![Three-dimensional femur model (adapted from Cheung et al., 2004). The bone was modeled as linearly elastic (E=12.8 GPa, ⌫=0.4) (Ramos et al., 2006). Two rigid pins were alternately placed in the depicted positions. Modeled forces (averages) included hip contact acting at the pin center (F H = [0.540, 0.328, 2.292] BW), and abductor force F H (F A = [0.580, 0.043, 1.040] BW) for body weight of BW=800 N (Radcli↵e and Taylor, 2007). The femoral shaft was constrained from all movement at its base. 16](https://www.researchgate.net/profile/Todd-Pataky/publication/43533754/figure/fig2/AS:554930945048576@1509317255123/Three-dimensional-femur-model-adapted-from-Cheung-et-al-2004-The-bone-was-modeled-as_Q320.jpg)
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Generalized n-dimensional biomechanical field analysis using
statistical parametric mapping
Todd C. Pata ky
Department of Bioengineering, Shinshu University, Japan
Abstract
A variety of biomechanical data are sampled from smooth n-dimensional spatiotemporal fields.
These data are usually analyzed discretely, by extracting summary metrics from particular
points or regions in the continuum. It has been shown that, in certain situations, such schemes
can compromise the spatiotemporal integrity of the original fields. An alternative methodology
called Statistical Parametric Mapping (SPM), designed specifically for continuous field analy-
sis, constructs statistical images that lie in the original, biomechanically meaningful sampling
space. The current paper demonstrates how SPM can be used to analyze both experimental
and simulated biomechanical field data of arbitrary spatiotemporal dimensionality. Firstly, 0-,
1-, 2-, and 3-dimensional spatiotemporal datasets derived from a pedobarographic experiment
were analyzed using a common linear model to emphasize that SPM procedures are (practically)
identical irrespective of the data’s physical dimensionality. Secondly two probabilistic finite el-
ement simulation studies were conducted, examining heel pad stress and femoral strain fields,
respectively, to demonstrate how SPM can be used to probe the significance of field-wide sim-
ulation results in the presence of uncontrollable or induced modeling uncertainty. Results were
biomechanically intuitive and suggest that SPM may be suitable for a wide variety of mechan-
ical field applications. SPM’s main theoretical advantage is that it avoids problems associated
with aprioriassumptions regarding the spatiotemporal foci of field signals. SPM’s main prac-
tical advantage is that a unified framework, encapsulated by a single linear equation, a↵ords
comprehensive statistical analyses of smooth scalar fields in arbitrarily bounded n-dimensional
spaces.
1
1 Introduction
Many classes of biomechanical data share two mathematically non-trivial characteristics: (i)
spatiotemporal smoothness within (ii) regular discrete bounds. This is true of 1D trajectories like
vertical ground reaction forces (VGRF), and joint kinematics, 2D continua like contact pressure
and surface thermal distributions, and 3D continua like bone strain and cardiac flow velocity. It is
also true of simulated mechanical continua. All may be regarded as smooth scalar (or vector) fields
bounded by anatomy, temporal events, or both.
These data are smooth not only because they are sampled above the Nyquist frequency, but
ultimately because biological tissue viscoelasticity (Fung, 1981) causes biomechanical processes to
be spatiotemporally smooth by nature. Smoothness is statistically non-trivial because it implies
local data correlation and therefore that the number of independent processes is less, perhaps
far less than the number of sampled points. Regular spatiotemporal bounds are also non-trivial
because they imply registrability (Maintz and Viergever, 1998) and thus that a direct, continuous
comparison of multiple field observations may be possible.
Since experimental and probabilistically simulated nD fields can yield a large volume of data,
they are typically reduced through regionalization, by extracting multiple local VGRF optima
(e.g. Nilsson and Thorstensson, 1989) or by discretizing modeled anatomy (e.g. Radcli↵e and Taylor,
2007), for example. Such procedures permit statistical testing but also unfortunately create an
abstraction: to understand tabular VGRF values, for example, one must mentally project these
data back to their reported regions in the original sampling space. Discretization can occasionally
also have statistical consequences, missing (Appendix D) or even reversing trends (Pataky et al.,
2008).
A methodology called Statistical Parametric Mapping (SPM) (Friston et al., 2007) can partially
o↵set these limitations by providing a framework for the continuous statistical analysis of smooth
bounded nD fields. It was originally developed for the analysis of cerebral blood flow in 3D PET
and fMRI images (Friston et al., 1991; Worsley et al., 1992) but it has since migrated to a variety of
diverse applications (Worsley, 1995; Chauvin et al., 2005) including a biomechanical one (2D pedo-
barography) (Pataky and Goulermas, 2008). SPM’s suitability for a wider range of biomechanical
2
applications has not yet been investigated.
The main goals of this study were to: (1) Review the mathematical foundations of nD SPM,
(2) Demonstrate how SPM can be utilized for the analysis of 0-, 1-, 2-, and 3D experimental data,
and (3) Demonstrate how SPM can be used in probabilistic simulations of biomechanical continua.
While each demonstration is, independently, narrowly focussed it is hoped that they collectively
reveal a broader utility.
2 Statistical Parametric Mapping (SPM)
2.1 General linear model (GLM)
The relation between experimental observations Yand an experimental design Xcan be sum-
marized using a mass-univariate GLM (Friston et al., 1995):
Y=X+"(1)
where is a set of regressors (to be computed), and "is a matrix of residuals. Yand "are (I⇥K),
Xis appropriately scaled and (I⇥J), and is (J⇥K), where I,J, and Kare the numbers of
observations, experimental factors, and nodes, respectively. The term ‘node’ is used here to refer to
the number of discrete measurement points, and an experimental observation is an n-dimensional
sampling of a scalar field that may be flattened into a K-vector. A full experiment yields Iflattened
K-vectors. Least-squares estimates of can be obtained by:
ˆ
=X+Y=(XTX)1XTY(2)
yielding errors:
"=YXˆ
(3)
where X+is the Moore-Penrose pseudo-inverse of X. Like the original dataset Y,themodelfits
Xˆ
are (I⇥K). In general a large proportion of variability can be explained using this approach
3
(see Sect.4.1).
Having estimated parameters ˆ
and residuals ", the next task is to compute test statistic values.
The GLM (Eqn.1) a↵ords arbitrary linear testing including: ANOVA, ANCOVA, etc., (Friston et
al., 1995), but for brevity only the generalized ttest will be considered presently. First nodal
variance 2
kis estimated as:
ˆ2
k=("T")kk
⌫=("T")kk
Irank(X)(4)
where ("T")kk is the kth diagonal element of the (K⇥K) error sum of squares matrix "T"and
where ⌫is the error degrees of freedom. The nodal tstatistic can then be computed as:
tk=cTˆ
k
ˆkpcT(XTX)1c(5)
where cis a (J⇥1) contrast vector. The nodal values tkform a K-vector that can be reshaped
into the original nD sampling space and viewed in the context of the original data. For this reason
(5) is known as a statistical ‘map’ and is referred to in the literature as ‘SPM{t}’ (Friston et al.,
2007), a notation that shall be adopted henceforth. The contrast vector cassigns weights to the J
experimental factors and thus represents the experimental hypothesis (see Sect.3.1).
2.2 Statistical inference
SPM uses random field theory (RFT) (Adler, 1981) to assess the field-wide significance of an
SPM{t}. A technical summary of RFT procedures is provided in Appendix A. Briefly, for n>0,
RFT is charged with solving the problem of multiple comparisons. That is, one could expect to
observe higher tkvalues, simply by chance, when conducting multiple statistical tests. A Bonferroni
correction for Kmultiple comparisons is valid, but is overly-conservative (in general) because
spatiotemporal correlation (local smoothness) e↵ectively ensures that fewer than Kindependent
processes exist. RFT takes advantage of this fact to conduct inference topologically, based on the
height and size of connected clusters that remain following suitably high SPM{t}thresholding (e.g.
t>3.0). Precise probability computations additionally depend on field smoothness (Appendix B)
4
and search space morphology (Appendix C). A key point is that a large suprathreshold cluster
is the topological equivalent of a large univariate tvalue. For clarity, a numerical 1D example is
provided in Appendix D.
3 Methods
3.1 Experimental dataset
A single subject (male, 30 years, 172 cm, 73 kg) from a previous study (Pataky et al., 2008), veri-
fied in post-hoc analysis to be representative of the mean subject’s statistical trends, was re-analyzed
here. Since population inference was not a goal a single subject was considered appropriate. The
subject performed twenty randomized repetitions of each of ‘slow’, ‘normal’, and ‘fast’ walking.
VGRF and pedobarographic data were sampled at 500 Hz (Kistler 8281B, Winterthur, Switzer-
land; RSscan Footscan 3D, Olen, Belgium). Walking speed was measured at 100 Hz (ProReflex,
Qualisys, Gothenburg, Sweden) and was treated as a continuous variable for statistical purposes.
Prior to participation the subject gave informed consent according to the policies of the Research
Ethics Committee of the University of Liverpool.
Maximal VGRF (0D), VGRF time series (1D), and peak (i.e. spatially maximal) pressure im-
ages (2D) were extracted from the original spatiotemporal (3D) pedobarographic dataset. Here
the 3D data are 2D time series (and the 1D data are 0D time series), but the data were treated as
mathematically 3D for volumetric smoothness, clustering, and topological probability computations
(Appendix A). The 1D data were registered via linear interpolation between heel-strike and toe-o↵.
The 2D data were registered to the chronologically first ‘normal’ walking image using mutual infor-
mation maximization through optimal planar rigid body transformation (Pataky and Goulermas,
2008). The 3D data were registered using the optimal 2D spatial transformation followed by linear
temporal interpolation, as above.
The data were modeled with four factors: main factor ‘speed’, an intercept, and linear and
sinusoidal time drift nuisance factors (see Appendix E); the nuisance factors were included both to
account for small baseline electronic drifts observed in pilot studies and to emphasize the flexibility
5
of the GLM for experimental modeling. The contrast vector: c=1000
T
(5) represents
the current hypothesis: that speed is positively correlated with the outcome measure (maximal
VGRF, etc.), and that the other factors are not of empirical interest. SPM analyses proceeded as
described in Sect.2. All analyses were implemented in Python 2.5 using Numpy 1.3 and Scipy 0.8
as packaged with the Enthought Python Distribution 5.0 (Enthought Inc., Austin, USA).
3.2 Simulation A
An axisymmetric finite element (FE) model of heel pad indentation was constructed following
(Erdemir et al., 2006) (Fig.1). The model was originally used to compute hyperelastic material
properties of 20 diabetic (D) and 20 non-diabetic (ND) subjects through inverse FE simulation.
Di↵erences in the material parameters (Fig.1 caption) between the two groups failed to reach
significance.
Here the reported variability is explored using Monte Carlo simulations to determine the relation
between univariate material parameter significance and field-wide stress significance as assessed
using SPM. Firstly, 1000 simulations were conducted for each group (D and ND) using the mean
material parameters and their reported variances. Indenter depth was 8.0 mm for all simulations.
Secondly, the mean ↵Dparameter was varied between 7.0 and 7.7 in steps of 0.05 to span the
range of the reported value (↵D=7.02) and the univariate significance threshold (↵D=7.585).
For each ↵D1000 simulations were repeated, holding variance constant.
Finally SPM was used to compare the probabilistic Von Mises stress fields resulting from each
(↵ND,↵D) combination using two-sample ttests (Appendix D). Only ↵Dwas varied because: (a)
the implicit hypothesis of Erdemir et al. (2006) and related studies was that diabetic subjects
had sti↵er heel pads, and (b) this parameter a↵ects the material’s high-strain response, which is
potentially of greater clinical interest than low strain. All FE problems were solved using ABAQUS
6.7 (Simulia, Providence, USA).
6
3.3 Simulation B
A 3D human femur model (Fig.2) was used to explore strain field changes associated with hip
replacement pin placement (a simplification of Radcli↵e and Taylor, 2007). Geometry was borrowed
from the third-generation standardized femur model (Cheung et al., 2004). Linear material param-
eters (Ramos and Sim˜oes, 2006), hip contact and abductor muscle forces (Radcli↵e and Taylor,
2007), and two rigid circular pin postures were modeled (see Fig.2 caption). Rather than o↵er-
ing orthopedic realism, this pin scheme was meant to demonstrate that SPM follows mechanical
expectation.
Since hip forces are highly variable (Bergmann et al., 1993) Monte Carlo simulations were
conducted (1000 repetitions for each pin configuration) by varying all force components with a
standard deviation of 20% (R¨ohrle et al., 1984). Di↵erences in bone strain fields between the two
pin configurations were assessed using two-sample ttests (Appendix D) and irregular-lattice RFT
inference procedures (Worsley et al., 1999).
4 Results
4.1 Experimental data
Experimental data (Fig.3) exhibited systematic changes with walking speed. Statistical analy-
sis of the 0D dataset (Fig.3a) yielded t=25.1 (p=0.000), indicating significant positive correlation
between walking speed and maximal VGRF. The 1D VGRF data (Figs.3b,4) were positively corre-
lated with speed in early (0-30%) and late (75-100%) stance and were negatively correlated in mid
(35-70%) stance. 2D peak pressures (Figs.3c,5) were positively correlated with walking speed at
the heel and distal forefoot and negatively correlated over the distal midfoot and proximal forefoot.
All e↵ects were significant (p=0.000).
Spatiotemporal 3D pressures (Figs.3d,6) were positively correlated with walking speed under
the heel, midfoot and proximal forefoot in early (0-25%) stance and under the medial forefoot and
phalanges in late (75-100%) stance. Negative correlation was found throughout mid- (35-85%)
stance for all areas excluding the phalanges. All three spatiotemporal clusters reached significant
7
(p=0.000). For reference, Table 1 lists the computed field smoothness and geometrical characteris-
tics of all experimental datasets.
4.2 Simulation A
While mean Non-Diabetic and Diabetic heel pad stress fields exhibited di↵erences (Figs.7a,b),
SPM supported the findings of Erdemir et al. (2006) by detecting no significant field-wide ef-
fects (Fig.7c); the maximum SPM{t}(t=0.8) was unsuitably low for thresholding. However, SPM
found significant broadly spanning stress responses (t>2.0, p=0.031; Fig.7d) considerably sooner
(↵D=7.300) than univariate parameter testing (↵D=7.585). This increased signal sensitivity (and
spatial detail evident in Fig.7d) resulted from SPM’s topological treatment of the field data (Friston
et al., 2007, Ch.19).
4.3 Simulation B
Mean bone strain fields were qualitatively di↵erent for the two pin configurations (Fig.8a), and
SPM found these di↵erences to be statistically significant (Fig.8b). The e↵ects were limited to areas
surrounding the pins, were maximal under the pins (the direction of the largest force component),
and were biased away from the non-involved pin, as could be expected.
5 Discussion
This paper has demonstrated that SPM can be used to conduct statistical inference in a contin-
uous and field-wide manner on nD registered biomechanical datasets. While SPM has previously
been applied to 2D pedobarographic datasets (Pataky and Goulermas, 2008), the main new findings
were that: (i) SPM can also be applied to smooth biomechanical field data of any physical dimen-
sionality, and (ii) SPM can be used to probe probabilistic simulations of biomechanical continua.
5.1 Present results
The 0-2D experimental results (Figs.3-5) have been previously described elsewhere (Nilsson and
Thorstensson, 1989; Rosenbaum et al., 1994; Keller et al., 1996; Taylor et al., 2004; Pataky et al.,
8
2008), but the 3D results (Fig.6) have not. Specifically, pressures under the midfoot and proximal
forefoot appear to remain lower during slower walking throughout mid-to-late (35-85%) stance.
While these results are arguably new, and although post- hoc analyses revealed qualitative consis-
tency with the average subject, no population inferences are made because only a single-subject’s
data were presented (for clarity). The main points are that SPM appears to yield results that
are biomechanically consistent with other approaches and that SPM can handle data of arbitrary
dimensionality.
The present FE simulation results reveal, firstly, that SPM produces results that are consistent
with mechanical expectation (Fig.8b). Secondly, SPM revealed, non-trivially, that broad continuum
responses can reach statistical significance well prior to the point at which parameters governing
those responses reach significance. It may therefore be prudent for future investigations to consider
field-wide e↵ects when interpreting mechanical parameter variance. These FE results collectively
imply that SPM provides a suitable framework for analyses of simulated continua, and thus that
it may be a useful compliment to existing probabilistic techniques (e.g. Dar et al., 2002; Laz et al.,
2007).
5.2 SPM
As a field analysis method SPM has a variety of scientific merits. It is highly generalized, and
thus highly flexible, and it a↵ords both field-wide and spatiotemporally focussed hypothesis testing
(Friston et al., 2007). It is mathematically very well developed and has been validated extensively
(e.g. Worsley et al., 1992; Worsley, 1995). When compared with regionalization approaches, its
main advantage is that it avoids two potential sources of discretization-induced bias: (i) regional
conflation (Pataky et al., 2008), and (ii) aprioriassumptions regarding spatiotemporal foci (e.g.
Appendix D).
Another key advantage is that statistics are conducted on a field-wide basis, so investigators need
neither devise nor adapt regionalization schemes for or to particular problems, nor would particular
schemes require justification during scientific review. Along these lines, the existence of various
open-source SPM packages (e.g. SPM8, Wellcome Trust Centre for Neuroimaging, University
9
College London) promotes a greater degree of scientific transparency than is possible with ad hoc
or manual regionalization.
A final advantage is that statistical results lie directly in the original nD sampled continua.
For example, given the context of original VGRF time series (Fig.3b), it is straightforward to
interpret the corresponding SPM results (Fig.4), and one could argue that discrete extrema analysis
(e.g. Fig.3a) o↵ers a somewhat incomplete impression of the field-wide changes associated with
experimental intervention.
As an aside, while SPM demands greater computational resources than regional techniques, it
should be noted that analyses can still be conducted with clinically feasible speed. The current
(non-compiled) Python implementation yielded statistical computation durations of: 1.48⇥103,
2.95⇥102, 3.31, and 14.3 s (0-3D experimental datasets, respectively); the 2- and 3D durations
would likely decrease substantially via optimized compiled implementation. These durations ex-
clude data organization coding, but a high-level automated interface could easily be constructed
(e.g. click on the five pre-surgery trials, click on the five post-surgery files, GO). Registration could
be performed automatically as data are collected; a recent 2D pedobarographic implementation
(Oliveira et al., in press) required only ⇠50 ms per image pair. Therefore SPM’s computational
demand isn’t necessarily a practical limitation.
5.3 Limitations
SPM’s procedures, especially those of RFT-based inference (Appendix A), are mathematically
more complex than those of regional univariate (or discrete multivariate) approaches. These com-
plexities potentially pose barriers to general adoption in investigations of biomechanical continua.
However, both the availability of open-source SPM packages and the highly matured neuroimaging
lead (Friston et al., 2007) could help to lower these barriers if SPM is deemed to o↵er empirical
advantages.
From a biomechanical perspective SPM’s greatest limitation is potentially its requirement for
co-registration of 1D and higher dimensional datasets. One could argue, for example, that the
current VGRF registration scheme yielded a misregistration of early stance peaks (Fig.3b). In
10
this particular case the apparent misregistration had no biomechanical consequences because the
suprathreshold SPM{t}spanned broadly across early stance (Fig.4). There may be situations where
SPM{t}extent is not order-of-magnitude larger than registration inaccuracy.
In such cases nonlinear registration may help (Sadeghi et al., 2003; Goulermas et al., 2005),
but there are also undoubtedly situations where registration is not biomechanically feasible. Elec-
tromyographical signals with poorly defined temporal bounds or gross tissue deformity, for exam-
ple, may pose practical registration problems. Qualitative geometrical manipulations of FE models
(e.g. Lin et al., 2007) could also render simulation datasets unregistrable. Nevertheless, these are
limitations of registration and not of SPM per se. Continued biomechanical registration scrutiny
(Sadeghi et al., 2000; Sadeghi et al., 2003; Duhamel et al., 2004; Page and Epifanio, 2007) may
help to clarify SPM’s appropriateness for specific applications.
5.4 Summary
SPM a↵ords topological statistical analysis of smooth, registrable n-dimensional scalar fields.
The present results suggest that SPM may be suitable for both laboratory and probabilistic simu-
lation studies involving a wide variety of biomechanical continua. SPM’s main advantages are that
statistical results lie directly in the original continuum and that potential problems associated with
ad hoc discretization are avoided.
Acknowledgments
Funding for this work was provided by Special Coordination Funds for Promoting Science and
Technology from the Japanese Ministry of Education, Culture, Sports, Science and Technology.
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Table 1: Smoothness (FWHM) and geometry (resel counts Rd) of the current experimental
datasets. The FWHM estimates assume isotropic and field-wide constant smoothness (see
Appendix A); in the 3D case the FWHM combines spatial (5 mm) and temporal (% stance)
dimensions from the current (57 ⇥23 ⇥100) (x, y, t) sampling lattice.
Dataset FWHM R0R1R2R3
0D - 1 - - -
1D 9.81 1 10.1 - -
2D 4.34 1 19.1 33.8 -
3D 7.59 2 24.9 108.5 67.7
14
Figure 1: Axisymmetric model of heel pad indentation (adapted from
Erdemir et al., 2006), 8mm indentation. The dashed rectangle depicts the
undeformed geometry. The authors reported Ogden hyperelastic material
parameters (means ±st.dev.) for non-diabetic (ND) and diabetic (D)
groups as: µND =16.45 (±8.27), µD=16.88 (±6.70) and ↵ND=6.82 (±1.57),
↵D=7.02 (±1.43), respectively.
15
Figure 2: Three-dimensional femur model (adapted from Cheung et al.,
2004). The bone was modeled as linearly elastic (E=12.8 GPa, ⌫=0.4)
(Ramos et al., 2006). Two rigid pins were alternately placed in the
depicted positions. Modeled forces (averages) included hip contact acting
at the pin center (FH=[0.540,0.328,2.292] BW), and abductor force
FH(FA=[0.580,0.043,1.040] BW) for body weight of BW=800 N
(Radcli↵e and Taylor, 2007). The femoral shaft was constrained from all
movement at its base.
16
Figure 3: Experimental data. A single subject performed 20 trials of each
of Slow, Normal, and Fast walking. (a) 0D raw dataset: maximal vertical
ground reaction force (VGRF), normalized by body weight (BW). (b) 1D
temporally registered dataset: VGRF time series. (c) 2D spatially
registered dataset (means): maximal (peak) pressure. (d) 3D
spatiotemporally registered dataset (means): pressure image time series.
17
Figure 4: SPM results, 1D experimental dataset, thresholded at t>3.5.
Probability (p) values indicate the likelihood that a suprathreshold cluster
of the same spatiotemporal extent could have resulted from a random field
process of the same smoothness as the observed residuals (Eqn.3).
18
Figure 5: SPM results, 2D experimental dataset, t>3.5.
19
Figure 6: SPM results, 3D experimental dataset, t>3.5.
20
Figure 7: Heel pad simulation results, undeformed geometry. (a) Mean
Non-Diabetic (ND) Von Mises ()field(↵ND =6.82). (b) Mean Diabetic
(D) field (↵D=7.02). (c) SPM{t}field for mean (↵ND =6.82, ↵D=7.02);
SPM{t}max=0.8. (d) Inference image for ↵D=7.300, t>2.0.
21
(a)
(b)
SPM{t} e (Max.Principal)
+5
0
-5
2.0
1.5
1.0
0.5
2.5
1e-9
p = 0.030
p = 0.028
Figure 8: Femur simulation results. (a) Maximal principal strain fields
for the pin1 (left) and pin2 (right) configurations under mean force vector
loading. (b) SPM{t}field, (pin2-pin1), |t|>1.0. Inference results for
|t|>2.0 are noted.
22
Appendix A. Statistical inference
Random field theory (RFT) (Adler, 1981) provides the mathematical
foundation for conducting topological statistical inference on an SPM. Given
⌫, the expected topological characteristics of an SPM depend on field smooth-
ness and search space geometry. Field smoothness can be estimated at each
node by first computing normalized residuals u(Kiebel et al., 1999):
ui="i
q"T
i"i
(A.1)
where iindexes the observations, then assembling an (I⇥n) gradient matrix
at each pixel (Worsley, 2007):
˙
uk⌘
2
6
6
6
6
6
6
6
4
r(uk)1
r(uk)2
.
.
.
r(uk)I
3
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
6
4
@(uk)1
@1
@(uk)1
@2... @(uk)1
@n
@(uk)2
@1
@(uk)2
@2... @(uk)2
@n
.
.
..
.
..
.
.
@(uk)I
@1
@(uk)I
@2... @(uk)I
@n
3
7
7
7
7
7
7
7
5
(A.2)
where r(uk)iis the gradient of the ith residual’s kth node, and @(uk)i
@dis
the dth component of that gradient vector. Finally nodal smoothness can
be estimated as:
ˆ
Wk= (4log2)1
2|˙
uT
k˙
uk|1
2n(A.3)
Here ˆ
Wkestimates the full-width at half-maximum (FWHM) of a Gaus-
sian kernel that, when convolved with uncorrelated Gaussian random field
data, would produce the same smoothness as was observed in the normalized
residuals ui. As ˆ
Wkincreases the expected size of suprathreshold SPM{t}
clusters also increases, a fact that RFT exploits.
The expected topological characteristics of an SPM{t}also depend on
the geometry of the search space A,thenD space in which the data lie.
Assuming a 3D dataset, the first step is to count the number of nodes (⇢0),
edges (⇢d), faces (⇢dd0), and cubes (⇢123)inA. This task can be rapidly
implemented using morphological erosion (Nixon and Aguago, 2008):
⇢0=|A B0|
⇢d=|A Bd|
⇢dd0=|A Bdd0|
⇢123 =|A B123|(A.4)
where the Bmatrices are directional connectivity structuring elements (Ap-
pendix B). Having assembled these basic morphological characteristics of A,
its global geometry can now be summarized by ‘resel’ or ‘resolution element’
counts Rd(Worsley et al., 1996):
R0=⇢0(⇢1+⇢2+⇢3)+(⇢12 +⇢13 +⇢23)⇢123
R1=1
ˆ
Wh(⇢1+⇢2+⇢3)2(⇢12 +⇢13 +⇢23)+3⇢123i
R2=1
ˆ
W2h(⇢12 +⇢13 +⇢23)3⇢123i
R3=1
ˆ
W3[⇢123] (A.5)
For simplicity A.3 assumes isotropic smoothness and A.5 assumes position-
independent smoothness ˆ
W=⌃ˆ
Wk/K, but these restrictions can easily be
lifted (Worsley et al., 1999). Each Rdis associated with an independent
probability density function (Worsley et al., 1996) (Appendix C) that di-
rectly depends only on the tthreshold.
These density functions can be used to compute a variety of topological
expectations, like the number of supra-threshold nodes and clusters, for
example (Friston et al., 1994). The final steps in RFT-based inference are
thus to threshold an observed SPM{t}at a suitably high value (e.g. t>3.0)
and then corroborate the observed topology with topological expectation,
computing pvalues for each cluster according to Friston et al. (1994), for
example. The logic of RFT is that smooth random fields are expected to
produce spatially broad suprathreshold clusters, but very broad and/or very
high clusters are expected to occur with low probability. The key message
is that a large suprathreshold cluster is the topological equivalent of a large
univariate tvalue.
Appendix B. Search space geometry
To rapidly compute the geometrical characteristic ⇢of an nD search
space defined by binary image A, one may use morphological erosion (Nixon
and Aguado, 2008):
⇢=|A B|(B.1)
where the Bmatrices are structuring elements that describe the neighbor-
hood connectivity of interest. In three dimensions single nodes (B0), adja-
cent nodes (Bd), faces (Bdd0), and cubes (B123) are given by the sets:
B0={0,0,0}
B1={{0,0,0},{1,0,0}}
B2={{0,0,0},{0,1,0}}
B3={{0,0,0},{0,0,1}}
B12 ={{0,0,0},{1,0,0},{0,1,0},{1,1,0}}
B13 ={{0,0,0},{1,0,0},{0,0,1},{1,0,1}}
B23 ={{0,0,0},{0,1,0},{0,0,1},{0,1,1}}
B123 ={{0,0,0},{1,0,0},{0,1,0},{1,1,0},...
{0,0,1},{1,0,1},{0,1,1},{1,1,1}} (B.2)
Here A Bis an eroded binary image whose elements are ones if the B
pattern exists at a given node and zeros otherwise. |A B|is the set size
of A Bor, equivalently, the number of ones in the eroded image.
Appendix C. Euler characteristic densities
Each ‘resel count’ Rd(A.5) is associated with an independent proba-
bility distribution function, or Euler characteristic density, pd(t). To three
dimensions, directly from Worsley et al. (1996, Table 2), the densities are:
p0(t)=Z1
t
(⌫+1
2)
⌫⇡1
2(⌫
2)(1 + u2
⌫)1
2(⌫+1)du
p1(t)=(4log2)1
2
2⇡✓1+t2
⌫◆1
2(⌫1)
p2(t)=(4log2)
2⇡
(⌫+1
2)
(⌫
2)1
2(⌫
2)t✓1+t2
⌫◆1
2(⌫1)
p3(t)=(4log2)3
2
(2⇡)2✓1+t2
⌫◆1
2(⌫1) ✓⌫1
⌫t21◆
where ⌫is the degrees of freedom. Note that p0is the univariate Student’s t
distribution. The general nD form of these distributions is given in Worsley
(1994, Corollary 5.3).
Appendix D. Numerical example
Consider five fictional force trajectories from each of two experimen-
tal conditions ‘A’ and ‘B’. (Fig.D.a) on a normalized time interval 0-100%
(K=100). Condition A data were created by adding smooth Gaussian
noise (FWHM=10%) to yA(t) = 800 N. Condition B data were created
by first adding positive Gaussian signals to yB(t) = 800 N at t=75% and
15% (see Fig.D.b), and then subsequently adding filtered Gaussian noise
(also at FWHM=10%). These two simulated experimental conditions were
then compared using a two-sample ttest (Eqn.5, main manuscript) where
c=⇥11
⇤Tand:
X=2
6
4
1111100000
0000011111
3
7
5
T
(D.1)
After thresholding the resultant SPM{t}at t>3.0, the significance of
the suprathreshold clusters (Fig.D.c) was assessed using the RFT procedures
described above, where the average FWHM was estimated to be 10.4% using
Eqn.A.3. This simulation highlights three concepts: (i) SPM can be used to
analyze continuous field data in a topological manner. (ii) A Bonferroni ap-
proach (K=100, tcritical = 5.192) would fail to identify significance anywhere
in the temporal field. (iii) A discrete approach that focusses only on the re-
gion t=75% would fail to identify the other signal at t=15%. While this
example has been tailored to emphasize these concepts, the methodology (i)
and dangers (ii,iii) clearly also apply to real experimental data.
Figure D: Example 1D SPM analysis. (a) Simulated raw data for two experimental con-
ditions ‘A’ and ‘B’. (b) Mean curves with standard deviation clouds. (c) SPM{t}with
threshold t>3.0. The pvalues were computed according to Appendix A-C and indicate
the probability that the specific suprathreshold cluster could have occurred by chance.
Appendix E. Model visualization
The general linear model (GLM) (Eqn.1, main manuscript) consists of:
experimental observations (Y), experimental design (X), regression coeffi-
cients (), and model errors ("). Since these matrices can be quite large,
numerical probing of their elements is inconvenient. The elements may, how-
ever, be conveniently probed qualitatively using matrix visualization tech-
niques. The most important matrix to visualize is Xbecause it represents
the experimenter’s statistical modeling decisions: it describes all modeled
experimental factors, it reveals the randomness of the design, and together
with a contrast vector (Eqn.5, main manuscript) it explicitly describes the
experimental hypothesis.
Fig.E.1 depicts the design matrix that was used to analyze the current
pedobarographic data. Rows correspond to trials, and columns to modeled
experimental factors. The main factor of interest was ‘speed’, and the first
column of Fig.E.1 reveals that walking speeds were randomized. Three
factors of non-interest were also also modeled: an intercept ‘y0’ and two
low-frequency time-drift nuisance factors. Visualizing these factors as a
matrix image can be helpful to understand the experimental design, so X
renderings like Fig.E.1 are often presented in scientific papers.
It is also instructive to visualize the entire GLM (Fig.E.2). The obser-
vations Ymay be regarded as a bird’s eye view of Fig.3a. The model fits
(Xˆ
) closely resemble the experimental data (Y), with only relatively mi-
nor di↵erences ("). This indicates, anecdotally, that the GLM can explain
a large proportion of the experimental variability.
Figure E.1: Design matrix X(Eqn.1, main manuscript). The matrix is (60⇥4): 60 trials
and 4 modeled experimental factors. The color scale is normalized within columns.
Figure E.2: Grayscale renderings of the general linear model matrices (Eqn.1, main
manuscript) for the 1D VGRF dataset. Each matrix is (60⇥100): 60 trials and 100
VGRF trajectory nodes per trial. Absolute errors |"|are presented.
References
Adler, R. J. 1981. The geometry of random fields, Wiley, Chichester.
Friston, K. J., Worsley, K. J., Frackowiak, R. S. J., Mazziotta, J. C., and Evans, A. C.
1994. Assessing the significance of focal activations using their spatial extent, Human
Brain Mapping 1, 210–220.
Kiebel, S. J., Poline, J. B., Friston, K. J., Holmes, A. P., and Worsley, K. J. 1999. Robust
smoothness estimation in statistical parametric maps using standardized residuals from
the general linear model, Neuroimage 10, 756–766.
Nixon, M. and Aguado, A. 2008. Feature extraction & image processing, second edition,
2nd ed., Academic Press.
Worsley, K. J. 1994. Local maxima and the expected Euler characteristic of excursion sets
of 2,Fand tfields, Advances in Applied Probability 26, 13–42.
Worsley, K. J. 2007. Random field theory. In: Statistical parametric mapping: the analysis
of functional brain images (K.J. et al. Friston, ed.), Elsevier/Academic Press, Amster-
dam.
Worsley, K. J., Andermann, M., Koulis, T., Macdonald, D., and Evans, A. C. 1999. De-
tecting changes in nonisotropic images, Human Brain Mapping 8, 98–101.
Worsley, K. J., Marrett, S., Neelin, P., Vandal, A. C., Friston, K. J., and Evans, A. C. 1996.
A unified statistical approach for determining significant signals in images of cerebral
activation, Human Brain Mapping 4, 58–73.
Appendix A. Statistical inference
Random field theory (RFT) (Adler, 1981) provides the mathematical
foundation for conducting topological statistical inference on an SPM. Given
⌫, the expected topological characteristics of an SPM depend on field smooth-
ness and search space geometry. Field smoothness can be estimated at each
node by first computing normalized residuals u(Kiebel et al., 1999):
ui="i
q"T
i"i
(A.1)
where iindexes the observations, then assembling an (I⇥n) gradient matrix
at each pixel (Worsley, 2007):
˙
uk⌘
2
6
6
6
6
6
6
6
4
r(uk)1
r(uk)2
.
.
.
r(uk)I
3
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
6
4
@(uk)1
@1
@(uk)1
@2... @(uk)1
@n
@(uk)2
@1
@(uk)2
@2... @(uk)2
@n
.
.
..
.
..
.
.
@(uk)I
@1
@(uk)I
@2... @(uk)I
@n
3
7
7
7
7
7
7
7
5
(A.2)
where r(uk)iis the gradient of the ith residual’s kth node, and @(uk)i
@dis
the dth component of that gradient vector. Finally nodal smoothness can
be estimated as:
ˆ
Wk= (4log2)1
2|˙
uT
k˙
uk|1
2n(A.3)
Here ˆ
Wkestimates the full-width at half-maximum (FWHM) of a Gaus-
sian kernel that, when convolved with uncorrelated Gaussian random field
data, would produce the same smoothness as was observed in the normalized
residuals ui. As ˆ
Wkincreases the expected size of suprathreshold SPM{t}
clusters also increases, a fact that RFT exploits.
The expected topological characteristics of an SPM{t}also depend on
the geometry of the search space A,thenD space in which the data lie.
Assuming a 3D dataset, the first step is to count the number of nodes (⇢0),
edges (⇢d), faces (⇢dd0), and cubes (⇢123)inA. This task can be rapidly
implemented using morphological erosion (Nixon and Aguago, 2008):
⇢0=|A B0|
⇢d=|A Bd|
⇢dd0=|A Bdd0|
⇢123 =|A B123|(A.4)
where the Bmatrices are directional connectivity structuring elements (Ap-
pendix B). Having assembled these basic morphological characteristics of A,
its global geometry can now be summarized by ‘resel’ or ‘resolution element’
counts Rd(Worsley et al., 1996):
R0=⇢0(⇢1+⇢2+⇢3)+(⇢12 +⇢13 +⇢23)⇢123
R1=1
ˆ
Wh(⇢1+⇢2+⇢3)2(⇢12 +⇢13 +⇢23)+3⇢123i
R2=1
ˆ
W2h(⇢12 +⇢13 +⇢23)3⇢123i
R3=1
ˆ
W3[⇢123] (A.5)
For simplicity A.3 assumes isotropic smoothness and A.5 assumes position-
independent smoothness ˆ
W=⌃ˆ
Wk/K, but these restrictions can easily be
lifted (Worsley et al., 1999). Each Rdis associated with an independent
probability density function (Worsley et al., 1996) (Appendix C) that di-
rectly depends only on the tthreshold.
These density functions can be used to compute a variety of topological
expectations, like the number of supra-threshold nodes and clusters, for
example (Friston et al., 1994). The final steps in RFT-based inference are
thus to threshold an observed SPM{t}at a suitably high value (e.g. t>3.0)
and then corroborate the observed topology with topological expectation,
computing pvalues for each cluster according to Friston et al. (1994), for
example. The logic of RFT is that smooth random fields are expected to
produce spatially broad suprathreshold clusters, but very broad and/or very
high clusters are expected to occur with low probability. The key message
is that a large suprathreshold cluster is the topological equivalent of a large
univariate tvalue.
Appendix B. Search space geometry
To rapidly compute the geometrical characteristic ⇢of an nD search
space defined by binary image A, one may use morphological erosion (Nixon
and Aguado, 2008):
⇢=|A B|(B.1)
where the Bmatrices are structuring elements that describe the neighbor-
hood connectivity of interest. In three dimensions single nodes (B0), adja-
cent nodes (Bd), faces (Bdd0), and cubes (B123) are given by the sets:
B0={0,0,0}
B1={{0,0,0},{1,0,0}}
B2={{0,0,0},{0,1,0}}
B3={{0,0,0},{0,0,1}}
B12 ={{0,0,0},{1,0,0},{0,1,0},{1,1,0}}
B13 ={{0,0,0},{1,0,0},{0,0,1},{1,0,1}}
B23 ={{0,0,0},{0,1,0},{0,0,1},{0,1,1}}
B123 ={{0,0,0},{1,0,0},{0,1,0},{1,1,0},...
{0,0,1},{1,0,1},{0,1,1},{1,1,1}} (B.2)
Here A Bis an eroded binary image whose elements are ones if the B
pattern exists at a given node and zeros otherwise. |A B|is the set size
of A Bor, equivalently, the number of ones in the eroded image.
Appendix C. Euler characteristic densities
Each ‘resel count’ Rd(A.5) is associated with an independent proba-
bility distribution function, or Euler characteristic density, pd(t). To three
dimensions, directly from Worsley et al. (1996, Table 2), the densities are:
p0(t)=Z1
t
(⌫+1
2)
⌫⇡1
2(⌫
2)(1 + u2
⌫)1
2(⌫+1)du
p1(t)=(4log2)1
2
2⇡✓1+t2
⌫◆1
2(⌫1)
p2(t)=(4log2)
2⇡
(⌫+1
2)
(⌫
2)1
2(⌫
2)t✓1+t2
⌫◆1
2(⌫1)
p3(t)=(4log2)3
2
(2⇡)2✓1+t2
⌫◆1
2(⌫1) ✓⌫1
⌫t21◆
where ⌫is the degrees of freedom. Note that p0is the univariate Student’s t
distribution. The general nD form of these distributions is given in Worsley
(1994, Corollary 5.3).
Appendix D. Numerical example
Consider five fictional force trajectories from each of two experimen-
tal conditions ‘A’ and ‘B’. (Fig.D.a) on a normalized time interval 0-100%
(K=100). Condition A data were created by adding smooth Gaussian
noise (FWHM=10%) to yA(t) = 800 N. Condition B data were created
by first adding positive Gaussian signals to yB(t) = 800 N at t=75% and
15% (see Fig.D.b), and then subsequently adding filtered Gaussian noise
(also at FWHM=10%). These two simulated experimental conditions were
then compared using a two-sample ttest (Eqn.5, main manuscript) where
c=⇥11
⇤Tand:
X=2
6
4
1111100000
0000011111
3
7
5
T
(D.1)
After thresholding the resultant SPM{t}at t>3.0, the significance of
the suprathreshold clusters (Fig.D.c) was assessed using the RFT procedures
described above, where the average FWHM was estimated to be 10.4% using
Eqn.A.3. This simulation highlights three concepts: (i) SPM can be used to
analyze continuous field data in a topological manner. (ii) A Bonferroni ap-
proach (K=100, tcritical = 5.192) would fail to identify significance anywhere
in the temporal field. (iii) A discrete approach that focusses only on the re-
gion t=75% would fail to identify the other signal at t=15%. While this
example has been tailored to emphasize these concepts, the methodology (i)
and dangers (ii,iii) clearly also apply to real experimental data.
Figure D: Example 1D SPM analysis. (a) Simulated raw data for two experimental con-
ditions ‘A’ and ‘B’. (b) Mean curves with standard deviation clouds. (c) SPM{t}with
threshold t>3.0. The pvalues were computed according to Appendix A-C and indicate
the probability that the specific suprathreshold cluster could have occurred by chance.
Appendix E. Model visualization
The general linear model (GLM) (Eqn.1, main manuscript) consists of:
experimental observations (Y), experimental design (X), regression coeffi-
cients (), and model errors ("). Since these matrices can be quite large,
numerical probing of their elements is inconvenient. The elements may, how-
ever, be conveniently probed qualitatively using matrix visualization tech-
niques. The most important matrix to visualize is Xbecause it represents
the experimenter’s statistical modeling decisions: it describes all modeled
experimental factors, it reveals the randomness of the design, and together
with a contrast vector (Eqn.5, main manuscript) it explicitly describes the
experimental hypothesis.
Fig.E.1 depicts the design matrix that was used to analyze the current
pedobarographic data. Rows correspond to trials, and columns to modeled
experimental factors. The main factor of interest was ‘speed’, and the first
column of Fig.E.1 reveals that walking speeds were randomized. Three
factors of non-interest were also also modeled: an intercept ‘y0’ and two
low-frequency time-drift nuisance factors. Visualizing these factors as a
matrix image can be helpful to understand the experimental design, so X
renderings like Fig.E.1 are often presented in scientific papers.
It is also instructive to visualize the entire GLM (Fig.E.2). The obser-
vations Ymay be regarded as a bird’s eye view of Fig.3a. The model fits
(Xˆ
) closely resemble the experimental data (Y), with only relatively mi-
nor di↵erences ("). This indicates, anecdotally, that the GLM can explain
a large proportion of the experimental variability.
Figure E.1: Design matrix X(Eqn.1, main manuscript). The matrix is (60⇥4): 60 trials
and 4 modeled experimental factors. The color scale is normalized within columns.
Figure E.2: Grayscale renderings of the general linear model matrices (Eqn.1, main
manuscript) for the 1D VGRF dataset. Each matrix is (60⇥100): 60 trials and 100
VGRF trajectory nodes per trial. Absolute errors |"|are presented.
References
Adler, R. J. 1981. The geometry of random fields, Wiley, Chichester.
Friston, K. J., Worsley, K. J., Frackowiak, R. S. J., Mazziotta, J. C., and Evans, A. C.
1994. Assessing the significance of focal activations using their spatial extent, Human
Brain Mapping 1, 210–220.
Kiebel, S. J., Poline, J. B., Friston, K. J., Holmes, A. P., and Worsley, K. J. 1999. Robust
smoothness estimation in statistical parametric maps using standardized residuals from
the general linear model, Neuroimage 10, 756–766.
Nixon, M. and Aguado, A. 2008. Feature extraction & image processing, second edition,
2nd ed., Academic Press.
Worsley, K. J. 1994. Local maxima and the expected Euler characteristic of excursion sets
of 2,Fand tfields, Advances in Applied Probability 26, 13–42.
Worsley, K. J. 2007. Random field theory. In: Statistical parametric mapping: the analysis
of functional brain images (K.J. et al. Friston, ed.), Elsevier/Academic Press, Amster-
dam.
Worsley, K. J., Andermann, M., Koulis, T., Macdonald, D., and Evans, A. C. 1999. De-
tecting changes in nonisotropic images, Human Brain Mapping 8, 98–101.
Worsley, K. J., Marrett, S., Neelin, P., Vandal, A. C., Friston, K. J., and Evans, A. C. 1996.
A unified statistical approach for determining significant signals in images of cerebral
activation, Human Brain Mapping 4, 58–73.
