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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).

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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

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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.