Assuming the posterior density to have a Gaussian distribution, we find

p ( x | y ) = p ( x ) p ( y | x ) p ( y ) = exp ⁡ [ − F α ( x ) ] / z p ( y ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiCaaNaeiikaGIaeCiEaGNaeiiFaWNaeCyEaKNaeiykaKIaeyypa0tcfa4aaSaaaeaacqWGWbaCcqGGOaakcqWH4baEcqGGPaqkcqWGWbaCcqGGOaakcqWH5bqEcqGG8baFcqWH4baEcqGGPaqkaeaacqWGWbaCcqGGOaakcqWH5bqEcqGGPaqkaaGccqGH9aqpjuaGdaWcaaqaaiGbcwgaLjabcIha4jabcchaWjabcUfaBjabgkHiTiabdAeagnaaBaaabaGaeqySdegabeaacqGGOaakcqWH4baEcqGGPaqkcqGGDbqxcqGGVaWlcqWG6bGEaeaacqWGWbaCcqGGOaakcqWH5bqEcqGGPaqkaaaaaa@5C77@

where z is a normalization constant called the partition function, F_α(x) = U₁(x) + αL(x) where U₁(x) and L(x) are energy functions associated with p(y | x) and p(x) respectively, and α (a positive scalar) is a tuning or regularization parameter. Then

x ˆ = min ⁡ x ( F α ( x ) ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiEaGNbaKaacqGH9aqpdaWfqaqaaiGbc2gaTjabcMgaPjabc6gaUbWcbaGaeCiEaGhabeaakiabcIcaOiabdAeagnaaBaaaleaacqaHXoqyaeqaaOGaeiikaGIaeCiEaGNaeiykaKIaeiykaKIaeiOla4caaa@3D47@

If measurement noise is assumed to be white, Gaussian and zero-mean, one can write U₁(x) as

U₁(x) = ||Kx - y||²

where K is a compact linear operator 716 (representing the forward solution) and ||.|| is the usual L₂ norm. L(x) may be written as U_s(x) + U_t(x) where U_s(x) introduces spatial (anatomical) priors and U_t(x) temporal ones 415. Combining the data attachment term with the prior term,

x ˆ = min ⁡ x ( F α ( x ) ) = min ⁡ x ( | | K x − y | | 2 + α L ( x ) ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiEaGNbaKaacqGH9aqpdaWfqaqaaiGbc2gaTjabcMgaPjabc6gaUbWcbaGaeCiEaGhabeaakiabcIcaOiabdAeagnaaBaaaleaacqaHXoqyaeqaaOGaeiikaGIaeCiEaGNaeiykaKIaeiykaKIaeyypa0ZaaCbeaeaacyGGTbqBcqGGPbqAcqGGUbGBaSqaaiabhIha4bqabaGccqGGOaakcqGG8baFcqGG8baFcqWHlbWscqWH4baEcqGHsislcqWH5bqEcqGG8baFcqGG8baFdaahaaWcbeqaaiabikdaYaaakiabgUcaRiabeg7aHjabdYeamjabcIcaOiabhIha4jabcMcaPiabcMcaPiabc6caUaaa@58E7@

This equation reflects a trade off between fidelity to the data and spatial/temporal smoothness depending on the α.

In the above, p(y | x) ∝ exp(-X^T.X) where X = Kx - y. More generally, p(y | x) ∝ exp(-Tr(X^T.σ^-1.X)), where σ^-1 is the data covariance matrix and 'Tr' denotes the trace of a matrix.

Even more generally, p(y | x) ∝ exp(-Tr((X - μ)^T.σ^-1.(X - μ))), where μ is the mean value of X. Suppose R is the variance-covariance matrix when a Gaussian noise component is assumed and Y is the matrix corresponding to the measurements y. The R-norm is defined as follows:

| | Y − K X | | R 2 = T r [ ( Y − K X ) T R − 1 ( Y − K X ) ] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiiFaWNaeiiFaWNaeCywaKLaeyOeI0IaeC4saSKaeCiwaGLaeiiFaWNaeiiFaW3aa0baaSqaaiabhkfasbqaaiabikdaYaaakiabg2da9iabdsfaujabdkhaYjabcUfaBjabcIcaOiabhMfazjabgkHiTiabhUealjabhIfayjabcMcaPmaaCaaaleqabaGaemivaqfaaOGaeCOuai1aaWbaaSqabeaacqGHsislcqaIXaqmaaGccqGGOaakcqWHzbqwcqGHsislcqWHlbWscqWHybawcqGGPaqkcqGGDbqxaaa@5056@

Non-Gaussian priors include entropy metrics and L_p norms with p < 2 i.e. L(x) = ||x||_p.

Entropy is a probabilistic concept appearing in information theory and statistical mechanics. Assuming x ∈ ℝⁿ consists of positive entries x_i > 0, i = 1, ..., n the entropy is defined as

ℰ ( x ) = − ∑ i = 1 n x i log ⁡ ( x i x i ∗ ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8hmHuKaeiikaGIaeCiEaGNaeiykaKIaeyypa0JaeyOeI0YaaabCaeaacqWG4baEdaWgaaWcbaGaemyAaKgabeaakiGbcYgaSjabc+gaVjabcEgaNnaabmaajuaGbaWaaSaaaeaacqWG4baEdaWgaaqaaiabdMgaPbqabaaabaGaemiEaG3aa0baaeaacqWGPbqAaeaacqGHxiIkaaaaaaGccaGLOaGaayzkaaaaleaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGUbGBa0GaeyyeIuoaaaa@5348@

where xi∗ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiEaG3aa0baaSqaaiabdMgaPbqaaiabgEHiQaaaaaa@2FC5@ > 0 is a is a given constant. The information contained in x relative to xi∗ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemiEaG3aa0baaSqaaiabdMgaPbqaaiabgEHiQaaaaaa@2FC5@ is the negative of the entropy. If it is required to find x such that only the data Kx = y is used, the information subject to the data needs to be minimized, that is, the entropy has to be maximized. The mathematical justification for the choice L(x) = -ℰ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8hmHueaaa@36AD@(x) is that it yields the solution which is most 'objective' with respect to missing information. The maximum entropy method has been used with success in image restoration problems where prominent features from noisy data are to be determined.

As regards L_p norms with p < 2, we start by defining these norms. For a matrix A, ||A||p=∑i,j|aij|pp MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiiFaWNaeiiFaWNaeCyqaeKaeiiFaWNaeiiFaW3aaSbaaSqaaiabdchaWbqabaGccqGH9aqpdaGcbaqaamaaqababaGaeiiFaWNaemyyae2aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGG8baFdaahaaWcbeqaaiabdchaWbaaaeaacqWGPbqAcqGGSaalcqWGQbGAaeqaniabggHiLdaaleaacqWGWbaCaaaaaa@454C@ where a_ij are the elements of A. The defining feature of these prior models is that they are concentrated on images with low average amplitude with few outliers standing out. Thus, they are suitable when the prior information is that the image contains small and well localized objects as, for example, in the localization of cortical activity by electric measurements.

As p is reduced the solutions will become increasingly sparse. When p = 1 17 the problem can be modified slightly to be recast as a linear program which can be solved by a simplex method. In this case it is the sum of the absolute values of the solution components that is minimized. Although the solutions obtained with this norm are sparser than those obtained with the L₂ norm, the orientation results were found to be less clear 17. Another difference is that while the localization results improve if the number of electrodes is increased in the case of the L₂ approach, this is not the case with the L₁ approach which requires an increase in the number of grid points for correct localization. A third difference is that while both approaches perform badly in the presence of noisy data, the L₁ approach performs even worse than the L₂ approach. For p < 1 it is possible to show that there exists a value 0 <p < 1 for which the solution is maximally sparse. The non-quadratic formulation of the priors may be linked to previous works using Markov Random Fields 1819. Experiments in 20 show that the L₁ approach demands more computational effort in comparision with L₂ approaches. It also produced some spurious sources and the source distribution of the solution was very different from the simulated distribution.

Regularization is the approximation of an ill-posed problem by a family of neighbouring well-posed problems. There are various regularization methods found in the literature depending on the choice of L(x). The aim is to find the best-approximate solution x^δ of Kx = y in the situation that the 'noiseless data' y are not known precisely but that only a noisy representation y^δ with ||y^δ - y|| ≤ δ is available. Typically y^δ would be the real (noisy) signal. In general, an xαδ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCiEaG3aa0baaSqaaiabeg7aHbqaaiabes7aKbaaaaa@30C3@ is found which minimizes

F_α(x) = ||Kx - y^δ||² + αL(x).

In Tikhonov regularization, L(x) = ||x||² so that an xαδ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCiEaG3aa0baaSqaaiabeg7aHbqaaiabes7aKbaaaaa@30C3@ is found which minimizes

F_α(x) = ||Kx - y^δ||² + α||x||².

It can be shown (in Appendix) that

x α ( δ ) δ = ( K ∗ K + α I ) − 1 K ∗ y δ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCiEaG3aa0baaSqaaiabeg7aHjabcIcaOiabes7aKjabcMcaPaqaaiabes7aKbaakiabg2da9iabcIcaOiabhUealnaaCaaaleqabaGaey4fIOcaaOGaeC4saSKaey4kaSIaeqySdeMaeCysaKKaeiykaKYaaWbaaSqabeaacqGHsislcqaIXaqmaaGccqWHlbWsdaahaaWcbeqaaiabgEHiQaaakiabhMha5naaCaaaleqabaGaeqiTdqgaaaaa@45E4@

where K* is the adjoint of K. Since (K*K + αI)^-1K* = K*(KK* + αI)^-1 (proof in Appendix),

x α ( δ ) δ = K ∗ ( K K ∗ + α I ) − 1 y δ . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCiEaG3aa0baaSqaaiabeg7aHjabcIcaOiabes7aKjabcMcaPaqaaiabes7aKbaakiabg2da9iabhUealnaaCaaaleqabaGaey4fIOcaaOGaeeikaGIaeC4saSKaeC4saS0aaWbaaSqabeaacqGHxiIkaaGccqGHRaWkcqaHXoqycqWHjbqscqGGPaqkdaahaaWcbeqaaiabgkHiTiabigdaXaaakiabhMha5naaCaaaleqabaGaeqiTdqgaaOGaeiOla4caaa@46D1@

Another choice of L(x) is

L(x) = ||Ax||²

where A is a linear operator. The minimum is obtained when

x α ( δ ) δ = ( K ∗ K + α A ∗ A ) − 1 K ∗ y MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCiEaG3aa0baaSqaaiabeg7aHjabcIcaOiabes7aKjabcMcaPaqaaiabes7aKbaakiabg2da9iabcIcaOiabhUealnaaCaaaleqabaGaey4fIOcaaOGaeC4saSKaey4kaSIaeqySdeMaeCyqae0aaWbaaSqabeaacqGHxiIkaaGccqWHbbqqcqGGPaqkdaahaaWcbeqaaiabgkHiTiabigdaXaaakiabhUealnaaCaaaleqabaGaey4fIOcaaOGaeCyEaKhaaa@4637@

In particular, if A = ∇ where ∇ is the gradient operator, then xα(δ)δ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCiEaG3aa0baaSqaaiabeg7aHjabcIcaOiabes7aKjabcMcaPaqaaiabes7aKbaaaaa@341A@ = (K*K + α∇^T∇)^-1K*y. If A = ΔB, where Δ is the Laplacian operator, then xα(δ)δ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCiEaG3aa0baaSqaaiabeg7aHjabcIcaOiabes7aKjabcMcaPaqaaiabes7aKbaaaaa@341A@ = (K*K + αB*Δ^TΔB)^-1K*y.

The regularization parameter α must find a good compromise between the residual norm ||Kx - y^δ|| and the norm of the solution ||Ax||. In other words it must find a balance between the perturbation error in y and the regularization error in the regularized solution.

Various methods 789 exist to estimate the optimal regularization parameter and these fall mainly in two categories:

1. Those based on a good estimate of ||ϵ|| where ϵ is the noise in the measured vector y^δ.

2. Those that do not require an estimate of ||ϵ||.

The discrepancy principle is the main method based on ||ϵ||. In effect it chooses α such that the residual norm for the regularized solution satisfies the following condition:

||Kx - y^δ|| = ||ϵ||

As expected, failure to obtain a good estimate of ϵ will yield a value for α which is not optimal for the expected solution.

Various other methods of estimating the regularization parameter exist and these fall mainly within the second category. These include, amongst others, the

1. L-curve method

2. General-Cross Validation method

3. Composite Residual and Smoothing Operator (CRESO)

4. Minimal Product method

5. Zero crossing

The L-curve method 212223 provides a log-log plot of the semi-norm ||Ax|| of the regularized solution against the corresponding residual norm ||Kx - y^δ|| (Figure 2a). The resulting curve has the shape of an 'L', hence its name, and it clearly displays the compromise between minimizing these two quantities. Thus, the best choice of alpha is that corresponding to the corner of the curve. When the regularization method is continuous, as is the case in Tikhonov regularization, the L-curve is a continuous curve. When, however, the regularization method is discrete, the L-curve is also discrete and is then typically represented by a spline curve in order to find the corner of the curve.

Similar to the L-curve method, the Minimal Product method 24 aims at minimizing the upper bound of the solution and the residual simultaneously (Figure 2b). In this case the optimum regularization parameter is that corresponding to the minimum value of function P which gives the product between the norm of the solution and the norm of the residual. This approach can be adopted to both continuous and discrete regularization.

P(α) = ||Ax(α)||.||Kx(α) - y^δ||

Another well known regularization method is the Generalized Cross Validation (GCV) method 2125 which is based on the assumption that y is affected by normally distributed noise. The optimum alpha for GCV is that corresponding to the minimum value for the function G:

G = | | K x ( α ) − y δ | | 2 ( T r ( I − K T ) ) 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4raCKaeyypa0tcfa4aaSaaaeaacqGG8baFcqGG8baFcqWHlbWscqWH4baEcqGGOaakcqaHXoqycqGGPaqkcqGHsislcqWH5bqEdaahaaqabeaacqaH0oazaaGaeiiFaWNaeiiFaW3aaWbaaeqabaGaeGOmaidaaaqaaiabcIcaOiabdsfaujabdkhaYjabcIcaOiabhMeajjabgkHiTiabhUealjabhsfaujabcMcaPiabcMcaPmaaCaaabeqaaiabikdaYaaaaaaaaa@4B90@

where T is the inverse operator of matrix K. Hence the numerator measures the discrepancy between the estimated and measured signal y^δ while the denominator measures the discrepancy of matrix KT from the identity matrix.

The regularization parameter as estimated by the Composite Residual and Smoothing Operator (CRESO) 2324 is that which maximizes the derivative of the difference between the residual norm and the semi-norm i.e. the derivative of B(α):

B(α) = α²||Ax(α)||² - ||Kx(α) - y^δ||²

Unlike the other described methods for finding the regularization parameter, this method works only for continuous regularization such as Tikhonov.

The final approach to be discussed here is the zero-crossing method 23 which finds the optimum regularization parameter by solving B(α) = 0 where B is as defined in Equation (7). Thus the zero-crossing is basically another way of obtaining the L-curve corner.

One must note that the above estimators for xα(δ)δ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCiEaG3aa0baaSqaaiabeg7aHjabcIcaOiabes7aKjabcMcaPaqaaiabes7aKbaaaaa@341A@ are the same as those that result from the minimization of ||Ax|| subject to Kx = y. In this case x = K^(*)(KK^(*))^-1y where K^(*) = (AA*)^-1K* is found with respect to the inner product ⟨⟨x, y⟩⟩ = ⟨Ax, Ay⟩. This leads to the estimator,

x = (A*A)^-1K*(K(AA*)^-1K*)^-1y

which, if regularized, can be shown to be equivalent to (6).

As regards the EEG inverse problem, using the notation used in the description of the forward problem in Section ??, the Bayesian methods find an estimate Dˆ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaaaaa@2CFA@ of D such that

D ˆ = min ⁡ ( U ( D ) ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaacqGH9aqpcyGGTbqBcqGGPbqAcqGGUbGBcqGGOaakcqWGvbqvcqGGOaakcqWHebarcqGGPaqkcqGGPaqkaaa@381C@

where

U ( D ) = | | M − G D | | R 2 + α L ( D ) . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyvauLaeiikaGIaeCiraqKaeiykaKIaeyypa0JaeiiFaWNaeiiFaWNaeCyta0KaeyOeI0IaeC4raCKaeCiraqKaeiiFaWNaeiiFaW3aa0baaSqaaiabhkfasbqaaiabikdaYaaakiabgUcaRiabeg7aHjabdYeamjabcIcaOiabhseaejabcMcaPiabc6caUaaa@450E@

As an example, in 26 one finds that the linear operator A in Equation (5) is taken to be a matrix A whose rows represent the averages (linear combinations) of the true sources. One choice of the matrix A is given by

A i j = A 3 ( p − 1 ) + k , 3 ( q − 1 ) + m = w j exp ⁡ − d p q 2 / σ i 2 for k = m and zero otherwise MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7725@

In the above equation, the subscripts p, q are used to indicate grid points in the volume representing the brain and the subscripts k, m are used to represent Cartesian coordinates x, y and z (i.e. they take values 1,2,3), d_pq represents the Euclidean distances between the pth and qth grid points. The coefficients w_j can be used to describe a column scaling by a diagonal matrix while σ_i controls the spatial resolution. In particular, if σ_i → 0 and w_j = 1 the minimum norm solution described below is obtained.

In the next subsections we review some of the most common choices for L(D).

Minimum norm estimates 52728 are based on a search for the solution with minimum power and correspond to Tikhonov regularization. This kind of estimate is well suited to distributed source models where the dipole activity is likely to extend over some areas of the cortical surface.

L(D) = ||D||²

D ˆ M N E = ( G T G + α I p ) − 1 G T M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemyta0KaemOta4Kaemyraueabeaakiabg2da9iabcIcaOiabhEeahnaaCaaaleqabaGaemivaqfaaOGaeC4raCKaey4kaSIaeqySdeMaeCysaK0aaSbaaSqaaiabdchaWbqabaGccqGGPaqkdaahaaWcbeqaaiabgkHiTiabigdaXaaakiabhEeahnaaCaaaleqabaGaemivaqfaaOGaeCyta0eaaa@422C@

or

D ˆ M N E = G T ( G G T + α I N ) − 1 M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemyta0KaemOta4Kaemyraueabeaakiabg2da9iabhEeahnaaCaaaleqabaGaemivaqfaaOGaeiikaGIaeC4raCKaeC4raC0aaWbaaSqabeaacqWGubavaaGccqGHRaWkcqaHXoqycqWHjbqsdaWgaaWcbaGaemOta4eabeaakiabcMcaPmaaCaaaleqabaGaeyOeI0IaeGymaedaaOGaeCyta0eaaa@41E8@

The first equation is more suitable when N > p while the second equation is more suitable when p > N. If we let T_MNE be the inverse operator G^T(GG^T + αI_N)^-1, then T_MNEG is called the resolution matrix and this would ideally the identity matrix. It is claimed 527 that MNEs produce very poor estimation of the true source locations with both the realistic and sphere models.

A more general minimum-norm inverse solution assumes that both the noise vector n and the dipole strength D are normally distributed with zero mean and their covariance matrices are proportional to the identity matrix and are denoted by C and R respectively. The inverse solution is given in 14:

D ˆ M N E = R G T ( G R G T + C ) − 1 M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemyta0KaemOta4Kaemyraueabeaakiabg2da9iabhkfasjabhEeahnaaCaaaleqabaGaemivaqfaaOGaeiikaGIaeC4raCKaeCOuaiLaeC4raC0aaWbaaSqabeaacqWGubavaaGccqGHRaWkcqWHdbWqcqGGPaqkdaahaaWcbeqaaiabgkHiTiabigdaXaaakiabh2eanbaa@4144@

R_ij can also be taken to be equal to σ_iσ_jCorr(i, j) where σi2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeq4Wdm3aa0baaSqaaiabdMgaPbqaaiabikdaYaaaaaa@3012@ is the variance of the strength of the ith dipole and Corr(i, j) is the correlation between the strengths of the ith and jth dipoles. Thus any a priori information about correlation between the dipole strengths at different locations can be used as a constraint. R can also be taken as RiiRjj(Corr(i,j)) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWaaOaaaeaacqWGsbGudaWgaaWcbaGaemyAaKMaemyAaKgabeaakiabdkfasnaaBaaaleaacqWGQbGAcqWGQbGAaeqaaaqabaGccqGGOaakcqWGdbWqcqWGVbWBcqWGYbGCcqWGYbGCcqGGOaakcqWGPbqAcqGGSaalcqWGQbGAcqGGPaqkcqGGPaqkaaa@4067@ where Rii=f(1ζi) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOuai1aaSbaaSqaaiabdMgaPjabdMgaPbqabaGccqGH9aqpcqWGMbGzdaqadaqcfayaamaalaaabaGaeGymaedabaGaeqOTdO3aaSbaaeaacqWGPbqAaeqaaaaaaOGaayjkaiaawMcaaaaa@38A3@ is such that it is large when the measure ζ_i of projection onto the noise subspace is small. The matrix C can be taken as σ²I if it is assumed that the sensor noise is additive and white with constant variance σ². R can also be constructed in such a way that it is equal to UU^T where U is an orthonormal set of arbitrary basis vectors 12. The new inverse operator using these arbitrary basis functions is the original forward solution projected onto the new basis functions.

The Weighted Minimum Norm algorithm compensates for the tendency of MNEs to favour weak and surface sources. This is done by introducing a 3p × 3p weighting matrix W:

L ( D ) = | | W D | | 2 D ^ W M N E = ( G T G + α W T W ) − 1 G T M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabiqaaaqaaiabdYeamjabcIcaOiabhseaejabcMcaPiabg2da9iabcYha8jabcYha8jabhEfaxjabhseaejabcYha8jabcYha8naaCaaaleqabaGaeGOmaidaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaem4vaCLaemyta0KaemOta4Kaemyraueabeaakiabg2da9iabcIcaOiabhEeahnaaCaaaleqabaGaemivaqfaaOGaeC4raCKaey4kaSIaeqySdeMaeC4vaC1aaWbaaSqabeaacqWGubavaaGccqWHxbWvcqGGPaqkdaahaaWcbeqaaiabgkHiTiabigdaXaaakiabhEeahnaaCaaaleqabaGaemivaqfaaOGaeCyta0eaaaaa@52F7@

or

D ^ W M N E = ( W T W ) − 1 G T ( G ( W T W ) − 1 G T + α I N ) − 1 M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaem4vaCLaemyta0KaemOta4Kaemyraueabeaakiabg2da9iabcIcaOiabhEfaxnaaCaaaleqabaGaemivaqfaaOGaeC4vaCLaeiykaKYaaWbaaSqabeaacqGHsislcqaIXaqmaaGccqWHhbWrdaahaaWcbeqaaiabdsfaubaakiabcIcaOiabhEeahjabcIcaOiabhEfaxnaaCaaaleqabaGaemivaqfaaOGaeC4vaCLaeiykaKYaaWbaaSqabeaacqGHsislcqaIXaqmaaGccqWHhbWrdaahaaWcbeqaaiabdsfaubaakiabgUcaRiabeg7aHjabhMeajnaaBaaaleaacqWGobGtaeqaaOGaeiykaKYaaWbaaSqabeaacqGHsislcqaIXaqmaaGccqWHnbqtaaa@5267@

W can have different forms but the simplest one is based on the norm of the columns of the matrix G: W = Ω ⊗ I₃, where ⊗ denotes the Kronecker product and Ω is a diagonal p × p matrix with Ωββ=∑α=1Ng(rα,rdipβ)⋅g(rα,rdipβ)T MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuyQdC1aaSbaaSqaaiabek7aIjabek7aIbqabaGccqGH9aqpdaGcaaqaamaaqahabaGaeC4zaCMaeiikaGIaeCOCai3aaSbaaSqaaiabeg7aHbqabaGccqGGSaalcqWHYbGCdaWgaaWcbaGaemizaqMaemyAaKMaemiCaa3aaSbaaWqaaiabek7aIbqabaaaleqaaOGaeiykaKIaeyyXICTaeC4zaCMaeiikaGIaeCOCai3aaSbaaSqaaiabeg7aHbqabaGccqGGSaalcqWHYbGCdaWgaaWcbaGaemizaqMaemyAaKMaemiCaa3aaSbaaWqaaiabek7aIbqabaaaleqaaOGaeiykaKYaaWbaaSqabeaacqWGubavaaaabaGaeqySdeMaeyypa0JaeGymaedabaGaemOta4eaniabggHiLdaaleqaaaaa@5A25@, for β = 1, ..., p.

This is a recursive procedure of weighted minimum norm estimations, developed to give some focal resolution to linear estimators on distributed source models 5272930. Weighting of the columns of G is based on the mag nitudes of the sources of the previous iteration. The Weighted Minimum Norm compensates for the lower gains of deeper sources by using lead-field normalization.

D ^ F O C U S S | i = W i W i T G T ( G W i W i T G T + α I N ) − 1 M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemOrayKaem4ta8Kaem4qamKaemyvauLaem4uamLaem4uamLaeiiFaWNaemyAaKgabeaakiabg2da9iabhEfaxnaaBaaaleaacqWGPbqAaeqaaOGaeC4vaC1aa0baaSqaaiabdMgaPbqaaiabdsfaubaakiabhEeahnaaCaaaleqabaGaemivaqfaaOGaeiikaGIaeC4raCKaeC4vaC1aaSbaaSqaaiabdMgaPbqabaGccqWHxbWvdaqhaaWcbaGaemyAaKgabaGaemivaqfaaOGaeC4raC0aaWbaaSqabeaacqWGubavaaGccqGHRaWkcqaHXoqycqWHjbqsdaWgaaWcbaGaemOta4eabeaakiabcMcaPmaaCaaaleqabaGaeyOeI0IaeGymaedaaOGaeCyta0eaaa@55D8@

where i is the index of the iteration and W_i is a diagonal matrix computed using

W i = w i W i − 1  diag  ( D ^ i − 1 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeC4vaC1aaSbaaSqaaiabdMgaPbqabaGccqGH9aqpcqWH3bWDdaWgaaWcbaGaemyAaKgabeaakiabhEfaxnaaBaaaleaacqWGPbqAcqGHsislcqaIXaqmaeqaaOGaeeiiaaIaeeizaqMaeeyAaKMaeeyyaeMaee4zaCMaeeiiaaIaeiikaGIafCiraqKbaKaadaWgaaWcbaGaemyAaKMaeyOeI0IaeGymaedabeaakiabcMcaPaaa@44C3@

wi=diag(1|G(:,j)||) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeC4DaC3aaSbaaSqaaiabdMgaPbqabaGccqGH9aqpcqqGKbazcqqGPbqAcqqGHbqycqqGNbWzdaqadaqcfayaamaalaaabaGaeGymaedabaGaeiiFaWNaeC4raCKaeiikaGIaeiOoaOJaeiilaWIaemOAaOMaeiykaKIaeiiFaWNaeiiFaWhaaaGccaGLOaGaayzkaaaaaa@42D4@, j ∈ [1, 2, ..., p] is a diagonal matrix for deeper source compensation. G(:, j) is the jth column of G. The algorithm is initialized with the minimum norm solution D^MNE MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemyta0KaemOta4Kaemyraueabeaaaaa@3081@, that is, W0=diag(D^MNE)=diag(D^0(1),D^0(2),...,D^0(3p)) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeC4vaC1aaSbaaSqaaiabicdaWaqabaGccqGH9aqpcqqGKbazcqqGPbqAcqqGHbqycqqGNbWzcqGGOaakcuWHebargaqcamaaBaaaleaacqWGnbqtcqWGobGtcqWGfbqraeqaaOGaeiykaKIaeyypa0JaeeizaqMaeeyAaKMaeeyyaeMaee4zaCMaeiikaGIafCiraqKbaKaadaWgaaWcbaGaeGimaadabeaakiabcIcaOiabigdaXiabcMcaPiabcYcaSiqbhseaezaajaWaaSbaaSqaaiabicdaWaqabaGccqGGOaakcqaIYaGmcqGGPaqkcqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcuWHebargaqcamaaBaaaleaacqaIWaamaeqaaOGaeiikaGIaeG4mamJaemiCaaNaeiykaKIaeiykaKcaaa@5862@, where D^0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaeGimaadabeaaaaa@2E14@(n) represents the nth element of vector D^0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaeGimaadabeaaaaa@2E14@. If continued long enough, FOCUSS converges to a set of concentrated solutions equal in number to the number of electrodes.

The localization accuracy is claimed to be impressively improved in comparison to MNE. However, localization of deeper sources cannot be properly estimated. In addition to Minimum Norm, FOCUSS has also been used in conjunction with LORETA 31 as discussed below.

LORETA 527 combines the lead-field normalization with the Laplacian operator, thus, gives the depth-compensated inverse solution under the constraint of smoothly distributed sources. It is based on the maximum smoothness of the solution. It normalizes the columns of G to give all sources (close to the surface and deeper ones) the same opportunity of being reconstructed. This is better than minimum-norm methods in which deeper sources cannot be recovered because dipoles located at the surface of the source space with smaller magnitudes are priveleged. In LORETA, sources are distributed in the whole inner head volume. In this case, L(D) = ||ΔB.D||² and B = Ω ⊗ I₃ is a diagonal matrix for the column normalization of G.

D ^ L O R = ( G T G + α B Δ T Δ B ) − 1 G T M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemitaWKaem4ta8KaemOuaifabeaakiabg2da9iabcIcaOiabhEeahnaaCaaaleqabaGaemivaqfaaOGaeC4raCKaey4kaSIaeqySdeMaeCOqaiKaeuiLdq0aaWbaaSqabeaacqWGubavaaGccqqHuoarcqWHcbGqcqGGPaqkdaahaaWcbeqaaiabgkHiTiabigdaXaaakiabhEeahnaaCaaaleqabaGaemivaqfaaOGaeCyta0eaaa@45DE@

or

D ^ L O R = ( B Δ T Δ B ) − 1 G T ( G ( B Δ T Δ B ) − 1 G T + α I N ) − 1 M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@563A@

Experiments using LORETA 27 showed that some spurious activity was likely to appear and that this technique was not well suited for focal source estimation.

This approach is similar to MNE with FOCUSS but based on LORETA rather than MNE. It is a combination of LORETA and FOCUSS, according to the following steps:

1. The current density is computed using LORETA to get D^LOR MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemitaWKaem4ta8KaemOuaifabeaaaaa@309B@.

2. The weighting matrix W is constructed using (10), the initial matrix being given by W0=diag(D^LOR)=diag(D^0(1),D^0(2),...,D^0(3p)) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeC4vaC1aaSbaaSqaaiabicdaWaqabaGccqGH9aqpcqqGKbazcqqGPbqAcqqGHbqycqqGNbWzcqGGOaakcuWHebargaqcamaaBaaaleaacqWGmbatcqWGpbWtcqWGsbGuaeqaaOGaeiykaKIaeyypa0JaeeizaqMaeeyAaKMaeeyyaeMaee4zaCMaeiikaGIafCiraqKbaKaadaWgaaWcbaGaeGimaadabeaakiabcIcaOiabigdaXiabcMcaPiabcYcaSiqbhseaezaajaWaaSbaaSqaaiabicdaWaqabaGccqGGOaakcqaIYaGmcqGGPaqkcqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcuWHebargaqcamaaBaaaleaacqaIWaamaeqaaOGaeiikaGIaeG4mamJaemiCaaNaeiykaKIaeiykaKcaaa@587C@, where D^0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaeGimaadabeaaaaa@2E14@(n) represents the nth element of vector D^0 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaeGimaadabeaaaaa@2E14@.

3. The current density D^i MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemyAaKgabeaaaaa@2E81@ is computed using (9).

4. Steps (2) and (3) are repeated until convergence.

Standardized low resolution brain electromagnetic tomography (sLORETA) 32 sounds like a modification of LORETA but the concept is quite different and it does not use the Laplacian operator. It is a method in which localization is based on images of standardized current density. It uses the current density estimate given by the minimum norm estimate D^MNE MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemyta0KaemOta4Kaemyraueabeaaaaa@3081@ and standardizes it by using its variance, which is hypothesized to be due to the actual source variance S_D = I_3p, and variation due to noisy measurements SMnoise MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeC4uam1aa0baaSqaaiabh2eanbqaaiabd6gaUjabd+gaVjabdMgaPjabdohaZjabdwgaLbaaaaa@3545@ = αI_N. The electrical potential variance is S_M = GS_DG^T + SMnoise MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeC4uam1aa0baaSqaaiabh2eanbqaaiabd6gaUjabd+gaVjabdMgaPjabdohaZjabdwgaLbaaaaa@3545@ and the variance of the estimated current density is SD^=TMNESMTMNET=GT[GGT+αIN]−1G MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeC4uam1aaSbaaSqaaiqbhseaezaajaaabeaakiabg2da9iabhsfaunaaBaaaleaacqWGnbqtcqWGobGtcqWGfbqraeqaaOGaeC4uam1aaSbaaSqaaiabh2eanbqabaGccqWHubavdaqhaaWcbaGaemyta0KaemOta4KaemyraueabaGaemivaqfaaOGaeyypa0JaeC4raC0aaWbaaSqabeaacqWGubavaaGccqGGBbWwcqWHhbWrcqWHhbWrdaahaaWcbeqaaiabdsfaubaakiabgUcaRiabeg7aHjabhMeajnaaBaaaleaacqWGobGtaeqaaOGaeiyxa01aaWbaaSqabeaacqGHsislcqaIXaqmaaGccqWHhbWraaa@4E88@. This is equivalent to the resolution matrix T_MNEG. For the case of EEG with unknown current density vector, sLORETA gives the following estimate of standardized current density power:

D ^ M N E , l T { [ S D ^ ] l l } − 1 D ^ M N E , l MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaqhaaWcbaGaemyta0KaemOta4KaemyrauKaeiilaWIaemiBaWgabaGaemivaqfaaOGaei4EaSNaei4waSLaeC4uam1aaSbaaSqaaiqbhseaezaajaaabeaakiabc2faDnaaBaaaleaacqWGSbaBcqWGSbaBaeqaaOGaeiyFa03aaWbaaSqabeaacqGHsislcqaIXaqmaaGccuWHebargaqcamaaBaaaleaacqWGnbqtcqWGobGtcqWGfbqrcqGGSaalcqWGSbaBaeqaaaaa@4853@

where D^MNE,l MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemyta0KaemOta4KaemyrauKaeiilaWIaemiBaWgabeaaaaa@32C2@ ∈ ℝ^{3 × 1} is the current density estimate at the lth voxel given by the minimum norm estimate and [SD^ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeC4uam1aaSbaaSqaaiqbhseaezaajaaabeaaaaa@2E59@]_ll ∈ ℝ^{3 × 3} is the lth diagonal block of the resolution matrix SD^ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeC4uam1aaSbaaSqaaiqbhseaezaajaaabeaaaaa@2E59@. It was found 32 that in all noise free simulations, although the image was blurred, sLORETA had exact, zero error localization when reconstructing single sources, that is, the maximum of the current density power estimate coincided with the exact dipole location. In all noisy simulations, it had the lowest localization errors when compared with the minimum norm solution and the Dale method 33. The Dale method is similar to the sLORETA method in that the current density estimate given by the minimum norm solution is used and source localization is based on standardized values of the current density estimates. However, the variance of the current density estimate is based only on the measurement noise, in contrast to sLORETA, which takes into account the actual source variance as well.

VARETA 34 is a weighted minimum norm solution in which the regularization parameter varies spatially at each point of the solution grid. At points at which the regularization parameter is small, the source is treated as concentrated When the regularization parameter is large the source is estimated to be zero.

D ^ V A R = arg ⁡ min ⁡ D , Λ ( | | M − G D | | 2 + | | Λ L 3 . W . D | | 2 + τ 2 | | L . ln ⁡ ( Λ ) − α | | 2 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7048@

where L is a nonsingular univariate discrete Laplacian, L₃ = L ⊗ I_{3 × 3}, where ⊗ denotes the Kronecker product, W is a certain weight matrix defined in the weighted minimum norm solution, Λ is a diagonal matrix of regularizing parameters, and parameters τ and α are introduced. τ controls the amount of smoothness and α the relative importance of each grid point. Estimators are calculated iteratively, starting with a given initial estimate D₀ (which may be taken to be D^LOR MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemitaWKaem4ta8KaemOuaifabeaaaaa@309B@), Λ_i is estimated from D_{i - 1}, then D_i from Λ_i until one of them converges.

Simulations carried out with VARETA indicate the necessity of very fine grid spacing 34.

In Quadratic regularization using dipole intensity gradients 4, L(D) = ||∇D||² which results in a source estimator given by

D ^ Q R = ( G T G + α ∇ T ∇ ) − 1 G T M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemyuaeLaemOuaifabeaakiabg2da9iabcIcaOiabhEeahnaaCaaaleqabaGaemivaqfaaOGaeC4raCKaey4kaSIaeqySdeMaey4bIe9aaWbaaSqabeaacqWGubavaaGccqGHhis0cqGGPaqkdaahaaWcbeqaaiabgkHiTiabigdaXaaakiabhEeahnaaCaaaleqabaGaemivaqfaaOGaeCyta0eaaa@42DF@

or

D ^ Q R = ( ∇ T ∇ ) − 1 G T ( G ( ∇ T ∇ ) − 1 ) G T + α I N ) − 1 M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemyuaeLaemOuaifabeaakiabg2da9iabcIcaOiabgEGirpaaCaaaleqabaGaemivaqfaaOGaey4bIeTaeiykaKYaaWbaaSqabeaacqGHsislcqaIXaqmaaGccqWHhbWrdaahaaWcbeqaaiabdsfaubaakiabcIcaOiabhEeahjabcIcaOiabgEGirpaaCaaaleqabaGaemivaqfaaOGaey4bIeTaeiykaKYaaWbaaSqabeaacqGHsislcqaIXaqmaaGccqGGPaqkcqWHhbWrdaahaaWcbeqaaiabdsfaubaakiabgUcaRiabeg7aHjabhMeajnaaBaaaleaacqWGobGtaeqaaOGaeiykaKYaaWbaaSqabeaacqGHsislcqaIXaqmaaGccqWHnbqtaaa@5233@

The use of dipole intensity gradients gives rise to smooth variations in the solution.

Spatial regularization is a modification of Quadratic regularization. It is an inversion procedure based on a non-quadratic choice for L(D) which makes the estimator become non-linear and more suitable to detect intensity jumps 27.

L ( D ) = ∑ n = 1 N v Φ v ( ∇ D | v ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemitaWKaeiikaGIaeCiraqKaeiykaKIaeyypa0ZaaabCaeaacqqHMoGrdaWgaaWcbaGaemODayhabeaakiabcIcaOiabgEGirlabhseaenaaBaaaleaacqGG8baFcqWG2bGDaeqaaOGaeiykaKcaleaacqWGUbGBcqGH9aqpcqaIXaqmaeaacqWGobGtdaWgaaadbaGaemODayhabeaaa0GaeyyeIuoaaaa@4412@

where N_v = p × N_n and N_n is the number of neighbours for each source j, ∇D_|v is the vth element of the gradient vector and Φv(u)=u2/[1+(uKv)2] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuOPdy0aaSbaaSqaaiabdAha2bqabaGccqGGOaakcqWG1bqDcqGGPaqkcqGH9aqpjuaGdaWcgaqaaiabdwha1naaCaaabeqaaiabikdaYaaaaeaadaWadaqaaiabigdaXiabgUcaRmaabmaabaWaaSaaaeaacqWG1bqDaeaacqWGlbWsdaWgaaqaaiabdAha2bqabaaaaaGaayjkaiaawMcaamaaCaaabeqaaiabikdaYaaaaiaawUfacaGLDbaaaaaaaa@40E9@. K_v = α_v × β_v where α_v depends on the distance between a source and its current neighbour and β_v depends on the discrepancy regarding orientations of two sources considered. For small gradients the local cost is quadratic, thus producing areas with smooth spatial changes in intensity, whereas for higher gradients, the associated cost is finite: Φ_v(u) ≈ Kv2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4saS0aa0baaSqaaiabdAha2bqaaiabikdaYaaaaaa@2F88@, thus allowing the preservation of discontinuities. The estimator at the ith iteration is of the form

D ^ i = Θ ( G , L ( D ^ i − 1 ) ) M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemyAaKgabeaakiabg2da9iabfI5arjabcIcaOiabhEeahjabcYcaSiabdYeamjabcIcaOiqbhseaezaajaWaaSbaaSqaaiabdMgaPjabgkHiTiabigdaXaqabaGccqGGPaqkcqGGPaqkcqWHnbqtaaa@3D90@

where Θ is a p by N matrix depending on G and priors computed from the previous source estimate D^i−1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemyAaKMaeyOeI0IaeGymaedabeaaaaa@305E@.

Time is taken into account in this model whereby the assumption is made that dipole magnitudes are evolving slowly with regard to the sampling frequency 415. For a measurement taken at time t, assuming that D^t−1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemiDaqNaeyOeI0IaeGymaedabeaaaaa@3074@ and D^t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemiDaqhabeaaaaa@2E97@ may be very close to each other means that the orthogonal projection of D^t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemiDaqhabeaaaaa@2E97@ on the hyperplane ED^t−1⊥ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyrau0aa0baaSqaaiqbhseaezaajaWaaSbaaWqaaiabdsha0jabgkHiTiabigdaXaqabaaaleaarmqr1ngBPrgitLxBI9gBaGabaiab=vQiEbaaaaa@3876@ perpendicular to D^t−1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemiDaqNaeyOeI0IaeGymaedabeaaaaa@3074@ is 'small'. The following nonlinear equation is obtained:

G T G + α ( Δ t + β P t − 1 ⊥ T P t − 1 ⊥ ) D t = G T M t MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeC4raC0aaWbaaSqabeaacqWGubavaaGccqWHhbWrcqGHRaWkcqaHXoqycqGGOaakcqqHuoardaahaaWcbeqaaiabdsha0baakiabgUcaRiabek7aIjabhcfaqnaaDaaaleaacqWG0baDcqGHsislcqaIXaqmaeaarmqr1ngBPrgitLxBI9gBaGabaiab=vQiEnaaCaaameqabaGaemivaqfaaaaakiabhcfaqnaaDaaaleaacqWG0baDcqGHsislcqaIXaqmaeaacqWFLkIxaaGccqGGPaqkcqWHebardaWgaaWcbaGaemiDaqhabeaakiabg2da9iabhEeahnaaCaaaleqabaGaemivaqfaaOGaeCyta00aaSbaaSqaaiabdsha0bqabaaaaa@55B1@

where

Δ t = − ∇ x T B x t ∇ x T ∇ x − ∇ y T B y t ∇ y MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuiLdq0aaWbaaSqabeaacqWG0baDaaGccqGH9aqpcqGHsislcqGHhis0daqhaaWcbaGaemiEaGhabaGaemivaqfaaOGaeCOqai0aa0baaSqaaiabdIha4bqaaiabdsha0baakiabgEGirpaaDaaaleaacqWG4baEaeaacqWGubavaaGccqGHhis0daWgaaWcbaGaemiEaGhabeaakiabgkHiTiabgEGirpaaDaaaleaacqWG5bqEaeaacqWGubavaaGccqWHcbGqdaqhaaWcbaGaemyEaKhabaGaemiDaqhaaOGaey4bIe9aaSbaaSqaaiabdMha5bqabaaaaa@4E10@

is a weighted Laplacian and

B x t = diag . [ b x t | k ] k = 1 , ... , p MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCOqai0aa0baaSqaaiabdIha4bqaaiabdsha0baakiabg2da9iabbsgaKjabbMgaPjabbggaHjabbEgaNjabc6caUiabcUfaBjabdkgaInaaDaaaleaacqWG4baEaeaacqWG0baDaaGccqGG8baFdaWgaaWcbaGaem4AaSgabeaakiabc2faDnaaBaaaleaacqWGRbWAcqGH9aqpcqaIXaqmcqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcqWGWbaCaeqaaaaa@4ADE@

with

b x t | k = Φ ′ ( ∇ x D t | k ) 2 ∇ x D t | k . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemOyai2aa0baaSqaaiabdIha4bqaaiabdsha0baakiabcYha8naaBaaaleaacqWGRbWAaeqaaOGaeyypa0tcfa4aaSaaaeaacuqHMoGrgaqbaiabcIcaOiabgEGirpaaBaaabaGaemiEaGhabeaacqWHebardaWgaaqaaiabdsha0jabcYha8jabdUgaRbqabaGaeiykaKcabaGaeGOmaiJaey4bIe9aaSbaaeaacqWG4baEaeqaaiabhseaenaaBaaabaGaemiDaqNaeiiFaWNaem4AaSgabeaaaaGccqGGUaGlaaa@4BAE@

Pt−1⊥ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCiuaa1aa0baaSqaaiabdsha0jabgkHiTiabigdaXaqaaeXafv3ySLgzGmvETj2BSbaceaGae8xPI4faaaaa@3733@ is the projector onto ED^t−1⊥ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyrau0aa0baaSqaaiqbhseaezaajaWaaSbaaWqaaiabdsha0jabgkHiTiabigdaXaqabaaaleaarmqr1ngBPrgitLxBI9gBaGabaiab=vQiEbaaaaa@3876@.

Apart from imposing temporal smoothness constraints, Galka et. al. 35 solved the inverse problem by recasting it as a spatio-temporal state space model which they solve by using Kalman filtering. The computational complexity of this approach that arises due to the high dimensionality of the state vector was addressed by decomposing the model into a set of coupled low-dimensional problems requiring a moderate computational effort. The initial state estimates for the Kalman filter are provided by LORETA. It is shown that by choosing appropriate dynamical models, better solutions than those obtained by the instantaneous inverse solutions (such as LORETA) are obtained.

The Backus-Gilbert method 5736 consists of finding an approximate inverse operator T of G that projects the EEG data M onto the solution space in such a way that the estimated primary current density D^BG MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemOqaiKaem4raCeabeaaaaa@2F4A@ = TM, is closest to the real primary current density inside the brain, in a least square sense. This is done by making the 1 × p vector RuvγT=TuγTGv MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCOuai1aa0baaSqaaiabdwha1jabdAha2jabeo7aNbqaaiabdsfaubaakiabg2da9iabhsfaunaaDaaaleaacqWG1bqDcqaHZoWzaeaacqWGubavaaGccqWHhbWrdaWgaaWcbaGaemODayhabeaaaaa@3C76@ (u, v = 1, 2, 3 and γ = 1, ..., p) as close as possible to δuvIγT MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeqiTdq2aaSbaaSqaaiabdwha1jabdAha2bqabaGccqWHjbqsdaqhaaWcbaGaeq4SdCgabaGaemivaqfaaaaa@34BC@ where δ is the Kronecker delta and I_γ is the γ th column of the p × p identity matrix. G_v is a N × p matrix derived from G in such a way that in each row, only the elements in G corresponding to the vth direction are kept. The Backus-Gilbert method seeks to minimize the spread of the resolution matrix R, that is to maximize the resolving power. The generalized inverse matrix T optimizes, in a weighted sense, the resolution matrix.

We reproduce the discrete version of the Backus-Gilbert problem as given in 5:

min ⁡ T u γ { [ I γ − G u T T u γ ] T W γ B G [ I γ − G u T T u γ ] + ∑ v = 1 3 ( 1 − δ v u ) T u γ T G v G v T T u γ } MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7D47@

under the normalization constraint: TuγTGu1p=1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCivaq1aa0baaSqaaiabdwha1jabeo7aNbqaaiabdsfaubaakiabhEeahnaaBaaaleaacqWG1bqDaeqaaOGaeCymaeZaaSbaaSqaaiabdchaWbqabaGccqGH9aqpcqaIXaqmaaa@38D4@. 1_p is a p × 1 matrix consisting of ones.

One choice for the p × p diagonal matrix WγBG MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeC4vaC1aa0baaSqaaiabeo7aNbqaaiabdkeacjabdEeahbaaaaa@3108@ is:

[ W γ B G ] α α = | | v α − v γ | | 2 , ∀ α , γ = 1 , ... , p MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabeGaaaqaaiabcUfaBjabhEfaxnaaDaaaleaacqaHZoWzaeaacqWGcbGqcqWGhbWraaGccqGGDbqxdaWgaaWcbaGaeqySdeMaeqySdegabeaakiabg2da9iabcYha8jabcYha8jabhAha2naaBaaaleaacqaHXoqyaeqaaOGaeyOeI0IaeCODay3aaSbaaSqaaiabeo7aNbqabaGccqGG8baFcqGG8baFdaahaaWcbeqaaiabikdaYaaakiabcYcaSaqaaiabgcGiIiabeg7aHjabcYcaSiabeo7aNjabg2da9iabigdaXiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiabdchaWbaaaaa@54C2@

where v_i is the position vector of grid point i in the head model. Note that the first part of the functional to be minimized attempts to ensure correct position of the localized dipoles while the second part ensures their correct orientation.

The solution for this EEG Backus-Gilbert inverse operator is:

T u γ = E u γ † L u L u T E u γ † L u MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCivaq1aaSbaaSqaaiabdwha1jabeo7aNbqabaGccqGH9aqpjuaGdaWcaaqaaiabhweafnaaDaaabaGaemyDauNaeq4SdCgabaGaeiiiGyiaaiabhYeamnaaBaaabaGaemyDauhabeaaaeaacqWHmbatdaqhaaqaaiabdwha1bqaaiabdsfaubaacqWHfbqrdaqhaaqaaiabdwha1jabeo7aNbqaaiabcccigcaacqWHmbatdaWgaaqaaiabdwha1bqabaaaaaaa@46EB@

where:

L u = G u 1 p , E u γ = C u γ + ∑ v = 1 3 ( 1 − δ u v ) F v , C u γ = G u W γ B G G u T , F v = G v G v T . MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@7120@

'^†' denotes the Moore-Penrose pseudoinverse.

As stated in 39 in a norm minimization approach we make several assumptions in order to choose the optimal mathematical solution (since the inverse problem is underdetermined). Therefore the validity of the assumptions determine the success of the inverse solution. Unfortunately, in most approaches, criteria are purely mathematical and do not incorporate biophysical and psychological constraints. LAURA (Local AUtoRegressive Average) 40 attempts to incorporate biophysical laws into the minimum norm solution.

According to Maxwell's laws of electromagnetic field, the strength of each source falls off with the reciprocal of the cubic distance for vector fields and with the reciprocal of the squared distance for potential fields. LAURA method assumes that the electromagnetic activity will occur according to these two laws.

In LAURA the current estimate is given by the following equation:

D ^ L A U R A = W j G T ( G W j − 1 G T + α I N ) − 1 M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemitaWKaemyqaeKaemyvauLaemOuaiLaemyqaeeabeaakiabg2da9iabhEfaxnaaBaaaleaacqWGQbGAaeqaaOGaeC4raC0aaWbaaSqabeaacqWGubavaaGccqGGOaakcqWHhbWrcqWHxbWvdaqhaaWcbaGaemOAaOgabaGaeyOeI0IaeGymaedaaOGaeC4raC0aaWbaaSqabeaacqWGubavaaGccqGHRaWkcqaHXoqycqWHjbqsdaWgaaWcbaGaemOta4eabeaakiabcMcaPmaaCaaaleqabaGaeyOeI0IaeGymaedaaOGaeCyta0eaaa@4B9E@

The W_j matrix is constructed as follows:

1. Denote by Vi MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8xfXB1aaSbaaSqaaiabdMgaPbqabaaaaa@38FA@ the vicinity of each solution point defined as the hexahedron centred at the point and comprising at most Vmax MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGae8xfXB1aaSbaaSqaaiabd2gaTjabdggaHjabdIha4bqabaaaaa@3BC6@ = 26 points.

2. For each solution point denote by N_k the number of neighbours of that point and by d_ki the Euclidean distance from point k to point i (and vice versa).

3. Compute the matrix A using e_i = 2 for scalar fields and e_i = 3 for vector fields

A i i = V m a x N i ∑ k ∈ V i d k i − e i MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyqae0aaSbaaSqaaiabdMgaPjabdMgaPbqabaGccqGH9aqpjuaGdaWcaaqaamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaGabaiab=vr8wnaaBaaabaGaemyBa0MaemyyaeMaemiEaGhabeaaaeaacqWGobGtdaWgaaqaaiabdMgaPbqabaaaaOWaaabuaeaacqWGKbazdaqhaaWcbaGaem4AaSMaemyAaKgabaGaeyOeI0Iaemyzau2aaSbaaWqaaiabdMgaPbqabaaaaaWcbaGaem4AaSMaeyicI4Sae8xfXB1aaSbaaWqaaiabdMgaPbqabaaaleqaniabggHiLdaaaa@54C9@

and

A i k = − d k i − e i MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaemyqae0aaSbaaSqaaiabdMgaPjabdUgaRbqabaGccqGH9aqpcqGHsislcqWGKbazdaqhaaWcbaGaem4AaSMaemyAaKgabaGaeyOeI0Iaemyzau2aaSbaaWqaaiabdMgaPbqabaaaaaaa@3A11@

4. The weight matrix W_j is defined by:

W_j = P^TP

where:

P = W_mA ⊗ I₃

where I₃ is the 3 × 3 identity matrix and ⊗ denotes the Kronecker product. W_m is a diagonal matrix formed by the mean of the norm of the three columns of the lead field matrix associated with the ith point.

This modified version 27 of Spatial Regularization is dedicated to the recovery of focal sources when the spatial sampling of the cortical surface is sparse. The source space dimension is reduced by iterative focusing on the regions that have been previously estimated with significant dipole activity. An energy criterion is used which takes into consideration both the source intensities and its contribution to data:

E = 2E_c + E_a

where E_c measures the contribution of every dipole source to the data and E_a is an indicator of dipole relative magnitudes. Sources with energy greater than a certain threshold are selected for the next iteration. The estimator at the ith iteration is given by

D ^ i = Θ ( G i − 1 , L ( D ^ i − 1 ) ) . M MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemyAaKgabeaakiabg2da9GGabiab=H5arjabcIcaOiabhEeahnaaBaaaleaacqWGPbqAcqGHsislcqaIXaqmaeqaaOGaeiilaWIaeCitaWKaeiikaGIafCiraqKbaKaadaWgaaWcbaGaemyAaKMaeyOeI0IaeGymaedabeaakiabcMcaPiabcMcaPiabc6caUiabh2eanbaa@41EB@

where G_i is the column-reduced version of G and Θ is a p_i ≤ p by N matrix depending on the G_i and priors computed from the previous source estimate D^i−1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemyAaKMaeyOeI0IaeGymaedabeaaaaa@305E@. A similar approach was used in 31 where the source region was contracted several times but at each iteration, LORETA was used to estimate the source tomography.

This algorithm combines the ideas of LORETA and FOCUSS and makes iterative adjustments to the solution space in order to reduce computation time and increase source resolution [?, 20]. Starting from the smooth LORETA solution, it enhances the strength of some prominent dipoles in the solution and diminishes the strength of other dipoles. The steps 20 are as follows:

1. The current density is computed using LORETA to get D^LOR MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemitaWKaem4ta8KaemOuaifabeaaaaa@309B@.

2. The weighting matrix W is constructed using (10), its initial value being given by W0=diag(D^LOR)=diag(D^0(1),D^0(2),...,D^0(3p)) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeC4vaC1aaSbaaSqaaiabicdaWaqabaGccqGH9aqpcqqGKbazcqqGPbqAcqqGHbqycqqGNbWzcqGGOaakcuWHebargaqcamaaBaaaleaacqWGmbatcqWGpbWtcqWGsbGuaeqaaOGaeiykaKIaeyypa0JaeeizaqMaeeyAaKMaeeyyaeMaee4zaCMaeiikaGIafCiraqKbaKaadaWgaaWcbaGaeGimaadabeaakiabcIcaOiabigdaXiabcMcaPiabcYcaSiqbhseaezaajaWaaSbaaSqaaiabicdaWaqabaGccqGGOaakcqaIYaGmcqGGPaqkcqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcuWHebargaqcamaaBaaaleaacqaIWaamaeqaaOGaeiikaGIaeG4mamJaemiCaaNaeiykaKIaeiykaKcaaa@587C@.

3. The current density D^i MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemyAaKgabeaaaaa@2E81@ is computed using (9).

4. (Smoothing operation) The prominent nodes (e.g. those with values larger than 1% of the maximum value) and their neighbours are retained. The current density values on these prominent nodes and their neighbours are readjusted by smoothing, the new values being given by

1 s l + 1 ( D ^ ( l ) + ∑ u D ^ ( u ) ) ∀ u  under constraint | | r l − r u | | = d MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeqabeWaaaqaaKqbaoaalaaabaGaeGymaedabaGaem4Cam3aaSbaaeaacqWGSbaBaeqaaiabgUcaRiabigdaXaaakmaabmaabaGafCiraqKbaKaacqGGOaakcqWGSbaBcqGGPaqkcqGHRaWkdaaeqbqaaiqbhseaezaajaGaeiikaGIaemyDauNaeiykaKcaleaacqWG1bqDaeqaniabggHiLdaakiaawIcacaGLPaaaaeaacqGHaiIicqWG1bqDcqqGGaaicqqG1bqDcqqGUbGBcqqGKbazcqqGLbqzcqqGYbGCcqqGGaaicqqGJbWycqqGVbWBcqqGUbGBcqqGZbWCcqqG0baDcqqGYbGCcqqGHbqycqqGPbqAcqqGUbGBcqqG0baDaeaacqGG8baFcqGG8baFcqWGYbGCdaWgaaWcbaGaemiBaWgabeaakiabgkHiTiabdkhaYnaaBaaaleaacqWG1bqDaeqaaOGaeiiFaWNaeiiFaWNaeyypa0Jaemizaqgaaaaa@68F8@

where r_l is the position vector of the lth node and s_l is the number of neighbouring nodes around the lth node with distance equal to the minimum inter-node distance d.

5. (Shrinking operation) The corresponding elements in Dˆ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaaaaa@2CFA@ and G are retained and the matrix M = DDˆ MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaaaaa@2CFA@ is computed.

6. Steps (2) to (5) are repeated until convergence.

7. The solution of the last iteration before smoothing is the final solution.

Steps (4) and (5) are stopped if the new solution space has fewer nodes than the number of electrodes or the solution of the current iteration is less sparse than that estimated by the previous iteration. Once steps (4) and (5) are stopped, the algorithm becomes a FOCUSS process. Results 20 using simulated noiseless data show that Shrinking LORETA-FOCUSS is able to reconstruct a three-dimensional source distribution with smaller localization and energy errors compared to Weighted Minimum Norm, the L₁ approach and LORETA with FOCUSS. It is also 10 times faster than LORETA with FOCUSS and several hundred times faster than the L₁ approach.

SSLOFO 41 combines the features of high resolution (FOCUSS) and low resolution (WMN, sLORETA) methods. In this way, it can extract regions of dominant activity as well as localize multiple sources within those regions. The procedure is similar to that in Shrinking LORETA-FOCUSS with the exception of the first three steps which are:

1. The current density is computed using sLORETA to get D^sLOR MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaem4CamNaemitaWKaem4ta8KaemOuaifabeaaaaa@320A@.

2. The weighting matrix W is constructed using (10), its initial value being given by W0=diag(D^sLOR)=diag(D^0(1),D^0(2),...,D^0(3p)) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeC4vaC1aaSbaaSqaaiabicdaWaqabaGccqGH9aqpcqqGKbazcqqGPbqAcqqGHbqycqqGNbWzcqGGOaakcuWHebargaqcamaaBaaaleaacqWGZbWCcqWGmbatcqWGpbWtcqWGsbGuaeqaaOGaeiykaKIaeyypa0JaeeizaqMaeeyAaKMaeeyyaeMaee4zaCMaeiikaGIafCiraqKbaKaadaWgaaWcbaGaeGimaadabeaakiabcIcaOiabigdaXiabcMcaPiabcYcaSiqbhseaezaajaWaaSbaaSqaaiabicdaWaqabaGccqGGOaakcqaIYaGmcqGGPaqkcqGGSaalcqGGUaGlcqGGUaGlcqGGUaGlcqGGSaalcuWHebargaqcamaaBaaaleaacqaIWaamaeqaaOGaeiikaGIaeG4mamJaemiCaaNaeiykaKIaeiykaKcaaa@59EB@.

3. The current density D^i MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaWgaaWcbaGaemyAaKgabeaaaaa@2E81@ is computed using (9). The power of the source estimation is then normalized as

D ^ i T ( l ) { [ R i ] l l } − 1 D i ( l ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCiraqKbaKaadaqhaaWcbaGaemyAaKgabaGaemivaqfaaOGaeiikaGIaemiBaWMaeiykaKIaei4EaSNaei4waSLaeCOuai1aaSbaaSqaaiabdMgaPbqabaGccqGGDbqxdaWgaaWcbaGaemiBaWMaemiBaWgabeaakiabc2ha9naaCaaaleqabaGaeyOeI0IaeGymaedaaOGaeCiraq0aaSbaaSqaaiabdMgaPbqabaGccqGGOaakcqWGSbaBcqGGPaqkaaa@4625@

where Ri=WiWiTGT(GWiWiT+αI)†G MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCOuai1aaSbaaSqaaiabdMgaPbqabaGccqGH9aqpcqWHxbWvdaWgaaWcbaGaemyAaKgabeaakiabhEfaxnaaDaaaleaacqWGPbqAaeaacqWGubavaaGccqWHhbWrdaahaaWcbeqaaiabdsfaubaakiabcIcaOiabhEeahjabhEfaxnaaBaaaleaacqWGPbqAaeqaaOGaeC4vaC1aa0baaSqaaiabdMgaPbqaaiabdsfaubaakiabgUcaRiabeg7aHjabhMeajjabcMcaPmaaCaaaleqabaGaeiiiGyiaaOGaeC4raCeaaa@48C1@ and [R_i]_ll is the lth diagonal block of matrix R_i.

In 41, SSLOFO reconstructed different source configurations better than WMN and sLORETA. It also gave better results than FOCUSS when there were many extended sources. A spatio-temporal version of SSLOFO is also given in 41. An important feature of this algorithm is that the temporal waveforms of single/multiple sources in the simulation studies are clearly reconstructed, thus enabling estimation of neural dynamics directly from the cortical sources. Neither Shrinking LORETA-FOCUSS nor FOCUSS are able to accurately reconstruct the time series of source activities.

The algorithms described above require a full computation of the matrix G. On the other hand, ALF 42 requires only 6%–11% of this matrix. ALF localizes sources from a sparse sampling of the source space. It minimizes forward computations through an adaptive procedure that increases source resolution as the spatial extent is reduced. The algorithm has the following steps:

1. A set of successive decimation ratios on the set of possible sources is defined. These ratios determine successively higher resolutions, the first ratio being selected so as to produce a targeted number of sources chosen by the user and the last one produces the full resolution of the model.

2. Starting with the first decimation ratio, only the corresponding dipole locations and columns in G are retained.

3. sLORETA (Equation(11)) is used to achieve a smooth solution. The source with maximum normalized power is selected as the centre point for spatial refinement in the next iteration, in which the next decimation ratio is applied. Successive iterations include sources within a spherical region at successively higher resolutions.

4. Steps 2 and 3 are repeated until the last decimation ratio is reached. The solution produced by the final iteration of sLORETA is used as initialization of the FOCUSS algorithm. Standardization (Equation(12)) is incorporated into each FOCUSS iteration as well.

5. Iterations are continued until there is no change in solution.

It is shown in 42 that the localization accuracy achieved is not significantly different than that obtained when an exhaustive search in a fully-sampled source space is made. A multiresolution framework approach was also used in 15. At each iteration of the algorithm, the source space on the cortical surface was scanned at higher spatial resolution such that at every resolution but the highest, the number of source candidates was kept constant.

The multiple-signal Classification algorithm (MUSIC) 651 is a version of the spatio-temporal approach. The dipole model can consist of fixed orientation dipoles, rotating dipoles or a mixture of both. For the case of a model with fixed orientation dipoles, a signal subspace is first estimated from the data by finding the singular value decomposition (SVD) 8M = UΣV^T and letting U_S be the signal subspace spanned by the p first left singular vectors of U. Two other methods of estimating the signal subspace, claimed to be better because they are less affected by spatial covariance in the noise, are given in 52. The first method involves prewhitening of the data matrix making use of an estimate of the spatial noise covariance matrix. This means that the data matrix M is transformed so that the spatial covariance matrix of the transformed noise matrix is the identity matrix. The second method is based on an eigen decomposition of a matrix product of stochastically independent sweeps. The MUSIC algorithm then scans a single dipole model through the head volume and computes projections onto this subspace. The MUSIC cost function to be minimized is

| | P S ⊥ g ( r , e ) | | 2 | | g ( r , e ) | | 2 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqcfa4aaSaaaeaacqGG8baFcqGG8baFcqWHqbaudaqhaaqaaiabdofatbqaaeXafv3ySLgzGmvETj2BSbaceaGae8xPI4faaiabhEgaNjabcIcaOiabhkhaYjabcYcaSiabhwgaLjabcMcaPiabcYha8jabcYha8naaCaaabeqaaiabikdaYaaaaeaacqGG8baFcqGG8baFcqWHNbWzcqGGOaakcqWHYbGCcqGGSaalcqWHLbqzcqGGPaqkcqGG8baFcqGG8baFdaahaaqabeaacqaIYaGmaaaaaaaa@5187@

where PS⊥=I−(USUST) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCiuaa1aa0baaSqaaiabdofatbqaaeXafv3ySLgzGmvETj2BSbaceaGae8xPI4faaOGaeyypa0JaeCysaKKaeyOeI0IaeiikaGIaeCyvau1aaSbaaSqaaiabdofatbqabaGccqWHvbqvdaqhaaWcbaGaem4uamfabaGaemivaqfaaOGaeiykaKcaaa@404C@ is the orthogonal projector onto the noise subspace, r and e are position and orientation vectors, respectively. This cost function is zero when g(r, e) corresponds to one of the true source locations and orientations, r = rdipi MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCOCai3aaSbaaSqaaiabdsgaKjabdMgaPjabdchaWnaaBaaameaacqWGPbqAaeqaaaWcbeaaaaa@331A@ and e = edi MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeCyzau2aaSbaaSqaaiabhsgaKnaaBaaameaacqWGPbqAaeqaaaWcbeaaaaa@3040@, i = 1, ..., p. An advantage over least-squares estimation is that each source is found in turn, rather than searching simultaneously for all sources.

In MUSIC, errors in the estimate of the signal subspace can make localization of multiple sources difficult (subjective) as regards distinguishing between 'true' and 'false' peaks. Moreover, finding several local maxima in the MUSIC metric becomes difficult as the dimension of the source space increases. Problems also arise when the subspace correlation is computed at only a finite set of grid points.

Recursive MUSIC (R-MUSIC) 53 automates the MUSIC search, extracting the location of the sources through a recursive use of subspace projection. It uses a modified source representation, referred to as the spatio-temporal independent topographies (IT) model, where a source is defined as one or more nonrotating dipoles with a single time course rather than an individual current dipole. It recursively builds up the IT model and compares this full model to the signal subspace.

In the recursively applied and projected MUSIC (RAP-MUSIC) extension 5455, each source is found as a global maximizer of a different cost function. Assuming g(r, e) = h(r)e, the first source is found as the source location that maximizes the metric

r ^ 1 = arg ⁡ max ⁡ r ( s u b c o r r ( h ( r ) , U S ) 1 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCOCaiNbaKaadaWgaaWcbaGaeGymaedabeaakiabg2da9iGbcggaHjabckhaYjabcEgaNnaaxababaGagiyBa0MaeiyyaeMaeiiEaGhaleaacqWHYbGCaeqaaOGaeiikaGIaem4CamNaemyDauNaemOyaiMaem4yamMaem4Ba8MaemOCaiNaemOCaiNaeiikaGIaeCiAaGMaeiikaGIaeCOCaiNaeiykaKIaeiilaWIaeCyvau1aaSbaaSqaaiabdofatbqabaGccqGGPaqkdaWgaaWcbaGaeGymaedabeaakiabcMcaPaaa@4FFD@

over the allowed source space, where r is the nonlinear location parameter. The function subcorr(h(r), U_S)₁ is the cosine of the first principal angle between the subspaces spanned by the columns of h(r) and U_S given by:

s u b c o r r ( h ( r ) , U S ) 1 2 = ( h ( r ) T U S , U S T h ( r ) ) ( h ( r ) T h ( r ) ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@6273@

The k-th recursion of RAP-MUSIC is

r ^ k = arg ⁡ max ⁡ r ( s u b c o r r ( Π G ^ k − 1 ⊥ h ( r ) , Π G ^ k − 1 ⊥ U S ) 1 ) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafCOCaiNbaKaadaWgaaWcbaGaem4AaSgabeaakiabg2da9iGbcggaHjabckhaYjabcEgaNnaaxababaGagiyBa0MaeiyyaeMaeiiEaGhaleaacqWHYbGCaeqaaOGaeiikaGIaem4CamNaemyDauNaemOyaiMaem4yamMaem4Ba8MaemOCaiNaemOCaiNaeiikaGIaeuiOda1aa0baaSqaaiqbdEeahzaajaWaaSbaaWqaaiabdUgaRjabgkHiTiabigdaXaqabaaaleaarmqr1ngBPrgitLxBI9gBaGabaiab=vQiEbaakiabhIgaOjabcIcaOiabhkhaYjabcMcaPiabcYcaSiabfc6aqnaaDaaaleaacuWGhbWrgaqcamaaBaaameaacqWGRbWAcqGHsislcqaIXaqmaeqaaaWcbaGae8xPI4faaOGaeCyvau1aaSbaaSqaaiabdofatbqabaGccqGGPaqkdaWgaaWcbaGaeGymaedabeaakiabcMcaPaaa@656C@

where G^k−1≡[g(r^1,e^1)...g(r^k−1,e^k−1)] MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafC4raCKbaKaadaWgaaWcbaGaem4AaSMaeyOeI0IaeGymaedabeaakiabggMi6kabcUfaBjabhEgaNjabcIcaOiqbhkhaYzaajaWaaSbaaSqaaiabigdaXaqabaGccqGGSaalcuWHLbqzgaqcamaaBaaaleaacqaIXaqmaeqaaOGaeiykaKIaeiOla4IaeiOla4IaeiOla4IaeC4zaCMaeiikaGIafCOCaiNbaKaadaWgaaWcbaGaem4AaSMaeyOeI0IaeGymaedabeaakiabcYcaSiqbhwgaLzaajaWaaSbaaSqaaiabdUgaRjabgkHiTiabigdaXaqabaGccqGGPaqkcqGGDbqxaaa@4E41@ is formed from the array manifold estimates (g(r^i,e^i)=h(r^i)e^i) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeiikaGIaeC4zaCMaeiikaGIafCOCaiNbaKaadaWgaaWcbaGaemyAaKgabeaakiabcYcaSiqbhwgaLzaajaWaaSbaaSqaaiabdMgaPbqabaGccqGGPaqkcqGH9aqpcqWHObaAcqGGOaakcuWHYbGCgaqcamaaBaaaleaacqWGPbqAaeqaaOGaeiykaKIafCyzauMbaKaadaWgaaWcbaGaemyAaKgabeaakiabcMcaPaaa@419D@ of the previous k - 1 recursions and ΠG^k−1⊥≡(I−G^k−1(G^k−1TG^k−1)−1G^k−1T) MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaeuiOda1aa0baaSqaaiqbhEeahzaajaWaaSbaaWqaaiabdUgaRjabgkHiTiabigdaXaqabaaaleaarmqr1ngBPrgitLxBI9gBaGabaiab=vQiEbaakiabggMi6kabcIcaOiabhMeajjabgkHiTiqbhEeahzaajaWaaSbaaSqaaiabdUgaRjabgkHiTiabigdaXaqabaGccqGGOaakcuWHhbWrgaqcamaaDaaaleaacqWGRbWAcqGHsislcqaIXaqmaeaacqWGubavaaGccuWHhbWrgaqcamaaBaaaleaacqWGRbWAcqGHsislcqaIXaqmaeqaaOGaeiykaKYaaWbaaSqabeaacqGHsislcqaIXaqmaaGccuWHhbWrgaqcamaaDaaaleaacqWGRbWAcqGHsislcqaIXaqmaeaacqWGubavaaGccqGGPaqkaaa@5704@ is the projector onto the left-null space of G^k−1 MathType@MTEF@5@5@+=feaagaart1ev2aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafC4raCKbaKaadaWgaaWcbaGaem4AaSMaeyOeI0IaeGymaedabeaaaaa@3068@. The recursions are stopped once the maximum of the subspace correlation in (13) drops below a minimum threshold.

A key feature of the RAP-MUSIC algorithm is the orthogonal projection operator which removes the subspace associated with previously located source activity. It uses each successively located source to form an intermediate array gain matrix and projects both the array manifold and the estimated signal subspace into its orthogonal complement, away from the subspace spanned by the sources that have already been found. The MUSIC projection to find the next source is then performed in this reduced subspace. Other sequential subspace methods besides R-MUSIC and RAP-MUSIC are S-MUSIC and IES-MUSIC 54. Although they all find the first source in the same way, in these latter methods the projection operator is applied just to the array manifold, rather than to both arguments as in the case of RAP-MUSIC.

An alternative signal subspace algorithm 56 is FINES (First Principal Vectors). This approach, used in order to estimate the source locations, employs projections onto a subspace spanned by a small set of particular vectors (FINES vector set) in the estimated noise-only subspace instead of the entire estimated noise-only subspace as in the case of classic MUSIC.

In FINES the principal angle between two subspaces is defined according to the closeness criterion 56. FINES creates a vector set for a region of the brain in order to form a projection operator and search for dipoles in this specific region.

An algorithmic description of the FINES algorithm can be found in 56. Simulation results in 56 show that FINES produces more distinguishable localization results than classic MUSIC and RAP-MUSIC even when two sources are very close spatially.

Since the inverse source localization problem can be considered a minimization problem – find the optimal coordinates and orientation for each dipole – the optimization can be performed with an artificial neural network (ANN) based system.

The main advantage of neural network approaches 57 is that once trained, no further iterative process is required. In addition, although iterative methods are shown to be better in noise free environments, ANN performs best in environments with low signal to noise ratio 58. Therefore ANNs seem to be more noise robust. In any case, many research works 596061626364656667 claim a localization error in ANN methods of less than 5%.

A general ANN system for EEG source localization is illustrated in Figure 3. According to 65, the number of neurons in the input layer is equal to the number of electrodes and the features at the input can be directly the values of the measured voltage. The network also consists of one or two hidden layers of N neurons each and an output layer made up of six neurons, 3 for the coordinates and 3 for dipole components. In addition each hidden layer neuron is connected to the output layer with weights equal to one in order to permit a non-zero threshold of the activation function. Weights of inter connections are determined after the training phase where the neural network is trained with