Manifold based analysis.
Projection to ambient space
Dr. Felipe Orihuela-Espina
Initial proposition: 2016
Current version: 2016
Análisis

Tema:
 Análisis de la codificación de tensores en
espacios ambientes

Hipótesis:
 Bajo determinadas condiciones, la proyección


Basado
en variedades
de subtensores a puntos de un espacio
contenedor ambiente puede garantizar la
obtención de una variedad.
Si se respetan estas condiciones, entonces es
posible diseñar proyecciones que de forma
concomitante lleven a cabo algún tipo de
procesamiento.
Objetivo:
 Establecer cómo codificar diferentes
operaciones de procesamiento de la
información durante la proyección sin “romper”
la formación de la variedad.

Contribuciones:
 Establecimiento de un conjunto de funciones
de proyección
(c) 2012-6 Dr. Felipe Orihuela Espina
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THE OBSERVATIONS MANIFOLD:
FROM THE IMAGE TO POINTS IN
A SPACE
© 2015-6. INAOE fNIRS group
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Temporal
From subtensors to points
Spatial
Single parameter
We need to take care that
={n} is conformable for the
product. We currently know
how to do this for matrices but
not for full n-dimensional
tensors.
Ambient Space
Reconstructed
Data
Subtensor

© 2015-6. INAOE fNIRS group
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From subtensors to points
 A subtensor is a multidimensional signal.
 It may have temporal, spatial and parameter
coordintes.
 Let our neuroimage be the tensor; I=h(X,T,P), it is
possible to refer to a subset of the neuroimage by
the subtensor:
 Where



is a subset of spatial locations
is a subset of time samples
is a subset of parameters
© 2015-6. INAOE fNIRS group
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From subtensors to points
 The subtensor may represent many things. Some
examples are:
 An experimental condition block or trial
 e.g. a subset of temporal samples, across all channels and
parameters
 A brain region for ROI analysis
 e.g. a subset of channels, across all time samples and
parameters
 A topographical view for spatial analysis
 e.g. a single temporal sample
 The isolation of a parameter behavior
 e.g. a single parameter
 You name it here…pick and mix!
© 2015-6. INAOE fNIRS group
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From subtensors to points
 Perhaps the simplest case is to choose the subtensor
such that we isolate a single channel xiX and
parameter pkP:
 In this case, the subtensor becomes a classical
timecourse signal.
 Please note that this is NOT the only analysis that
you can do or you might ever be interested in doing. It
just happens to be a departing example which is
convenient for the analysis of effective connectivity.
© 2015-6. INAOE fNIRS group
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From subtensors to points
 Let x(t)* be a continuous signal and x[t] its
sampled and quantized [digitized] version.
An example with impulse
sampling. Quantization not
illustrated.
* Watch out! I’m using an ambiguous notation here. This x(t) is any generic
signal and should not be confused with some specific signal coming from a
© 2015-6. INAOE fNIRS group
certain channel x.
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From signals to points
 It is customary to think of x[t] as a point
x=<x1,…,xn> in a n dimensional space,
where n is the number of samples.
 The value xi along the i-th dimensión
corresponds to the signal value or intensity at
the t-th sample; x[ti].
 In other words, there is a multivalued function
X:TRn such that univocally makes a
correspondence between sampled signals x[t]
and points in a space.
© 2015-6. INAOE fNIRS group
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From signals to points
 Let n be a function (of a coordinate
system) representing the n-th axis in Rn.
n:Tx[n]R
 It is posible to construct a coordinate
system ={n}* to project each sample to a
coordinate and thus matching the signal to
a point in the space.
*Again, there is an ambiguity in notation here.  does not represent a gradient or differential
change, but just a set of functions.  just happens to be the capital letter for .
© 2015-6. INAOE fNIRS group
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From signals to points
 Example: Let x[t]=t.
X
x[t]=t
x[3]=3
x[2]=2
x[1]=1
t=1
t=2
t=3
T
This example has been modified from: [http://cnx.org/contents/1Aio61Uf@1/Signals-and-vectors-are-points]
© 2015-6. INAOE fNIRS group
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From signals to points
 Example (Cont.): Let;
…
This example has been modified from: [http://cnx.org/contents/1Aio61Uf@1/Signals-and-vectors-are-points]
© 2015-6. INAOE fNIRS group
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From signals to points
 Example (Cont.): In the particular case of
the first 3 samples;
x[3]=3
3
2
x[2]=2
x[1]=1
1
This example has been modified from: [http://cnx.org/contents/1Aio61Uf@1/Signals-and-vectors-are-points]
© 2015-6. INAOE fNIRS group
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From signals to points
 Example (Cont.): So moving along x[t]
yields a point in Rn with the coordinate
system given by ={n}:
X
x[3]=3
x[t]=t
x[3]=3
={n}
3
x[2]=2
2
x[1]=1
x[2]=2
t=1 t=2 t=3
x[1]=1
1
T
This example has been modified from: [http://cnx.org/contents/1Aio61Uf@1/Signals-and-vectors-are-points]
© 2015-6. INAOE fNIRS group
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From signals to points
© 2015-6. INAOE fNIRS group
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From signals to points
 In fact, it hasn’t got to be a signal, but any(?)
function can be mapped to a point
Gustavo,Samuel;
¿Podeis pensar en
alguna función que no
se pueda? Si hay alguna
que no, ¿qué debe
cumplir la función para
que se pueda?
Electromagnetic spectrum as a discrete Spectrum Space
© 2015-6. INAOE fNIRS group
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[Orihuela-Espina, PhDThesis]
From signals to points
 Of course the above particular choice of
={n} is arbitrary and many other
coordinate systems can be chosen.
PENDIENTE: Tengo que
hacer un ejemplo con
un ={n} que no sea el
ortonormal (y
proyectar a 2D para
que se vea la
diferencia)
© 2015-6. INAOE fNIRS group
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From signals to points
 In principle, the choice of the coordinate system
should NOT affect the topology –short scale curvature(i.e. connectedness)
 e.g. an atlas and its equivalence class tell us that we

should “see” the same regardless of the coordinate system
However, the topology can be “seen” very differently upon
imposing a geometry –large scale curvature- (i.e. distance
function).
 HOWEVER This is only true if we can sample the
manifold very finely and thus reconstruct the ambient
space faithfully.
 In general terms, in neuroimaging this won’t be the case
as we will only have sparse sampling of the space, and
thus, the choice of the coordinate system affects the
topology
Can we do some tests to show this?
© 2015-6. INAOE fNIRS group
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From subtensors to points
 The next case is to choose the subtensor such that we
isolate a single channel xiX but observing all
parameters at once:
 In this case, the subtensor becomes a multivariate
timecourse signal.
  Again, please note that this is NOT the only
analysis that you can do or you might ever be
interested in doing. It just happens to be convenient
for the analysis of effective connectivity.
© 2015-6. INAOE fNIRS group
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From multivariate signals to points
 Multivariate signals (and complex –bivariate- signals in
particular) can also be projected to points. Two options are
available here.
 Concatenate or merge –not necessarrily by concatenation- the
univariate marginal projections of the signal along each
dimension, building an intermediate univariate signal, and then
proceed normally projecting the intermediate univariate signal
 For k dimensions, the intermediate signal will have kxn samples, and
the space where the point is projected will be Rkn.
 This is the approach taken in [Leff 2007]
 Treat the multivariate signal as it is generating “multidimensional”
samples.
 In the particular case of bivariate functions, each simple is a complex
number C, and thus the signal projects to Cn instead of Rn.
 …for higher k dimensional signals*, we will have something like a Kn
space.
 IMPORTANT: Even though C can be “represented” by R2, they still are
different!!! So will be Kn from Rkn.
* Note: There are no numbers sets that I’m aware of to represent this.
© 2015-6. INAOE fNIRS group
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[Orihuela-Espina, PhDThesis]
From multivariate signals to points
 Opt 1: Using an intermediate univariate
signal
Marginal
components
T
Univariate
proxy
© 2015-6. INAOE fNIRS group
2T
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From multivariate signals to points
 Opt 2: Multivariate proxy
Multivariate
samples
Re(x[n])
Re(x[1])
Im(x[1])© 2015-6. INAOE fNIRS group
Not sure of the
implications 
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From multivariate signals to points
 Opt 3: True multivariate
Multivariate
samples
C(x[n])
C(x[1])
C(x[1])
© 2015-6. INAOE fNIRS group
Requires complex
variable modelling
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From multivariate signals to points
 Opc 1:
 Opc 2:
 Opc 3:
© 2015-6. INAOE fNIRS group
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From multivariate signals to points
 Opc 3 (bis): Of course, no need to think of
time as the shift. Another possibility is;
© 2015-6. INAOE fNIRS group
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From multivariate signals to points
 Opt 1,2: Using an intermediate univariate signal or multivariate proxies
 Easier to model, but introduces “artefacts” at the “concatenation/merging” points
 Gustavo dijo que podemos concatenar de forma suave (PENDIENTE CHECAR)
 When projected as a point, dimensions can be “reorganized” with no consequences;
marginal values can be dissociated since there is no “link” to its original sample cocoordinates.
 Example: <x[1,k(1)], x[1,k(2)],…, x[n,k(k)]>
 Operations internal to a sample might become external.
 Consequences of this are still uninvestigated
 Opt 3: True multivariate
 Feasible for bivariate as Complex numbers, but difficult to generalize to higher



dimensions
 Note that this is the case for fNIRS
Since marginal values of each sample are still “binded” to its original sample cocoordinates, it is not possible to dissociate samples by reorganization
 Example: <K[1], …, K[n]>
Operations internal to a sample remain internal.
No previous example in literature that we are aware of. Not even sure how to address
this!.
© 2015-6. INAOE fNIRS group
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From multivariate signals to points
 IMPORTANT: Regardless of the choice; upon choosing the
ortonormal  for projecting to the ambient space, transforming
a signal (whether univariate or multivariate) to a point loses
the temporal information (we no longer know which sample
precedes/follows which other sample) unless the space
dimensions are somehow ordered.
 A posible way to circunvent this problema is to choose a  that in
each dimensión encodes not only a single sample, but a
“knowledge” of neighbour samples.
 For example: a  for a moving average filter. Note how each
dimension of the ambient space knows a bit of the “central”
sample but also of its neighbour sample.
© 2015-6. INAOE fNIRS group
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From neuroimages to points
 Since a neuroimage is a function f(X,T,),
it can ergo be mapped to a point in a
certain space.
f(X,T,)

X
T
3
2
1
© 2015-6. INAOE fNIRS group
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From neuroimages to points
 …but, of course, it is arbitrary to project the
FULL neuroimage to a single point, or take
specific subsets –perhaps having some
specific anatomic or experimental meaningand project each subset to a single point.
 Of course, the choice determine what
semantic is assigned to each point, and thus
which analysis can be performed and
neuroscientific question answered.
 Let’s see some other possibilities…
© 2015-6. INAOE fNIRS group
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From neuroimages to points
 Each channel is considered a separate
function fx(xi,T,), and projects to distinct
points
 The whole neuroimage projects to a cloud of
points.
fx(xi,T,)

X
T
3
2
1
© 2015-6. INAOE fNIRS group
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From neuroimages to points
 Blocks are splitted, resampled to a
ft(xi,[ti…ti+k],)

X
common length, then averaged and
for the averaged block, each channel
fx(xi,T,) is projected to a distinct point
 The whole neuroimage projects to a
cloud of points.
T
fx,t(xi,T’,)

X
3
2
T’
1
Average block
© 2015-6. INAOE fNIRS group
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