In layperson's terms, how does electrical impedance tomography work?

Figure 4.2(a) shows a typical tactile imaging measurement configuration at one snapshot in the sequence of measurements. A circular sensor is shown for illustration, although other shapes have been employed. The sensors can be considered two-dimensional. Usually 16 electrodes are employed [1–5] and roughly 200 measurements are made per image, systematically switching the positions of the injection, measurement, and ground electrodes. In EIT, the cost of the small number of electrodes placed solely at the perimeter is an increased complexity of the data acquisition electronics as well as of the subsequent signal processing. Figure 4.2(a) shows a bipolar or single current source injection system, also referred to as pair drive, which is the one most commonly employed in tactile imaging. It also shows a bipolar voltage measurement configuration. Software-controlled switches define the injection and measurement electrodes by changing connections throughout the sequence of measurements.

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To obtain one element of V_touch or V_ref, a current is injected at one electrode. It flows throughout the entire conductive sensor to ground, and voltages are measured between other electrode pairs. Various voltage measurement patterns have been used, but the dominant one has been the 'adjacent' pattern [5, 6] (also called neighboring), as shown in figure 4.2(a), in which voltage differences between neighboring electrodes are obtained. In practice the voltages are often recorded between each electrode and ground, and the differences between the electrodes are calculated in software.

Various electrode pairs can also be used to inject the current [1, 6]. In the adjacent injection pattern the current-injecting, or driving, electrodes are again immediate neighbors, such as electrodes 1 and 2, whereas in the 'polar' pattern (also called opposite) they are on opposite sides of the sensor (i.e. electrodes 1 and 9). Other, non-bipolar injection patterns, such as trigonometric [7], optimal (multiple current source) [8], and diagonal [9], have also been proposed, but the hardware complexity required for implementation [4, 10, 11], which may require multiple current sources, means that they are rarely used in practice [12, 13], so we do not treat them here.

To name the electrode pairs, we refer to the gaps or offsets between them, and the injection–measurement (I–M) pattern is named with the injection electrode pair followed by the measurement pair. Therefore the adjacent–adjacent I–M pattern is, for 16-electrode systems, called 1–1 and the polar-adjacent pattern is 8–1. Another commonly used pattern, pseudo-polar-adjacent, is 7–1. If polar injection is used with polar measurement, it is called 8–8. The latter configuration has a high symmetry since rotation by 90°, 180°, and 270° produces identical results for a uniform initial sensor conductivity. One disadvantage of this naming convention is that the names change if a different number of electrodes is used. For example, with 32 electrodes, 8–8 becomes 16–16. In this book, all the I–M pattern designations refer to a 16-electrode system unless otherwise noted.

For each pair of injection electrodes, voltages are measured between multiple other pairs. For adjacent current injection combined with adjacent measurement (1–1), when the injection pair is at positions 1, 2, the measurement pairs are first 3,4 then 4,5, etc, ending at 15,16. (In practice, the voltages may be measured simultaneously rather than sequentially [14].) Voltages are not measured (or they are recorded and discarded) between injecting electrodes and their immediate neighbors, as indicated in figure 4.2(a), to avoid contact resistance errors [12], so 13 voltage differences are obtained for the adjacent measurement pattern and 12 for most others, except for 8–8, for which 14 are collected. The injection electrodes are then rotated to the next position (e.g. for adjacent from 1.2 to 2.3, while for polar from 1.9 to 2.10), and voltages are once again obtained between the other electrodes. The process continues around the perimeter for a total of m measurements. For the 1–1 pattern this sequence leads to a total of m = 16 × 13 = 208 voltage measurements and for most other patterns, 192 measurements. This set of measurements is known as a 'frame'. Due to various symmetries, however, the number of independent measurements is only on the order of 100, further discussed below. The number of measurements in the frame increases approximately with the square of the number of electrodes: for 7–1, 32 electrodes requires 896 measurements.

In tactile imaging, constant DC current, rather than AC excitation, is typically used and the sensors can be considered two-dimensional. Figure 4.2(b) shows the voltage drop for a planar sensor between the injecting electrode and ground as a function of injecting electrode offset position; electrode 1 is kept at ground [15]. For a constant injection current, the voltage between the injecting electrodes is higher the further apart they are, since V = IR and R increases with distance. These simulation results agree with calculations of the resistance between points on a resistive circular area [16]. Since voltage differences between any other two points on the perimeter are fractions of this largest voltage difference, measured voltages will be highest overall for the polar injection pattern.

As mentioned above, to obtain an image of a conductivity change, such as due to a touch on a tactile sensor, the conductivity within the sensing area is compared to a reference image, which for tactile imaging is typically one obtained without a touch. Simulated equipotential lines are shown in figure 4.2(c) for I–M = 1–1 (adjacent–adjacent) and in figure 4.2(d) for 7–1 (pseudo-polar–adjacent) for a circular sensing area with uniform background conductivity. These are the reference images. Consistent with figure 4.2(b), the voltage drop between the current injecting electrodes is larger for pseudo-polar injection, as shown by the color scale. These figures also illustrate the greater voltage gradient (field strength) in the center of the sensor for pseudo-polar than for adjacent injection [17], since in the latter the current flows primarily at the periphery of the sensor. Figure 4.2(e) shows the equipotential lines for pseudo-polar injection for a sensor with a circular region of lower conductivity (higher resistance) than the background, indicated by the white line, which could be the result of a touch. The equipotential lines are distorted from those in figure 4.2(d) by the change in conductivity Δσ: more voltage is now dropped over that region of locally higher resistance and there are thus small changes in the perimeter voltages. EIT algorithms 'reconstruct' the conductivity changes that led to the voltage changes. Figure 4.2 is key for understanding why some injection patterns have been preferred in the literature: the further apart the injection electrodes, the greater the voltage drops between all other electrodes, and the larger the field gradients at the location of the perturbation, the larger the relative voltage changes.

Figure 4.3(a) shows the 208 voltages in the arrays V_ref and V_touch for simulations of the I–M = 1–1 pattern without and with a conductivity change near the perimeter, respectively, from a sensor with 16 electrodes. The voltages in the reference state are high when the measurement electrode is near the injection electrode, which occurs cyclically as the measurement pair position performs its rotation for each injection position in the frame. (For example, referring to figures 4.2(a) and (c), the voltage difference between the pair of measurement electrodes is highest when the injection electrode pair is 1,2 and measurement electrode pair is 3,4 or 15,16 and again when the injection electrode pair is 2,3 and measurement electrode pair is 4,5 or 16,1.) For 1–1, there are 13 points in each voltage measurement sub-cycle. The voltage difference decreases with distance from the injection pair, resulting in a series of 16 U-shaped features. (If the injection electrode pair is 1,2 the voltage differences between adjacent electrodes is smallest when the measurement electrode pair is 9,10.) For I–M = 7–1 and 7–7, on the other hand, there are regions where the voltage differences are negative (figure 4.4), because, referring to figure 4.2(d), the orientation of the measurement pair with respect to ground switches in half of the sub-cycle.

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In figure 4.3(b) V_ref is subtracted from V_touch, giving the difference array, Δ = V_touch − V_ref, which is the information used by the EIT algorithms (figure 4.1). (The values in V_touch and V_ref are voltage differences between pairs of measurement electrodes, while the values in Δ are differences in those same pairs at different times.) Note that in some places the differences are positive while in others they are negative; all contain information about the touch and contribute to the EIT image. The m values Δ_i in Δ are small, with many of them near zero (note the difference in y-axis scales between figures 4.3(a) and (b)). In this simulation the target was near the perimeter, closer to some of the electrodes, so the differences Δ_i are largest there. The Δ_i increase proportionally with the size of the stimulus, i.e. the strength of the touch.

Figure 4.3(c) shows Δ for the 7–1 pattern under identical conditions. The Δ_i are larger than in figure 4.3(b) (again note the difference in y-axis scales), as expected from figure 4.2(e). Also note that, as expected from figures 4.2(c) and (d), the maxima are more uniform across the vector instead of more focused at a few positions. Figure 4.3(c) also illustrates that Δ changes with the location of the target. The Δ_i for a target at x = 0 were smaller and showed periodicity.

The simulated data in figure 4.3 were noise-free. Experimentally, noise may be contributed by both V_touch and V_ref, and if these are in turn obtained from the difference of two measurements between an electrode and ground, then there are noise contributions in both of those measurements. In practice each voltage is typically obtained by averaging multiple readings on each electrode to reduce noise, although this increases the data collection time. Often V_ref is obtained by averaging a large number of frames, making it essentially noise-free. The standard deviation of the noise may be 1–300 μV [18]. At minimum the noise between an electrode and ground is at the level of the least significant bit of the analog-to-digital converter (ADC).

If the sensor area Ω is divided into spatial sub-regions Ω_k (figure 4.5(a)), the voltage V_i between the measurement electrodes is the sum of contributions from all the Ω_k. . The contribution to a change in V_i due to a change in the conductivity Δσ_k of Ω_k depends on the amount of current flowing through Ω_k , the distance of Ω_k to that electrode, and the field lines ( E = −∇ϕ) at Ω_k associated with the injection and measurement electrodes [19]. Specifically, the sensitivity to Δσ_k , or the amount that each element Ω_k contributes to the measured signal [17], is highest where the injection and measurement fields are parallel: it depends on their dot product, where the field due to the measurement electrodes is taken as if they had been used for driving [21–23]. The value of the dot product is the same upon reversing the injecting and measuring electrodes by the reciprocity theorem or lead theorem [23]. Expressing the above mathematically:

$$\begin{eqnarray}&&{J}_{i,k}=\,\nabla \,{\varphi }_{i,k}\cdot \,\nabla \,{\varphi }_{I,k}\end{eqnarray} \tag{ 4.3 }$$

where $\nabla \,{\varphi }_{i,k}$ is the potential gradient that would arise at element k if current were injected into the voltage-measuring electrode pair and $\nabla \,{\varphi }_{I,k}$ is the potential gradient arising from the current-injecting electrode pair.

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In finite element modeling (FEM) the regions Ω_k are mesh elements. The matrix J of terms

$$\begin{eqnarray}&&{J}_{i,k}={\frac{\partial {V}_{i}}{\partial {\sigma }_{k}}| }_{{\sigma }_{0}}\end{eqnarray} \tag{ 4.4 }$$

which depends on the sensor conductivity σ₀, is called the Jacobian or sensitivity matrix (figure 4.5(b)), where i goes from 1 to the m, the length of Δ (the frame size), and k goes from 1 to n, the number of mesh elements. In general the Jacobian of a matrix F is a matrix whose (i,k)th entry is $\partial {F}_{i}/\partial {{\rm{x}}}_{k}$. In EIT, the Jacobian relates changes in voltages at each electrode to changes in conductivity at each mesh element for every drive-measurement pair in the particular I–M pattern. This transformation matrix is a linearized mapping between the boundary voltages and the internal conductivity, assuming small variations for which the derivative (slope) is sufficient to describe the dependence. (J is m × n. For the I–M pattern 7–1 in a 16-electrode system, m = 16 × 12 = 192. Using the EIDORS 2-D mesh model j2d4c, the number of mesh elements n = 20,230; so J is 192 × 20,230.) The linear approximation can be used with voltage difference data because the changes in voltages are small. On the other hand, for a 'static' image reconstruction of a medium's absolute conductivity from a single set of data, the reconstruction problem is highly nonlinear [24].

We can write

$$\begin{eqnarray}&&J\sigma \,=\,\Delta \end{eqnarray} \tag{ 4.5 }$$

where σ is a vector of the conductivity changes Δσ_k of the mesh elements and Δ is the vector of voltage differences (figures 4.3(b) and (c)). The vector σ encodes an image. In EIT forward problem simulations, σ is the difference between the baseline conductivity reference image and the image with the target (figure 2.3). The assumption in equation (4.4) should be emphasized: inverse solution methods often assume that the conductivity changes Δσ_k in σ are small in order to linearize the problem. However, in tactile imaging this assumption may easily be violated. For example, near the edge of the sensor large-area or multiple high-contrast contacts result in incorrect reconstructions.

The Jacobian is found from the reference baseline conductivity and from the I–M pattern: it comes from the reference forward model (e.g. figures 4.2(c) and (d)). It contains no information about the target or the noise.

The sensitivity of individual injection and measurement electrode combinations in the data frame to different locations on the sensor can be visualized [61, 62], as shown in figure 4.6; each row of J is one such image (normalized by mesh element size). The first seven rows (i.e. J_1,: – J_7,:, figure 4.5) out of the total m are shown for the adjacent (1–1) and pseudo-polar (7–1) patterns, with red indicating regions of high positive sensitivity (positive slope in equation (4.4)) and blue high negative sensitivity. (The other rows can be inferred by symmetry.) The adjacent pattern has high sensitivity at the perimeter, highest when the injection and measurement electrode pairs are close together (e.g. J_1,:) as discussed above, and low sensitivity at the center, whereas the pseudo-polar pattern sensitivity is just as high or higher at the perimeter but extends into the center for some combinations, particularly J_1,: and J_7,:. (Other patterns are shown in appendix C.) Note that many values of the Jacobian are close to zero (represented as white in the images), which poses a challenge for the inverse problem, as discussed below.

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To go from voltages at the perimeter to a conductivity distribution over the region, i.e. to take the measured data and create a conductivity image, is called the inverse problem, discussed in section 4.4. This involves inverting the Jacobian to find σ = J⁻¹ Δ.

One way this inversion can be done, and which allows valuable insight into the reconstructions, is by factoring J into three matrices by performing a singular value decomposition (SVD) [20, 25, 26]:

$$\begin{eqnarray}&&{\bf{J}}\,{\boldsymbol{=}}\,{\bf{US}}{{\bf{T}}}^{{\rm{T}}}\end{eqnarray} \tag{ 4.6 }$$

where S is a diagonal matrix of so-called singular values s_i = S_ii , ordered from highest to lowest (see section 4.3.4). The columns of U, u_j, and the columns of V, ${\mathbf{v}}_{{\rm{k}}}$, are the left and the right singular vectors of J. The superscript T indicates the transpose, since the values are real. (For a lucid discussion of SVD in the context of image processing and 'eigenfaces' see [27], and for SVD discussed theoretically see [28].) The number of nonzero values in S, and also the number of nonzero columns in V, is given by the rank r of J, or the number of linearly independent columns. By the reciprocity theorem and because of various symmetries, depending on I–M, r is less than m (the matrices are rank deficient). (For I–M = 7–1, r = 104 while for I–M = 8–8, r = 28.)

What SVD does is re-express J, the linearized relationship between voltage changes and conductivity changes, in terms of two orthonormal bases: the u_j are the basis vectors for the voltage measurements for a particular I–M pattern ('eigen-Δ', so to speak, discussed in section 4.3.5) and the ${\mathbf{v}}_{{\rm{k}}}$ are the basis vectors, or eigen-images, for the conductivity changes (section 4.3.3). S, like J, is size m × n, U is m × m, and V is n × n. (For I–M = 7–1, S is 192 × 20,230 with 104 nonzero diagonal values and all other entries zero, or, due to numerical rounding errors, near zero. Note: for 7–1 the command svd in Matlab returns U as 192 × 192, but it returns S as a sparse 192 × 1 matrix and V as only 20,230 × 192.) Because the u_j and ${\mathbf{v}}_{{\rm{k}}}$ are orthonormal bases we can write:

$$\begin{eqnarray}&&{\bf{J}}=\,{{\rm{s}}}_{1}{{\bf{u}}}_{{\bf{1}}}{{\mathbf{v}}_{{\bf{1}}}}^{{\rm{T}}}\,+\,{{\rm{s}}}_{2}{{\bf{u}}}_{{\bf{2}}}{{\mathbf{v}}_{{\bf{2}}}}^{{\rm{T}}}+\,\cdots \,+\,{{\rm{s}}}_{{\rm{r}}}{{\bf{u}}}_{{\bf{r}}}{{\mathbf{v}}_{{\bf{r}}}}^{{\rm{T}}}\end{eqnarray} \tag{ 4.7 }$$

where each of the r terms s_i u_i ${\mathbf{v}}_{i}$ ^T is also m × n. (For 7–1, each s_i u_i ${\mathbf{v}}_{i}$ ^T consists of 104 image vectors.) From here forward we use the same subscript i for all three components to highlight that they are matched in this way.

U and V are spectral bases [29]. Just as a Fourier expansion of sines and cosines can be used to represent any voltage signal, sums of u_i can represent any measurement signal and sums of ${\mathbf{v}}_{i}$ any spatial conductivity change, and the singular values s_i are analogous to the coefficients of a Fourier expansion and show the overall importance of the ith component to J, in descending order. Recapitulating, the u _i form the basis for Δ (the perimeter voltage frames can be created by combining these orthogonal vectors) and the ${\mathbf{v}}_{i}$ form the basis for σ (the conductivity image can be created from these orthogonal basis functions). The terms with the largest s_i are those that contribute most of the information to the conductivity image for a given I–M pattern.

Matrix inversion is an important application of SVD. The inverse (or pseudo-inverse or Moore–Penrose generalized inverse or minimum 2-norm least squares inverse) is found by:

$$\begin{eqnarray}&&{{\bf{J}}}^{-1}={\bf{V}}{{\bf{S}}}^{-1}{{\bf{U}}}^{{\rm{T}}}\end{eqnarray} \tag{ 4.8 }$$

where S⁻¹ is formed simply by taking the reciprocals of the nonzero diagonal elements, 1/s_i , and transposing.

If the sensitivity matrix has values near zero, for example because little current passes through an element Ω_k , as is the case for the 1–1 pattern near the center of the sensor (figure 4.6), then the inversion will be unstable because of division by such small values [19]. This instability makes the inverse problem 'ill-posed'.

The eigen-images resemble standing waves or mode shapes of the head of a drum (figure 4.7). As explained in [30], these can be visualized by treating the vectors as image data in EIDORS. Figure 4.7(a) shows, for a 16-electrode system, examples of ${\mathbf{v}}_{i}$ that have circular symmetry with the index i increasing left to right; note how the spatial frequency increases with i. Note that the amplitudes of the ${\mathbf{v}}_{i}$ do not depend on current or sensor conductivity—that information is contained in the singular values, not the basis functions. Figure 4.7(b) shows examples of ${\mathbf{v}}_{i}$ with four lobes, two positive and two negative, and figure 4.7(c) illustrates a peripheral pattern as the index and angular frequencies increase. The first eight ${\mathbf{v}}_{i}$ (i = 1–8) for I–M = 1–1 and 7–1 are shown in figures 4.7(d) and (e), respectively. (All the ${\mathbf{v}}_{i}$ for these two I–M are shown appendix B). By comparing figures 4.7(d) and (e), it can be seen that the relative importance of the particular ${\mathbf{v}}_{i}$ differ for 1–1 and 7–1, as indicated by their order, and that while 1–1 and 7–1 share some of the mode shapes, some that are found in one are not found in the other.

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To obtain an image with fine resolution, a large number of ${\mathbf{v}}_{i}$ is needed, just as in a Fourier expansion a large number of terms is needed to recreate the corner of a step function from sine waves. (For an illustration of how low-frequency modes form the main features of the target and high-frequency components reveal its details, see [21].) To separate adjacent targets (figure 2.1) or to recreate pointed features, a full, or nearly full, complement of ${\mathbf{v}}_{i}$ is required. Importantly, a large number of ${\mathbf{v}}_{i}$ is also needed to image features at the center of the sensor [25, 31]: note in figure 4.7 how the higher i modes have fine structure at the image center, whereas the lower i modes do not.

Both 1–1 and 7–1 have a total of 104 ${\mathbf{v}}_{i}$ (the rank of J) in 16-electrode systems; for eight electrodes the equivalent patterns 1–1 and 3–1 have only 20, and for 32 electrodes the equivalent patterns 1–1 and 15–1 have 430. Other I–M patterns have fewer ${\mathbf{v}}_{i}$ due to symmetries. For example 8–8 has the highest symmetry in the 16-electrode system and has only 28 modes. Thus, 1–1 confers greater resolution than 8–8 under low noise conditions.

Combinations of the ${\mathbf{v}}_{i}$ can produce images that little resemble the individual ${\mathbf{v}}_{i}$, as shown in the reconstructions of figure 2.1. The process of combining eigen-images is illustrated in figure 4.8 for a triangular target. The top two rows show the ${\mathbf{v}}_{i}$ that make the most significant contributions for this target, up to i = 28. The images in the bottom row show the weighted running sums as the ${\mathbf{v}}_{i}$ are added, moving left to right, with the final EIT image on the right. Note that the weight for ${\mathbf{v}}_{1}$ is negative, so height is added to the center of the sensor and subtracted from the perimeter; the 'energy' is focused in the center of the sensor. Weights are further discussed in section 4.6.2. Adding the next two ${\mathbf{v}}_{i}$ with significant weights, ${\mathbf{v}}_{5}$ and ${\mathbf{v}}_{6}$, adds height to the right side, and so forth.

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A plot of the singular values s_i for 7–1 and 8–8 are shown in figure 4.9. The s_i are usually plotted on a log scale (the y-axis on the right). (Sometimes plots of s_i /s₁ are used instead.) Note the large range of values. The ratio of the largest to the smallest s_i is the condition number, and the large ratio makes S ill-conditioned for inversion, as will be discussed below. Also note the sudden fall-off by many decades for 7–1 above i = 104 and the fall-off for 8–8 above i = 28, which are the ranks of S for these two patterns. The tiny values for i > r are not zero due to numerical rounding errors in the FEM.

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The singular values depend on the I–M pattern and number of electrodes, and they scale with the current and conductivity, but they do not depend on target or on noise characteristics since J is based on the reference conductivity distribution. As discussed further below, the singular values are critical to understanding the effects of noise and regularization. Signal and image components (which are linked) with smaller singular values are more affected by noise and are preferentially removed by regularization.

As discussed above, the u_i form an orthonormal basis for Δ: sums of the modes u_i can create any voltage difference vector Δ (figures 4.3(b) and (c)), just as the eigen-images are summed to form the final EIT image (figure 4.8). A plot of u₁, u₂, and u₅ for 1–1 and 7–1 are plotted in figure 4.10, illustrating that the u_i also differ for various I–M patterns. For example, u₁ for 7–1 is not found anywhere in the set of modes for 1–1. The amplitudes of the u_i do not depend on current or sensor conductivity.

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Examples of the component matrices contributing to J are included in appendix D.1. The first 10 entries of s₅ u₅ ${\mathbf{v}}_{5}$ for I–M = 7–1 are shown.

Regularization approaches were reviewed in [22], and the advantages of various methods have receive much consideration [34–39]. We focus on Tikhonov regularization, a so-called penalty method, and in particular on the approach in [40]. Numerous prior treatments have already provided mathematical justifications and derivations, so the following discussion focuses first on the general idea in plain English, and then on the implementation mechanics.

Regularization essentially entails using known information about the physical reality of the conductivity and how it changes as a constraint on the solution. It involves balancing the measured data with that prior information (figure 4.11), employing a hyperparameter λ to act as a dial that controls the amount of regularization. The raw data may contain a lot of high frequency components, while the assumption about the physical reality may be that the conductivity changes are smooth, with conductivities not being greatly different in immediately neighboring elements Ω_k . Also, it can be assumed that the conductivity of the medium is everywhere real, positive, and nonzero [33]. The baseline conductivity is typically known.

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Regularization penalizes particular image features, as determined by a 'prior', discussed below, to eliminate destabilizing large variations and make the inverse problem better posed [20]. If high spatial frequency components are penalized, for example by down-weighting components of the solution for which s_i < λ [21, 41], the image is smoothed. Of course, the greater the smoothing, the lower the resolution. On the other hand, if λ is too small the inversion becomes numerically unstable or the image becomes dominated by noise.

The terms L1 and L2 norm are frequently encountered in discussions of penalizing high frequency contributions. In regularization based on the L1 norm, distances are calculated as the sum of the absolute values of terms ∣∣x∣∣ = $\sum | {x}_{i}|$. These approaches are also known as total variation [22]. Regularization using an L2 norm (also known as a Euclidian norm), as is used in Tikhonov regularization, calculates distances instead as Euclidean lengths ∣∣x∣∣ = $\sqrt{\sum {x}_{i}^{2}}$.

Much has been written about algorithms that can be used for the inverse problem (see for example [36, 42–44]). They have been grouped in various ways (use of L1 vs L2 norm, projection methods versus penalty methods, etc), but for tactile imaging the most important distinction is whether they are one-step or iterative. Higher accuracy and finer resolution can be achieved using iterative methods, but they take more time and computational power, and speed is essential for tactile imaging [4, 45]. Iterative methods are required for using the L1 norm [46]. Importantly, one-step methods can pre-calculate the regularized inverse of the sensitivity matrix [23, 47], so they require one matrix multiplication instead of matrix inversions [40]. In order to provide real-time imaging that allows immediate responses to tactile cues, we therefore limit the discussion here to the Gauss–Newton (GN) one-step method [35, 40], which uses an L2 norm, because of its speed [37, 48].

In the GN one-step inversion, the regularized inverse, equation (4.11), is obtained by finding ${{\bf{J}}}_{{\bf{reg}}}^{-1}$ this way:

$$\begin{eqnarray}&&{{\bf{J}}}_{{\bf{reg}}}^{-1}={({{\bf{J}}}^{{\bf{T}}}{\bf{W}}{\bf{J}}+{\lambda }^{2}{{\bf{R}}}^{{\bf{T}}}{\bf{R}})}^{-1}\,{{\bf{J}}}^{{\bf{T}}}{\bf{W}}\end{eqnarray} \tag{ 4.12 }$$

(For images based on element nodes, this step is performed in the Matlab file nodal_solve.m, and for elements, in inv_solve_diff_GN_one_step.m.) Before examining the components of equation (4.12), it should be noted that it is an example of generalized Tikhonov regularization, which solves the problem of an ill-posed inverse, in this case the near-singular matrix J^T J (meaning that the ratio of the largest to smallest singular values, s₁/s_r is large) by adding positive elements, given by the regularization term λ² R^T R. Equation (4.12) is further discussed in appendix D.3.

In equation (4.12), W is a weight matrix (calculated in calc_meas_icov.m), which is the identity matrix (m × m) by default if no measurement covariances are specified. For uncorrelated noise, W is a diagonal matrix with the noise variance at each measurement [46].

The important R is the regularization matrix, and R^T R (n × n) (obtained in calc_RtR_prior.m) acts as a 'filter', chosen to give preference to solutions that are considered to be desirable. For example, with the Tikhonov prior, R^T R = I. (If W = I and R^T R = I, then the method is called zeroth order Tikhonov regularization; for difference imaging assuming identical channels W = I [35]). Note that the Tikhonov prior should not be confused with Tikhonov regularization, with which it is used. The NOSER prior instead scales the identity matrix by the sensitivity [46] (table 4.1, table D.1) (specifically, the diagonal elements of R^T R are (J^T J)^p where p = 0 to 1, typically 0.5 [35]). The EIDORS default is the Laplace prior (implemented in prior_laplace.m), which is a second order high pass filter (defined for a general mesh as a factor of 3 for the element and −1 for each adjacent element); it adds matrix elements not only to the diagonal but also to places off the diagonal. The Laplace prior was compared to others in [39] and found to better preserve feature edges, while the performance of various priors with the GN algorithm was discussed in [37]. It is important to specify the prior if this type of approach to finding the inverse is used: the effect of different priors on EIT images is shown below, in section 4.5. R^T R is unchanged with forward model parameters such as current, conductivity, and I–M pattern.

In equation (4.12), λ² R^T R begins to noticeably contribute to the sum in the denominator, and thus to affect the image, when it is greater than J^T WJ (= J^T J, n × n, when W = I), which does depend on forward model parameters. Examples of matrix values are shown in appendix D.2 for a relatively large hyperparameter, i.e. λ² R^T R > J^T WJ(1,1), showing how this term affects the denominator.

In the truncated SVD (TSVD) method of approximating the inverse, the ill-conditioned problem σ = J⁻¹ Δ is handled differently. J can be approximated by using only terms with the highest singular values. Instead of using the full expansion of J into r terms of s_i u_i ${\mathbf{v}}_{i}$ ^T as in equation (4.7), in TSVD the expansion is truncated at j terms, where j < r [41]. This truncation removes small-value diagonal terms in the matrix, corresponding to high frequency components.

The optimal choice of hyperparameter has received much attention, as reviewed in [49, 50]. The effects of the hyperparameter are covered in depth in chapter 5, but are introduced here.

Figure 4.12 shows example reconstructions for a ring-shaped target at different positions between x = 0 and 0.8 for increasing λ. Greater regularization results in greater smoothing, while features broaden and fall, or 'defocus'. By λ = 1, almost all information about the contact has been lost. For a small hyperparameter, amplitudes and resolutions do not change appreciably with the target's position (figure 4.12(a)). For larger λ, however, the reconstructed feature does depend on position, being more 'defocused', at the center of the sensor (figure 4.12(c)). The images thus misrepresent the relative sizes and strengths of targets. This spatial dependence is detrimental for interpreting tactile contacts, since they are represented very differently at different sensor locations. Furthermore, when imaging multiple contacts, a touch at the center can be masked, or hidden, by touches at the edges. In addition, non-uniformity leads to distorted feature shapes and position errors. Spatial non-uniformity increases with λ. Since sensitivity to changes in conductivity, $\partial {V}_{i}/\partial {\sigma }_{k}={J}_{i,k}$ are smaller at the center of the sensor than at the edge, the J^T WJ terms are smaller there, and the prior information λ² R^T R makes a relatively larger contribution to the sum. Thus, to retain image information, the smallest possible λ must be used, and this, in turn, necessitates the smallest levels of noise.

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It should also be noted in figure 4.12 that at small hyperparameters, most pronouncedly at the periphery of the sensor, the ring is not of uniform amplitude. Rather, there are several points at which the amplitude is higher, shown by the darker red. This variation arises because the image is created from a sum of eigen-functions, as shown in figure 4.7.

The sum of eigen-functions is clearly apparent when looking at cross-sections along y = 0, through the center of the target. Figure 4.13 shows cross-sections for a range of target sizes r_T , at two target positions and two hyperparameters. As the target radius increases beyond r_T = 0.2, it becomes too wide to represent with a single peak; the reconstruction then employs by two peaks along x; the center of the feature becomes a local minimum. The number of peaks increases stepwise with r_T . Below r_T = 0.1, the reconstructed peak decreases in amplitude but not significantly in width, since the eigen-functions have a fixed width. (Simplifying the display to show only values above a threshold eliminates the rippled representation, but this may hinder distinguishability of closely positioned touches.) Another thing to note is that the width of the eigen-functions increases with λ.

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When the target is off center (figures 4.13(b) and (d)) the reconstructed feature becomes increasingly asymmetric, since the effects of the hyperparameter are spatially varying. Beyond a certain radius (in this example, at r_T = 0.6) the reconstruction can fail because the assumption of small conductivity changes used to linearize the problem no longer holds. Yet such stimuli might occur during tactile imaging.

The product J^T WJ in equation (4.12) depends on the sensor conductivity, the injection current, and the I–M pattern. Since J (equation (4.4)) has units of VΩm = Am S⁻² (current distance/conductance²), then J^T WJ has units of A²m²/S⁴. This term therefore increases as the square of the current and decreases as the 4th power of the conductivity.

Thus, to have the same level of influence on the sum in equation (4.12), λ must be increased or decreased commensurately with changes in current and conductivity (table 4.2). The higher the current, the larger λ² R^T R must be to challenge J^T WJ. For each 10× increase in current, the size of λ must be increased by 10× for it to have the same relative strength. In other words, the numerical value for λ entered into the simulation goes up, but the image resolution will be the same. This effect is illustrated by the cross-sections in figure 4.14 (blue versus black curves). Alternatively stated, using a higher current at the same hyperparameter results in finer resolution (see also appendix E.1). On the other hand, the higher the sensor conductivity, the smaller λ must be set to have the same influence. This relationship goes as the inverse square: for each 10× increase in conductivity, the size of λ must be decreased by 100× for it to have the same relative strength, as shown in figure 4.14 (red versus black curves). (See also appendix E.2.) Note that changing the contrast Δσ_Τ does not change the effective strength of λ, although it does improve noise immunity by increasing the SNR.

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We therefore introduce a scaled hyperparameter λ_s that allows comparison across experimental conditions:

$$\begin{eqnarray}&&{\lambda }_{s}=\lambda {\sigma }_{0}^{2}/I\end{eqnarray} \tag{ 4.13 }$$

Thus, we can consider the values of the unscaled λ to be per (S/cm)²/A. The λ_s in figure 4.14 are all the same: λ_s = 10⁻³. Another scaling that has been used is to give λ in terms of J^T J [31].

Increasing the current at a given σ₀ results in a larger signal Δ since V = IR. A larger Δ is beneficial in the presence of a fixed noise level. Increasing I is not usually possible in medical imaging, since the current injected into the patient must be limited for safety [47, 51]. In tactile imaging, on the other hand, it is feasible to increase I, but while the issue is not one of safety, the power consumption (P = I² R) increases, which is a problem. In addition, increasing I may result in additional noise on the signal, or may result in heating that alters the sensor's conductivity [52].

In a piezoresistive material formed by mixing conductive particles into an insulating host, σ₀ is determined by the 'loading', or fraction of particles. For the experimentalist wondering whether σ₀ should be high or low, decreasing σ₀ (increasing resistivity) generates a larger signal Δ for a fixed injection current, and thus improves susceptibility to noise, but at the cost of greater power consumption.

A higher conductivity change or contrast Δσ can be achieved with a larger gauge factor (GF), or sensor sensitivity: GF = (ΔR/R₀)/ε, where ΔR is the change in resistance and ε is the strain. (Stronger touches also result in greater Δσ, but that is not controlled by sensor design.)

Increasing the target contrast Δσ increases the peak amplitude, the relationship being linear. Unlike for changes to σ₀, there is no change in resolution for a fixed λ. The gauge factor should therefore be as high as possible, an intuitively obvious conclusion. A caveat applies, however: since the inverse is approximated by solving a linear system of equations, changes in resistance are recommended to be limited to approximately 20% [19].

In EIT, even tiny amounts of noise more than a thousand times smaller than the measured voltage differences Δ_i can overwhelm an image because of the ill-posed nature of the inverse problem. (This topic is taken up in detail in chapter 6.) The effects of noise are illustrated in figure 4.18. In general, as noise N increases the image becomes distorted, begins to show artefacts, and then breaks up completely. Artefacts are features that appear in the reconstruction that are not associated with the target. Given the importance of identifying the correct number, locations, and strengths of touches, which may be highly variable, the onset of artifacts is unacceptable. At the lowest noise level (figure 4.18(a)), the chosen hyperparameter λ = 10⁻² was large enough to correctly show a single target; the triangular shape, however, was lost at this combination of λ and noise. As the noise increased (figure 4.18(d)) it became increasingly unclear what the original target was, and finally (figure 4.18(f)) the noise dominated the image. This image deterioration is what necessitates the use of increasing levels of regularization with noise.

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To guide experimental decisions such as how many data points to record at each electrode to reduce noise by averaging, versus spending that time instead on employing a larger number of electrodes to try to increase resolution, one needs to understand SNR in the context of EIT. There are several conceptualizations of 'signal' that can be compared with noise. It has been stated that the smallest values in Δ determine the quality of the image since they are the most changed by noise [10] and large voltages can be measured more precisely than small ones [57]. This assertion turns out not to be the case, as can be shown by simulations that apply noise selectively to the largest or smallest Δ_i (see chapter 6), and for reasons that become evident below. Alternatively, the average amplitude of Δ has been used as a measure of signal strength, which works well, but only for a given system and target. For example, it has been suggested that I–M patterns having a larger average Δ should be more immune to noise [5, 10, 30], but that is also not necessarily the case.

It is instructive to examine ∣U^T Δ∣ spectra [20, 26, 58] to appreciate how small levels of noise can destabilize an image. The use of ∣U^T Δ∣ also has the advantage of allowing comparison of different systems and targets. To judge the impact of a certain level of noise on a measurement, the signal basis vectors u_i are multiplied by the measured voltage differences Δ [20]. Recall that the u_i (figure 4.10) depend only on the system forward-problem parameters, such as the I–M pattern and number of electrodes. The u_i with the smallest indices i correspond to the largest singular values s_i (figure 4.9) and thus have the most importance in determining the reconstructed image. To understand this product, return to the Fourier expansion analogy: the coefficient b_n in the sum over n of b_n sin(nπx/L) is found for a particular f(x) by integrating the product f(x)sin(nπx/L) over the interval –L to L. Likewise, the product U^T Δ (size m × 1) gives the amplitudes of coefficients in the expansion of Δ into eigen-Δs.

The absolute values P_i = ∣u_i ^T Δ∣ are known as Picard coefficients. The P_i have units of volts and can be directly compared to the noise. (The P_i can be found by adapting the Matlab EIDORS code; our codes are available upon request.) Using the Picard coefficients for SNR dates back to employing SVD for solving inverse problems and regularizing ill-conditioned matrices [59].

Figure 4.19(a) shows the first 50 P_i without noise for two targets, circular and triangular, plotted in order of their singular value index i on a logarithmic axis. Differences in Δ for the two targets are translated into differences in P_i . Likewise, differences in I–M pattern would also be manifested in the P_i .

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Figure 4.19(b) shows the first 130 P_i for the triangular target without noise and when two levels of noise were added, indicated by the dashed horizontal lines. Without noise the P_i decay with i and then drop abruptly above i = r. Terms with index greater than the rank of J, here beyond position 104, which are nominally zero but are actually small values near 10⁻¹⁷ V due to numerical noise, do not play a role in the reconstruction. Raising their values by 8 orders of magnitude by adding N = 1 × 10⁻⁹ V did not interfere with the creation of a good image. With noise, the P_i instead level off at the noise level. The index i at which the leveling begins is called the Picard threshold. Using the smallest hyperparameter that permitted reconstruction, λ = 10⁻⁷, at N = 3 × 10⁻⁹ V the target was well represented, as shown by the EIT inset. At N = 6 × 10⁻⁹ V the images broke up and showed artefacts. Even though 6 nV might seem like a tiny amount of noise compared to values in Δ, which may be 10 mV (figure 4.3), it is large compared to the amplitudes of the smaller u _i . The largest several u_i are, however, unaffected until the noise reaches mV levels.

Thus, noise creates inaccurate images because of the incorrect amplitudes of the affected u_i . Recalling from equation (4.7) that the s_i , u_i , and ${\mathbf{v}}_{i}$ at the singular value index i are linked, the corresponding eigen-images ${\mathbf{v}}_{i}$, which then also have corrupted amplitudes, are used in the reconstruction when they should be disregarded, as shown below. This is called 'over-fitting'. As noise increases, the number of eigen-images ${\mathbf{v}}_{i}$ that can be beneficially employed in the reconstruction decreases, being equal to the number of P_i above the noise level. The ${\mathbf{v}}_{i}$ corresponding to P_i in the noise plateau are excluded by increasing λ, the mechanism of which is shown in the next section. When using truncated SVD for regularization, the truncation is chosen to start at the index i at which the P_i begin to flatten due to noise [32]. In summary, the number of u_i that can be used to correctly represent the signal decreases with noise, and only the corresponding correct image components ${\mathbf{v}}_{i}$ should be retained.

Changing injection current, sensor conductivity, and contrast produces proportional up or down shifting of the ∣U^T Δ∣ spectra because Δ is linearly proportional to contrast and current and inversely proportional to conductivity (section 4.4.4). (The basis functions u_i are unaffected by these parameters.) Downward shifts make the system more susceptible to noise.

Noise has alternatively been compared to the singular values s_i . ∣U^T Δ∣ and S spectra have similar appearance (see appendices D.5 and D.6). However, the s_i pertain to J and therefore only characterize the measurement system, whereas the P_i also carry information about the current, the target, and other factors through the measured voltage differences Δ.

While ∣U^T Δ∣ spectra are related to the input, or signal, side of the EIT process, there is an analogous spectrum on the output, or image, side, which is ∣V^T σ∣. The two spectra strongly resemble each other, as would be expected from equation (4.7). However, ∣V^T σ∣ reflects the regularization used to obtain the inverse. Given the profound effects of regularization on the quality of the image (figure 4.12), it is instructive to examine its effects on the various image components. (This topic is taken up in detail in chapter 5.) Taking the product of V and the image vector σ [20], provides the coefficients, or weights, C_i of the ${\mathbf{v}}_{i}$ that sum to form the particular EIT image: C_i = ∣${\mathbf{v}}_{i}$ ^T σ ∣. Thus ∣V^T σ∣ shows the contributions of particular eigen-images ${\mathbf{v}}_{i}$ to the final reconstructed image, as was illustrated figure 4.8. (Note that to obtain the ∣V^T σ∣, the same mesh must be used for the forward and inverse problems, constituting an inverse crime.)

As a side note, the product SV^T σ can also be found. At small λ, C_i = P_i /s_i [26] since J σ = USV^T σ = Δ.

Examples of ∣V^T σ∣ spectra are shown in figure 4.20 (linear axis) at λ = 10⁻⁶ with increasing noise levels, as well as at λ = 10⁻⁵ at the top noise level. The most significant coefficients C_i are shown as peaks in the plot. (Again, C_i for i above r = 104 are due to numerical rounding errors and are near zero.) The corresponding EIT images are shown above the plot, and the contributions of eigen-image ${\mathbf{v}}_{101}$ are shown in the inset.

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Below index i = 80, the four spectra were identical. With negligible noise (black line), the larger C_i were of comparable amplitude up to i = 104, and the EIT image was a good approximation to the target. With increasing noise, the spectra began to show increasing, unpredictable contributions from the high-index ${\mathbf{v}}_{i}$, due to the corruption of the corresponding u_i (figure 4.19(b)): the ${\mathbf{v}}_{i}$ that are sensitive to noise is determined by the amplitude of the corresponding P_i . For example, the amplitude of ${\mathbf{v}}_{101}$ was virtually zero at N = 10⁻¹² V and increased in intensity with N. At N = 2 × 10⁻⁸ V (green line) the EIT image developed ringing but still had no artifacts. At N = 3 × 10⁻⁸ V, with the incorrect Picard coefficients being even larger, the EIT image developed artifacts. At an even higher noise level, contributions of the high-frequency eigen-images dominated the final image, resulting in a solution that was no longer approximately correct. Raising λ to 10⁻⁵, the amplitudes of terms above i = 87 were dampened, effectively eliminating those eigen-images from the sum and thereby removing artefacts from the image.

As discussed above, regularization results in smoothing because above a certain i, image information, carried by the ${\mathbf{v}}_{i}$, is removed. (As phrased in [29], regularization schemes are spectral filtering methods.) Raising λ increases the number of ${\mathbf{v}}_{i}$ that are diminished, and the damping effect of λ on the C_i moves from right to left as λ increases. The images are changed in different ways depending on the I–M pattern and the prior, and to different extents depending on target position. For very large hyperparameters, which for the simulation conditions used in this book means λ = 1, only a handful of low-frequency eigen-images remain and almost all information about the contacts with the sensor is lost (figure 4.12(d)).

Turning to examine the operation of the prior, noise-free ∣V^T σ∣ spectra for four regularization methods (section 4.4.1) at comparable levels of regularization, as used in figure 4.16, are illustrated in figure 4.21 for a target at x = 0.1. (These spectra have a small number of peaks because they are dominated by the circularly symmetric ${\mathbf{v}}_{i}$.) The un-regularized spectrum is shown in black. The TSVD method (figure 4.21(a), green) simply sets all the C_i above a particular index, here i = 52, to zero [41]; below that the terms were equal to those without regularization (the green and black lines in the figure overlap perfectly so that the back line is hidden). TSVD acts as a stepwise low-pass filter on i. The Tikhonov prior (blue) instead reduced the amplitudes gradually, the decrease being visually apparent starting from i = 40, where the blue line drops below the green one, and ending with near zero amplitudes at i = 62. It behaved as a ramped low-pass filter on i. The Laplace and NOSER priors (figure 4.21(b)), on the other hand, instead penalize different properties of the image damping C_i based on image criteria rather than i. Note how even at high i, some of the amplitudes did not go entirely to zero. Since the four filters have different effects on the C_i , the resulting EIT images better retain different types of information. (∣V^T σ∣ spectra for a range of hyperparameters are shown in appendix D.7.)

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∣V^T σ∣ spectra also provide insight into differences in performance among I–M patterns, as shown in figure 4.22. At small λ, all full-rank I–M patterns provide virtually identical reconstructions, since a similar number of ${\mathbf{v}}_{i}$ are used; differences in performance only arise when regularizing. For a circular target at x = 0.0, EIT images made using λ = 10⁻⁶ (blue lines) with full-rank patterns (here 1–1 and 7–1, figures 4.22(a) and (b)) were constructed from six circularly symmetric ${\mathbf{v}}_{i}$. The indices i for these ${\mathbf{v}}_{i}$ varied with the I–M pattern. Again, more ${\mathbf{v}}_{i}$ were needed for a target at x = 0.5 (black lines). For lower rank patterns, such as 4–4 (figure 4.22(c)), the spectra terminate at smaller i, so the final EIT images are constructed from fewer ${\mathbf{v}}_{i}$ and suffer from poorer resolution and diminished peak height. The spectra make clear the nature and origin of the deleterious effects of low rank due to symmetry. At λ = 1 (red lines, overlying the blue ones at small i) the C_i were diminished except at the smallest i. For I–M = 4–4, increasing λ had no effect on the image until λ was large enough to reach i = 62.

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Looking more closely at the spectrum for I–M = 1–1, its reputation for poor performance compared to other full-rank patterns becomes clear: the ${\mathbf{v}}_{i}$ at i = 1 and 2 were insignificant for targets away from the perimeter, and the first peak for the target at x = 0 was located not at i = 1 but at i = 8 and was small. Because the lowest index patterns are the largest contributors to an image and the last to be affected by regularization, it is clear that targets at the center of the sensor using 1–1 would suffer more from regularization than with other patterns. This is how the lower sensitivity of 1–1 than 7–1 at the center of the sensor (figure 4.6) manifests itself in the SVD.

The singular value indices i_f can be found at which the larger C_i (C_i > 2 × 10⁻⁴) were first lowered by more than 5% from their values at λ = 10⁻⁶, for hyperparameters between 10⁻⁵ and 10⁰. The points can be well fit to the form i = −alog(λ) + b as shown in figure 4.22(d) for the circular target at x = 0.5 (black curves in figures 4.22(a)–(c)). The fit for 1–1 had a steeper slope than the others, showing a more rapid advance over i with λ; it also had a negative intercept, showing that even C₁ was affected at λ = 1: the effect of regularization was more severe for 1–1 than other I–M patterns. (The line for 4–4 did not begin until λ = 10⁻³.)

To summarize, the I–M pattern affects performance in two ways. The first is symmetry, which plays a role primarily at small hyperparameters. Symmetry results in less information gathered despite an equal number of measurements, shown by a reduced rank r. Symmetry results in missing higher-order ${\mathbf{v}}_{i}$, leading to lower-quality reconstructions. (The highest-symmetry patterns also result in mirror image features, not shown here.) For full-rank patterns, at small hyperparameter it makes little difference which I–M is chosen. The second consideration applies at large hyperparameters: low sensitivity at the center of the sensor. This manifests itself in higher indices i of centrally-concentrated modes, so they are attenuated at relatively lower λ, and stronger dampening versus i for a given λ. The examination of the role of I–M pattern continues in sections 5.3, 6.2.1, and 7.2.

A critical concept is that the EIT image can be considered to be a sum of eigen-images, with a resolution determined by the number of independent terms in that series whose amplitudes are greater than the noise. High symmetry measurement configurations such as polar have fewer independent terms, and thus inherently lower possible resolution.

To understand the action of the hyperparameter it can thus be helpful to look at ∣V^T σ∣ spectra, which are alternative views of the EIT image in the form of fingerprints that show the coefficients C_i of the eigen-images that sum to produce the EIT image. The spectra reveal how the coefficients change with the particular coupling of injection and measurement pairs at different locations on the surface and with target properties. The hyperparameter controls the number of eigen-images contributing to the reconstruction; more specifically, λ controls the amplitudes of those eigen-images, which are systematically reduced in favor of prior information as λ increases. The prior information is typically a uniform background conductivity for a tactile sensor. Therefore, the features broaden and fall.

Since the ∣V^T σ∣ spectra reflect the action of the hyperparameter, through the final image σ, they demonstrate explicitly how smoothing occurs and show the relative rate versus index i at which increases in λ affect the coefficients. The eigen-images corresponding to the highest index singular values, which are associated with the highest spatial frequency information, are removed first. With increasing λ, the number of eigen-images contributing to the EIT image drops as logλ. The rule of thumb that regularization removes image components for which the singular values s_i ² < λ² holds for a broad range of i but finally breaks down at large λ. For large hyperparameters, which for the simulation conditions used in this chapter means λ = 1, only a handful of low-frequency eigen-images remain. Thus almost all information about the contacts is lost, including the radius. The scheme governing the details of coefficient reduction is determined by the prior that is chosen. While regularization acts broadly as a low-pass filter, where the index i plays the role of frequency, the details depend on the image filtering strategy.

The I–M pattern and the target characteristics determine the original coefficients in the ∣V^T σ∣ spectra. For the simulations in this chapter, the coefficients were unaffected below λ = 10⁻⁶. The particulars of which coefficients are large, and thus which associated ${\mathbf{v}}_{i}$ make a substantial contribution to the final image, provide insight into the reconstructions. Some I–M patterns yield no appreciable peaks at low index for a given target, while others have a peak immediately at i = 1. The former are thus affected at a smaller hyperparameter than the latter. These differences reflect the underlying physics: the current flow through different locations on the sensor depending on the injection pattern. For the same reason the I–M pattern determines the rate at which the hyperparameter advances across the spectrum of eigen-image coefficients. The steepness of that slope and the particular eigen-images contributing to forming the reconstructed feature determine the image metrics.