Multiresolution morphology of
cosmological fields
Tartu Observatoorium,
Observatòri Astronomic, Universitat de València,
DAPNIA/SEDI-SAP. Service d’Astrophysique,
Department of Statistic, Stanford University
Enn Saar, Vicent Martı́nez, Jean-Luc Starck, David Donoho
JENAM 2004, Granada
Multiresolution morphology of cosmological fields – p. 1/5
The 2dFGRS
Christmas tree.
2dF Image Gallery
Multiresolution morphology of cosmological fields – p. 2/5
Notes
A cube of the 2dF GRS galaxy distribution. The passport
pictures show the morphological types of individual
galaxies, and the semitransparent volume shows the
underlying density field. The density field is frequently a
more important physical descriptor than the list of galaxy
positions.
Source: (http://magnum.anu.edu.au/ TDFgg/)
Multiresolution morphology of cosmological fields – p. 3/5
Overview
The basic model
Adaptive methods
The à trous wavelet transform
The multiresolution world
Minkowski functionals
Multiresolution morphology
Multiresolution morphology of cosmological fields – p. 4/5
The basic model
Random (Gaussian) fields, realizations:
A Poisson process filling in:
The information content of the Universe
, Tegmark
Creation of information?
Cosmology, astronomy, physics etc.
Multiresolution morphology of cosmological fields – p. 5/5
Notes
Max Tegmark has written a paper arguing that the
Universe contains practically no information (see his home
page http://www.hep.upenn-edu/ max); the
structure we see is a result of chaotic amplification of
random fluctuations. This could be right, but our practice
has taught us that maps are also important, in many
branches of science.
Multiresolution morphology of cosmological fields – p. 6/5
Adaptive methods
is a distribution function:
where
Adaptive kernels
Multiresolution morphology of cosmological fields – p. 7/5
Techniques:
sample point, sandbox, gather estimators:
balloon, scatter estimators:
.
Bias
Var
to minimize
choose
MSE
Multiresolution morphology of cosmological fields – p. 8/5
Notes
Kernel methods for density estimation are better than
the usual histograms, where we choose the bin
locations at will. Adaptive kernels are better than
constant-width kernels, as they do not oversmooth
the density.
Note that statisticians minimize the total mean square
error (MSE), not only the variance.
Multiresolution morphology of cosmological fields – p. 9/5
Kd-trees
k-d tree example
Data structure (3D case)
Introduction to Pattern Recognition
Ricardo Gutierrez-Osuna
Wright State University
Partitioning (2D case)
13
Multiresolution morphology of cosmological fields – p. 10/5
Notes
Examples of adaptive density estimation methods.
K-d trees are formed by recursive division of the sample
space into two equal-probability (equal points number)
halves. It is a simplest version of adaptive histograms. For
a recent application of k-d trees to density estimation see
Y. Ascasibar and J. Binney, astro-ph/0409233.
Multiresolution morphology of cosmological fields – p. 11/5
kNN kernel
kNN Density Estimation, example 2 (a)
The performance of the kNN
density estimation technique on
two dimensions is illustrated in
these figures
The top figure shows the true density,
a mixture of two bivariate Gaussians
1
1
P(x) = N( 1, 1 ) + N( 2 , 2 )
2
2

 1 1
T
 1 = [0 5]
1 = 


 1 2
with
 1 − 1
 = [5 0]T
2 = 

 2
− 1 4 
The bottom figure shows the density
estimate for k=10 neighbors and
N=200 examples
In the next slide we show the
contours of the two distributions
overlapped with the training data
used to generate the estimate
Introduction to Pattern Recognition
Ricardo Gutierrez-Osuna
Wright State University
13
smooth (Washington University N-body shop,
Joachim Stadel)
Multiresolution morphology of cosmological fields – p. 12/5
Notes
Another well-known method – k-th nearest
neighbour.
Smooth is a public-domain adaptive kernel density
estimation program, written for N-body applications,
but useful also for general density estimation.
Recommended.
Multiresolution morphology of cosmological fields – p. 13/5
Wavelet cleaning
Left – Gaussian
Mpc smoothing, middle – wavelet cleaning,
Mpc smoothing.
right – Gaussian
Multiresolution morphology of cosmological fields – p. 14/5
Wavelet cleaning of a model galaxy distribution
(left – a successful attempt, right – overcleaning).
Multiresolution morphology of cosmological fields – p. 15/5
Notes
Wavelet cleaning is a popular image processing
methodology; we have tried to apply it to 3-D density
estimation. The first example shows that
wavelet-cleaned densities are adaptive, describing all
spatial scales.
3-D wavelet cleaning is also hard, due to the large
density range; negative or zero-density artefacts tend
to appear frequently.
Multiresolution morphology of cosmological fields – p. 16/5
The à trous wavelet transform
:
One-dimensional signals
Decomposition (and reconstruction)
The scaling function
Define the wavelet:
Multiresolution morphology of cosmological fields – p. 17/5
, and from
to
to
Smooth: Passage from
0
C
−4
−3
0
−1
−2
1
h(−2) h(−1) h(0) h(1)
2
3
4
2
3
4
3
4
h(2)
C1
−4
−3
h(−2)
−2
0
−1
h(−1)
1
h(0)
h(1)
h(2)
C2
−4
−3
−2
0
−1
1
2
The wavelet coefficients are:
Multiresolution morphology of cosmological fields – p. 18/5
Notes
Wavelet transforms can be formulated as convolution
of data with a zero-mean kernel. Changing the size of
the kernel, we can study details of different scale.
Wavelet transforms are local real space – frequency
(wavenumber) space transforms. The à trous (with
holes) transform is fast and breaks a data sample into
a sum of several “densities” of different
non-overlapping dyadic frequency ranges.
The slides give the rules to calculate the transform,
and to reconstruct the final density.
Multiresolution morphology of cosmological fields – p. 19/5
Three dimensions:
The scaling function
decomposition (and reconstruction)
Smooth:
data sets for all dimensions.
Multiresolution morphology of cosmological fields – p. 20/5
Notes
The à trous transform can be easily generalized to any
number of dimensions; the scaling function is a direct
product of one-dimensional scaling functions.
Interestingly, the wavelet itself is isotropic.
is the number of dyadic frequency ranges (the ratio
of the largest and smallest frequencies, and spatial
scales, is ).
See Jean-Luc Starck’s home page
(http://jstarck.free.fr) for a tutorial and
references.
Multiresolution morphology of cosmological fields – p. 21/5
spline
The
0.7
0.6
0.5
0.4
The
spline:
– the
scaling function, – the
wavelet.
ϕ
0.3
0.2
ψ
0.1
0
-0.1
-0.2
-0.3
-2
-1.5
-1
-0.5
0
x
0.5
1
1.5
2
Multiresolution morphology of cosmological fields – p. 22/5
Notes
The
spline is a popular scaling function in wavelet
applications.
The function is the wavelet; it can be calculated
from the scaling function and the weights above.
Note that it integrates to zero.
Multiresolution morphology of cosmological fields – p. 23/5
The 3-D
spline generated wavelet
Multiresolution morphology of cosmological fields – p. 24/5
DAPNIA/SEI-SAP
An à trous example
NGC2997
Data – NGC 2997
Multiresolution morphology of cosmological fields – p. 25/5
50
À trous transform of the NGC 2997 data
Multiresolution morphology of cosmological fields – p. 26/5
À trous transform, wavelet amplitudes
Multiresolution morphology of cosmological fields – p. 27/5
Notes
The galaxy image is transformed into several images – the
smoothest scaling image at the lower right (above in the
last slide), and into wavelet orders of different spatial
scales. These sum together to give the original image.
Multiresolution morphology of cosmological fields – p. 28/5
The multiresolution world
À trous density slices for an N-body model structure (logarithmic scale).
Multiresolution morphology of cosmological fields – p. 29/5
À trous potential (slices) for the N-body model above (linear scale).
Multiresolution morphology of cosmological fields – p. 30/5
À trous density (slices) for a model (GIF) galaxy sample (logarithmic scale).
Multiresolution morphology of cosmological fields – p. 31/5
À trous density (slices) for a Voronoi walls sample (square root scale).
Multiresolution morphology of cosmological fields – p. 32/5
Notes
Examples of à trous decomposition of 3-D fields.
N-body model densities – upper row shows scaling
(smoothed) distributions, lower row – wavelet orders.
Gravitational potential of the same model. Note that
although the potential is smoother than density, it has
strong small-scale features.
The GIF galaxy distribution is hoped to represent well
the spatial distribution of bright galaxies.
Voronoi walls are an extreme example of
non-Gaussian densities.
Multiresolution morphology of cosmological fields – p. 33/5
Minkowski functionals
Topology of Gaussian cubes of different smoothing
(left columns – empty regions, right columns – filled regions).
Multiresolution morphology of cosmological fields – p. 34/5
Topology of a (model) galaxy sample for different density levels.
Multiresolution morphology of cosmological fields – p. 35/5
Notes
Examples of density distributions of different connectivity
(“topology, morphology”).
Density distributions are cut with an isodensity
surface, lower density regions are called “empty”,
other regions – “filled”.
Realizations of Gaussian random fields have low
density – high density symmetry.
The isodensity surfaces for a galaxy sample seem to
differ from the Gaussian densities in the previous
slide.
Multiresolution morphology of cosmological fields – p. 36/5
Complete morphological description of scalar
fields is given by Minkowski functionalsa (
for -dimensional space).
We start for a:
point distribution – decorating the points
with balls( ),
continous distribution – slicing by density
isolevels.
a
Minkowski functionals: read K.R. Mecke, T. Buchert, H. Wagner, “Robust morphological measures for large-scale structure in
the Universe”, Astron. Astrophys. 288, 697-704 (1994).
Multiresolution morphology of cosmological fields – p. 37/5
Notes
“Complete morphological description” does not mean that
Minkowski functionals exhaust all the information about a
field. It only means that any additive, motion invariant
and conditionally continuous functional defined for any
(hyper)surface is a linear combination of its Minkowski
functionals.
Multiresolution morphology of cosmological fields – p. 38/5
Let
be the excursion set of a field
(all
). For a 3-D field the
points where
Minkowski functionals are:
(volume),
(surface area),
(integrated mean curvature of the
boundary),
Multiresolution morphology of cosmological fields – p. 39/5
(integrated
Gaussian curvature of the boundary).
of holes ( is the
of isolated regions
topological genus).
density deviation
Arguments:
,
:
Gaussianized volume fraction
.
Multiresolution morphology of cosmological fields – p. 40/5
Notes
The Gaussianized volume fraction is used to avoid the
influence of the “trivial” non-Gaussianity – the
deformation of the one-point density distribution,
inevitably caused by gravitational evolution of the density.
Minkowski functionals are usually used to search for
traces of primordial non-Gaussianity.
Multiresolution morphology of cosmological fields – p. 41/5
Lattice algorithm (Kendrick invariants):
a lattice of a step , with vertices.
In the excursion set:
vertices,
segments,
faces,
cells.
Multiresolution morphology of cosmological fields – p. 42/5
Notes
The use of Kendrick invariants gives extremely fast
algorithms to calculate Minkowski functionals. Numerical
integration of curvature integrals is also used; it is slower,
but is immune to sample border effects. Kendrick
invariants work well for simple sample geometries
(bricks).
Multiresolution morphology of cosmological fields – p. 43/5
where
Gaussian predictions for densities:
,
,
,
,
.
Multiresolution morphology of cosmological fields – p. 44/5
Notes
Note that the functional forms of the Minkowski
functionals do not depend on the power spectrum or the
correlation function of a Gaussian random field; only the
amplitudes do.
Multiresolution morphology of cosmological fields – p. 45/5
1
2.5
0.9
0.8
2
0.7
V1(νG)
1.5
V1
V0
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V0(νG)
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V0(νσ)
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V1(νσ)
0.1
0
0
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νσ , νG
2
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νσ , νG
15
150
100
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50
V2(νσ)
5
V3(νσ)
0
V2(νG)
V3
V2
1
0
V3(νG)
-50
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-15
-250
-3
-2
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νσ , νG
2
3
4
-3
-2
-1
0
1
νσ , ν G
Minkowski functionals for model (GIF) galaxies (dotted lines show predictions for Gaussian
random field MF-s).
Multiresolution morphology of cosmological fields – p. 46/5
Notes
The Minkowski functionals of the GIF galaxies are
close to the Gaussian field predictions, if studied as an
argument of the Gaussianized volume fraction. This
is a general result, obtained so far for most of the
observed data and simulated distributions.
When expressed in density deviation arguments, the
functionals differ strongly (for the Gaussian case both
arguments coincide).
Multiresolution morphology of cosmological fields – p. 47/5
30000
80000
60000
20000
40000
10000
20000
0
V3
V3
0
-10000
-20000
-40000
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-80000
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-3
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ν
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ν
Topological characteristics for dark matter densities (left – smoothed versions. right –
scale-separated data).
Multiresolution morphology of cosmological fields – p. 48/5
Notes
Examples of the scale-by-scale morphological analysis.
Gaussian predictions are shown by dot-dashed lines, other
lines represent different à trous orders. Both the scaling
(smoothed) distributions and the wavelet distributions are
close to the Gaussian shape only for large-scale orders.
Multiresolution morphology of cosmological fields – p. 49/5
80
5000
4000
60
3000
2000
1000
V3
V3
40
20
0
-1000
-2000
0
-3000
-4000
-20
-5000
-40
-2.5
-2
-1.5
-1
-0.5
0
0.5
ν
1
1.5
2
2.5
3
-6000
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
ν
Topological characteristics for dark matter potentials (left – smoothed versions. right –
scale-separated data).
Multiresolution morphology of cosmological fields – p. 50/5
Notes
The Minkowski functionals for the gravitational potential
also show strong non-Gaussian features for smaller scales.
The shift of the minima of to positive volume fractions
shows the so-called “spaghetti” morphology – regions of
high potential values are much strongly connected than in
the case of Gaussian densities.
Multiresolution morphology of cosmological fields – p. 51/5
12000
15000
10000
10000
8000
5000
V3
V3
6000
4000
0
2000
-5000
0
-2000
-10000
-3
-2
-1
0
1
ν
2
3
4
-3
-2
-1
0
1
2
3
ν
Topological characteristics for model galaxy densities (left – smoothed versions. right –
scale-separated data).
Multiresolution morphology of cosmological fields – p. 52/5
200
600
400
100
0
-100
-200
V3
V3
200
0
-200
-400
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-600
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-1600
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
ν
Topological characteristics for Voronoi wall densities (left – smoothed versions. right –
scale-separated data).
Multiresolution morphology of cosmological fields – p. 53/5
Notes
The functional for model galaxies differs strongly
from Gaussian predictions. As usual, it approaches
the Gaussian form for strong smoothing, only.
For the Voronoi walls the functional differs from
Gaussian predictions for any à trous orders, as
expected. We also used Gaussian smoothing for this
density distribution and found that this transforms
the morphology to Gaussian, hiding the true
non-Gaussianity. Multiscale analysis is a much more
transparent tool.
Multiresolution morphology of cosmological fields – p. 54/5
2dF example
2500
300
2000
200
2dF North, wavelet order 2
1500
2dF North, wavelet order 3
100
1000
0
500
V3
V3
-100
0
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-500
-300
-1000
-1500
-400
-2000
-500
-2500
-600
-3
-2
-1
0
ν
1
2
-3
3
-2
-1
0
1
2
3
2
3
ν
40
10
2dF North, scaling order 4
2dF North, wavelet order 4
0
0
V3
5
V3
20
-20
-5
-40
-10
-60
-15
-80
-20
-3
-2
-1
0
ν
1
2
3
-3
-2
-1
0
1
ν
The multiresolution topological characteristic for a 2dF Northern volume-limited sample.
Multiresolution morphology of cosmological fields – p. 55/5
Notes
These results were obtained a couple of weeks after
JENAM.
The 2df data used here is for a maximum-volume
brick, cut from the 2dF volume-limited sample for
, Northern sky.
The solid line shows the observational result, the
dotted lines show the 95% (“2-sigma”) confidence
regions for a Gaussian field.
The spatial scales for the wavelet order range from 1
;
to
,
(all in units of
to 5,
Mpc).
Multiresolution morphology of cosmological fields – p. 56/5
Notes
Only the largest-scales scaling function (
, scales
are larger than 40) and wavelet (scales from 20 to 40)
can be considered Gaussian.
Higher-order functions are already too noisy to
analyze; larger sample volumes than available at
present are needed for scales larger than 40.
Multiresolution morphology of cosmological fields – p. 57/5
Conclusions
Adaptive density estimation and wavelet
cleaning gives an estimate of the true density
field.
Multiresolution analysis allows
scale-by-scale study of cosmological fields.
Gaussianity is preserved in the present
galaxy density fields only at very large scales
(starting from 20–40
Mpc), and even at
these scales not too well.
Multiresolution morphology of cosmological fields – p. 58/5