CHAPTER 2. BASICS OF EEG-BASED BRAIN IMAGING 27
3.2 The Proposed Scheme
3.2.4 Upper Bound in the Solution Error
The twofold discretization process introduces errors that are conveyed to the approxi- mate solution. In order to ensure a stable reconstruction, the numerical solution error is required to lie within bounds. This subsection deals with the definition of these errors and the conditions that lead to a bounded solution error. The first approx- imation error is a result of the sampling operation, i.e., the mapping of the points along the tracing lines to the nearest reconstruction points. If f(xh, yh, zh) are the field values that correspond to the points with coordinates (xh, yh, zh) along the line Lwith orientation parametersφ, θ, then each assignment of the samplef(xh, yh, zh) to the valuef(i, j, k) of the discrete poing [i, j, k] introduces an error vectorδf(xh, yh, zh), that, analyzed in the three field coefficients, can be described by
fxh =fx(i, j, k) +δfxh
fyh =fyx(i, j, k) +δfyh (3.2.15) fzh =fz(i, j, k) +δfzh.
Assuming a segment of the lineLof lengthk∆Lijkkthat lies within the tile enumerated by [i, j, k] such that ∆Lijk ⊆L, and if the indexes [i, j, k] are substituted by an index g for simplicity the segment sum becomes
I∆Lg = X
fxh,fxh,fxh∈∆Lg
(fxhcosφsinθ+fyhsinφsinθ+fzhcosθ) ∆r
= (tgfx(g) +X
h
δfxh) cosφsinθ+ (tgfy(g) +X
h
δfyh) sinφsinθ+ (tgfz(g)
+X
h
δfzh) cosθ
∆r, (3.2.16)
where the sampling step ∆r is assumed constant and tg denotes the number of sam- pling points that lie within the cellg. An example of a segment ∆Lg is shown in Fig.
CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 40
3.3. The total error in a single tile g can be expressed in a vector form as
δfx[g], δfy[g], δfz[g]
= t2
X
t1
δfxh,
t2
X
t1
δfyh,
t2
X
t1
δfzh
, (3.2.17)
where t2 −t1 + 1 =tg. Taking also into account that the sum along the line L best approximates the corresponding line integral when the sampling step ∆r tends to zero, it can be written as the sum of all the line segments ∆Lg comprising L as
IL= lim
∆r→0
X
∆Lg
I∆Lg = lim
∆r→0
X
∆Lg
(tgfx(g) +δfx[g]) cosφsinθ + (tgfy(g) +δfy[g]) sinφsinθ+ (tg fz(g) +δfz[g]) cosθ
∆r.
Thus, if the sampling procedure error is defined as δf = lim
∆r→0
X
∆Lg
δfx[g] cosφsinθ+δfy[g] sinφsinθ+δfz[g]) cosθ
∆r, (3.2.18)
and a sampling error coefficienteg is introduced, so that|eg|<1 andtg∆r=k∆Lgk−
eg∆r as in Fig. 3.3, equation (3.2.18) turns into IL= lim
∆r→0
X
∆Lg
k∆Lgk −eg∆r
fx[g] cosφsinθ+fy[g] sinφsinθ+fz[g] cosθ +δf.
(3.2.19) Explicitly, the termtg∆rrepresents the line segment along which the sampling points are assigned to the tile with indexg. Since the sampling step is greater than zero, it generally holds that tg∆r6= k∆Lgk, excluding possible cases when the first and last sampling points lying within a cell coincide with the intersection points of the line with this cell. Moreover, the length of the line iskLk=P
gtg∆r=P
gk∆Lgk, hence, P
geg∆r= 0, simplifying (3.2.19). Consequently, one can obtain the line integral and
CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 41
Figure 3.3: An example of a line segment ∆Lg crossing a tile with indexg. The sampling points are denoted by asterisks, the distance between them is ∆r, while tg stands for the number of sampling points lying within g. For easier comparison, tg∆r is assumed to initiate from the same point as ∆Lg. Apparently, tg∆r 6= k∆Lgk, as the error coefficient eg is greater than zero. An example of the single integer tile enumeration is also shown, as g increases by one along the z-axis, and by N2 along the x-axis. In the case of this line, there is no change in the tile index j along the direction of y. Otherwise, the numbering would increase byN along the y-axis, and by N2+N along the x-axis.
the approximated line integral using IL =X
∆Lg
∆Lgk fx[g] cosφsinθ+fy[g] sinφsinθ+fz[g] cosθ
+δf, (3.2.20)
and
I˜L =X
∆Lg
k∆Lgk fx[g] cosφsinθ+fy[g] sinφsinθ+fz[g] cosθ
−∆r, (3.2.21)
CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 42
respectively, where the term
∆r=X
∆Lg
eg∆r fx[g] cosφsinθ+fy[g] sinφsinθ+fz[g] cosθ) (3.2.22)
expresses the error introduced by the sampling process. Finally, by subtracting (3.2.21) from (3.2.20)
IL−I˜L=δf +∆r, (3.2.23)
it is evident that the deviation of the approximated integral from the real one can be written as a sum of the two kinds of errors, the first rising from the reconstruction on discrete grid points within the domain and the second resulting from the sampling along the line.
The recovery of the field after the estimation of the coefficients of the system is performed using the least squares method. Having specified the method’s intrinsic errors, these coefficients, denoted as ¯kh,u can take the form
k¯h,u =
(k∆Lukh−ehu∆r) cosφhsinθh for 1≤u≤ n3
(k∆Lu−n3kh−ehu−n3∆r) sinφhsinθh
for n3 + 1 ≤u≤ 2n3 ,
(k∆Lu−2n
3 kh−ehu−2n
3 ∆r) cosθh for 2n3 + 1≤u≤n
(3.2.24)
if there is a line segment ∆Luh ⊆ Lφh,θh. Otherwise, they are equal to zero. Briefly, the matrix comprising of ¯kh,u can be expressed as ¯K=K−δK, where δKis the per- turbation introduced by the non zero sampling step ∆r. If ¯f,f0 stand for the solutions of the systemsv= ¯K¯fandv=Kf0+δf, respectively, whereδfis the discretization er- ror vector for all the line orientations used, the relative error of the numerical solution will be kf0 −¯fk/kf0k. This error is related to the size of the perturbations according to [188]
kf0−¯fk
kf0k ≤eKkK¯+k2kKk2+eKκ(K) + efkK¯+k2kvk
kf0k , (3.2.25)
CHAPTER 3. RECONSTRUCTION OF 3D IRROTATIONAL VFs 43
where κ(K) is the condition number, kKk2 is the L2 norm of the matrix and K+ denotes the Moore-Penrose pseudo-inverse matrix of K, kδKk2 = eKkKk2 6= 0 and kδfk = efkvk 6= 0. In the case of an approaching zero sampling step, the sampling error is negligible and (3.2.25) becomes
kf0−¯fk
kf0k ≤ efκ(K) 1−ef
. (3.2.26)
Clearly, from (3.2.26), the relative numerical solution error is bounded when the discretization error is greater than zero, with 0 < ef 1 and κ(K) is bounded, a condition that is satisfied, since the numerical system of equations is an approximation to the line integrals. Naturally, a very dense discretization of the volume will lead to an unbounded condition number of the system matrix, a rank deficient system and an unbounded solution error.