λ approaches 0 such that the water and fat images from magnitude-based reconstruction are used. On the other hand, when fat-fraction from the first step is near 50%, λ approaches 1 such that the fat fraction calculated from the complex-based reconstruction contributes most to the final results.

In Fig. 3a,b, the black dotted curves represent simulation results from the hybrid approach. The noise performance is more favorable at fat-fractions near 50% than the magnitude-based method. Although a small error is introduced near 50%, low- and high-fat-fractions are more accurately estimated, which is critical for liver fat quantification. However, the improved accuracy at low fat-fractions is at the cost of reduced noise performance in those ranges compared with the complex reconstruction.

Experiments

In vivo scanning was performed on GE 1.5 T and 3 T scanners (HDx, GE Healthcare, Waukesha, WI) to evaluate the various approaches described. Informed consent and permission from our Institutional Review Board were obtained. A 3D-SPGR multiecho sequence was used to collect six echoes with “fly-back” gradients. For each scan, water–fat separation was performed using complex-based (Tmath image-IDEAL), the magnitude-based, and the hybrid methods using the same source data. The fat-fraction images were generated from the separated water and fat images and evaluated for each approach.

Comparison With MR Spectroscopy

To demonstrate that the hybrid reconstruction improves the accuracy of the fat-fraction measurements, comparison of MRI measured fat-fraction with MR spectroscopy measured fat-fraction was made in 58 studies (55 patients with 3 patients scanned twice) at 1.5 T that do not include the in vivo scans mentioned earlier. All studies were performed with Institutional Review Board approval and informed written consent. As a reference standard, single voxel magnetic resonance spectroscopy (MRS) using stimulated echo acquisition mode was performed (26). Imaging parameters of MRI include: 256 × 128 × 24 matrix size, field of view = 35 × 35 cm, 6 echoes, TE1 = 1.3 ms, ΔTE = 2.0 ms, 5 degree flip angle, and 21-second breath-hold with parallel imaging acceleration. Imaging parameters of MRS included: voxel size = 20 × 20 × 25 mm3, pulse repetition time = 3.5 s to minimize T1 weighting, 2048 readout points, 1 signal average, receiver bandwidth = ±5 kHz, and 5 echo times at 10, 20, 30, 40, 50 ms (to facilitate T2 correction), requiring a 21 s breath-hold. The MRS voxel was placed in the right hepatic lobe avoiding large blood vessels and other nonliver tissues. All MRS data were postprocessed by an MR physicist blinded to the MRI results, using AMARES algorithm (27) in jMRUI (28). The comparison between MRS and imaging results using the hybrid reconstruction will also be included in a separate clinical study (29) based on an earlier report (30). In this work, we retrospectively reconstructed the imaging data of all patients using the complex-based Tmath image-IDEAL as well as the hybrid method. Fat-fraction measurements were obtained in ROIs drawn in the liver co-localized with the MRS voxel and perfectly co-registered between the two reconstructions and were compared with the MR spectroscopy measured fat-fraction.

RESULTS

Figure 1 shows results from a healthy volunteer at 1.5 T, reconstructed by the complex-based Tmath image-IDEAL, the magnitude-based reconstruction, and the hybrid approach, respectively. The complex-based Tmath image-IDEAL results in 8% of artifactual fat in liver, whereas spectroscopy (not shown) confirms no presence of fat in liver. With the magnitude-based reconstruction and the hybrid approach, the fat-fraction measured in liver is reduced to 3%. This improvement may be clinically significant as the diagnosis of steatosis is typically made when liver fat content exceeds 5.56% (22). Additionally, the magnitude and the hybrid reconstructions successfully produced separated water and fat images with the help of the results from the complex reconstruction in the first step.

Results from a patient with severe steatosis at 3 T are presented in Fig. 4. The complex-based reconstruction measures 49% fat in liver. As predicted by simulations shown in Fig. 3, the magnitude-based reconstruction leads to substantial noisy artifact in the liver due to its poor noise performance near 50%. This artifact is removed in the hybrid approach, where the weighted combination of the liver fat-fraction measurements is dominated by the results of the complex reconstruction.

Figure 4.

Results from a six-echo scan at 3 T in a patient with significant fatty infiltration of liver. Fat-fraction images of various approaches are shown. Images are cropped to the liver area of interest. The fat-fraction is 49% measured from complex-based Tmath image-IDEAL. As predicted by the simulations, the magnitude-based reconstruction demonstrates noisy fat-fraction estimated in liver, which is not present in the fat-fraction image produced by the hybrid reconstruction due to the weighted combination process. Imaging parameters include: 192 × 192 × 16 matrix size, field of view = 44 × 33 cm, 6 echoes interleaved in 2 repetitions, TE1 = 0.8 ms, ΔTE = 0.7 ms, and 26-s breath-hold.

In 58 patient scans at 1.5 T, the MRI measured fat-fraction is compared with the MRS measured fat-fraction. The in vivo scans shown in Figs. 1 and 4 were not part of this study. The correlation and Bland–Altman plots are shown in Fig. 5. The complex-based Tmath image-IDEAL results in fat-fraction bias at the low fat-fraction regime, which results in a slope of 0.91 and intercept of 2.17%. With the hybrid reconstruction, the agreement between MRS and MRI is substantially improved (slope = 1.00, intercept = 0.18%), demonstrating the importance of correcting phase errors for quantitative fat imaging applications. The improvement can also be seen from the Bland–Altman plots, where the hybrid reconstruction leads to a much tighter 95% confidence interval [(−1.3%, 1.7%) vs. (−2.9%, 6.2%)] from complex-based reconstruction). The sensitivity and specificity of the complex-based reconstruction are 0.95 (18/19) and 0.69 (27/39), respectively, compared with 1 (19/19) and 1 (39/39) from the new hybrid reconstruction. Note that the correlation as well as sensitivity and specificity of the hybrid reconstruction were also included in the results of Meisamy et al. (29). The limitation of the study is that no patient was found to possess fat-fraction higher than 30%. In addition, the magnitude-only reconstruction was not performed. The performance of the magnitude-only reconstruction is expected to be similar to the hybrid reconstruction as none of the patients included in this study has fat-fraction near 50%.

Figure 5.

Comparison of MR spectroscopy measured fat-fraction with imaging measured fat-fraction in 58 studies in the format of correlation plot and Bland–Altman plot. With complex-based Tmath image-IDEAL reconstruction (a), substantial bias is seen with the low fat-fraction measurement. From the Bland–Altman plot (b), the mean of the difference is 1.6% and the 95% confidence interval is (−2.9%, 6.2%). The correlation is improved with the hybrid reconstruction (c), particularly at low fat fraction regime, which is also reflected in the tighter Bland–Altman plot (d) [mean of the difference: 0.2%, 95% confidence interval: (−1.3%, 1.7%)]. In (a) and (c), the solid lines represent linear regression, whereas the dashed line denotes the identity line (Y = X).

DISCUSSION

Conventional multipoint chemical shift water–fat separation methods rely heavily on phase information to separate water and fat. As a result, they are sensitive to phase errors, such as those from eddy currents. For quantitative applications, we have shown that the phase errors can lead to a clinically relevant bias particularly at low fat-fractions. The challenge of accurate low fat-fraction measurement has also been observed in other studies using chemical shift based water–fat separation methods (4, 31, 32). Considering that one study concluded the cut-off fat-fraction for normal vs. abnormal liver is ∼5.56% (22), a clinically useful fat quantification technique must be capable of quantifying fat accurately, particularly at low fat-fractions.

In this work, we introduced a hybrid multipoint water–fat separation approach, where a Tmath image-corrected, spectrally modeled “complex based” water–fat separation method (Tmath image-IDEAL) is followed by a fitting algorithm based on magnitude multiecho signals. The results from the two steps are combined for the final estimate of water and fat to remove the effects of phase errors as well as to avoid poor noise performance at fat-fractions near 50%. The second step, based on magnitude signals, relies on the results from the first step for initial conditioning of the fit to achieve a full dynamic range from 0 to 100%.

As Tmath image decay contains no phase information, we have assumed that the Tmath image estimated from the complex water–fat separation is sufficiently accurate and can be used to correct the source signals, such that additional Tmath image correction is not required in step 2. This approximation allows much simpler nonlinear fitting because no exponential terms (emath image) are involved. In addition, the complex method provides a relatively accurate (although not perfect) initial estimate of water and fat signals, and therefore, the magnitude-based fitting is very fast, typically requiring only a few iterations for convergence. Therefore, minimal additional computational overhead is necessary for the hybrid approach compared with the complex-based method.

Despite better accuracy, our two-step implementation of the magnitude-based reconstruction has poor noise performance near 50%, which is likely a manifestation of water–fat ambiguity. For those magnitude-based reconstruction methods that only attempt to estimate fat-fraction in a 50% dynamic range (13), it is possible that they do not suffer from this noise problem near 50%. To alleviate this noise performance limitation, a weighted combination of the results from the two reconstructions was proposed. This combination weights the contribution from the magnitude-based reconstruction more heavily at low fat-fractions and the complex-based reconstruction is weighted more heavily near 50%. In our work, we chose a Fermi function to achieve smooth transition between the two regimes. However, there are many weighting functions that could be chosen to achieve the same goals. Despite the weighted combination, the improved accuracy at the low fat fractions with the hybrid reconstruction is at the cost of reduced noise performance compared with the complex-reconstruction.

The phase errors from eddy currents could also be removed with alternative methods. For example, if the phase errors were always in the first echo, it is possible to formulate a problem that fits the model using only the magnitude of the first echo signal in addition to complex signals of the rest 5 echoes. Another approach would be to perform an estimate from the last five echoes first, the results of which are used to extrapolate a phase for the first echo, replacing the initial erroneous phase. A six-echo Tmath image-IDEAL can then be performed. However, it is not ideal to discard the first echo as the first echo has the best SNR in methods that model Tmath image (18). The hybrid approach we proposed in this work can also be used for removing more general phase errors, for example, phase errors in water–fat separation methods with bipolar gradients (non-fly-back) (33, 34).

In conclusion, the proposed complex-based and magnitude-based hybrid water–fat separation method is an effective approach for removing undesired phase errors for fat-fraction measurement. Correction of unwanted phase shifts leads to significantly improved accuracy for quantification of low fat-fractions, which may be the most clinically relevant for methods used for screening and diagnosis of hepatic steatosis.

Acknowledgements

The authors thank Dr. Mark Bydder for helpful discussion.

APPENDIX

We describe a simple iterative nonlinear fitting algorithm to solve Eq. 2, which is shown in the following (Eq. A1) for convenience.

equation image(A1)

Step 1: Use initial estimates of water and fat based on those from the complex-based Tmath image-IDEAL (Wc, Fc) water–fat separation, i.e.:

equation image

Step 2: Calculate the signals corresponding to the current estimates:

equation image

Step 3: Calculate the error terms:

equation image

We now define the matrix B as:

equation image

Therefore, a linear least squares inversion will give an estimate of the error terms (ΔW and ΔF):

equation image

Step 4: Update the current estimates

equation image

Step 5: Go to step 2, unless iteration converges or the maximum number of iteration is reached.

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