SAMPLING SCHEMES FOR 2D SIGNALS WITH FINITE RATE OF INNOVATION

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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE
RATE OF INNOVATION
A transfer talk on
by
Pancham Shukla
supervisor
Dr P L Dragotti
Communications and Signal Processing
Group Imperial College London
? This research is supported by EPSRC.
1/3/2005
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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATIONOUTLINE

• INTRODUCTION
• Sampling Problem, Background, and Scope
• SIGNALS WITH FINITE RATE OF INNOVATION (FRI)
  (non-bandlimited)
• Definition, Extension in 2-D
• 2-D Sampling setup, Sampling kernels and their
  properties
• SAMPLING OF FRI SIGNALS
• SETS OF 2-D DIRACS
• Local reconstruction (amplitude and position)
• BILEVEL POLYGONS DIRACS using COMPLEX MOMENTS
• Global reconstruction (corner points)
• PLANAR POLYGONS using DIRECTIONAL DERIVATIVES
• Local reconstruction (corner points)
• CONCLUSION AND FUTURE WORK

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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION1. INTRODUCTION

• Why sampling? Many natural phenomena are
  continuous (e.g. Speech, Remote sensing) and
  required to be observed and processed by
  sampling.
• Many times
  we need reconstruction (perfect !) of the
  original phenomena.
• Continuous ?
  Discrete (samples) ? Continuous
• Sampling theory by Shannon (Kotelnikov,
  Whittaker)
• Why not always bandlimited-sinc? (Although
  powerful and widely used since 5 decades)
• 1. Real world signals are non-bandlimited.
• 2. Ideal low pass (anti-aliasing,
  reconstruction) filter does not exist.
  (Acquisition devices)
• 3. Shannons reconstruction formula is rarely
  used in practice with finite length signals (esp.
  images)
• due to infinite support and slow decay of
  sinc kernel. (do we achieve PR in practice?)

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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION1. INTRODUCTION contd.

• Extensions of Shannons theory
• So many papers but for comprehensive account,
  we refer to Jerry 1977, Unser 2000.
• Shift-invariant subspaces Unser et al.
• The classes of non-bandlimited signals (e.g.
  uniform splines) residing in the shift-invariant
  subspaces can be perfectly reconstructed. The
  other non-bandlimited signals are approximated
  through their projections.

We look into Non-bandlimited signals that do
not reside in shift-invariant subspace but have a
parametric representation. Non-traditional ways
of perfect reconstruction .from the projections
of such signals in the shift-invariant
subspace. Is it possible to perfectly
reconstruct such signals from their samples? Any
examples of such signals ? What type of kernels
? Sampling and reconstruction schemes?
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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION2. FRI SIGNALS

• Very recently such signals are identified and
  termed as
• Signals with Finite Rate of Innovation (or FRI
  signals) Vetterli et al. 2002
• Model Non-bandlimited signals that do not
  reside in shift-invariant subspace.
• Examples Streams of Diracs, non-uniform
  splines, and piecewise polynomials.
• Unique feature A finite number of degrees of
  freedom per time (rate of innovation ?)
• e.g. a Dirac in 1-D has a rate of innovation 2
  (i.e. amplitude and position).
• The sampling schemes for such signals in 1-D are
  given by Vetterli, Marziliano and Blu 2002.
• Extensions of these schemes in 2-D are given by
  Maravic and Vetterli 2004, however, focusing
  on Sampling kernels as sinc and Gaussian.
• Algorithms Little more involved reconstruction
  algorithms (solution of linear systems, root
  finding) based on Annihilating filter method
  from Spectral estimation, Error correction
  coding.
• Reconstruction Only a finite number of samples
  (?? ?) guarantees perfect reconstruction.

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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION2. FRI SIGNALS contd.

• Assortments of kernels Dragotti, Vetterli and
  Blu, ICASSP-2005
• For 1-D FRI signals, one can use varieties of
  kernels such as
• That reproduce polynomials (satisfy Strang and
  Fix conditions)
• Exponential Splines (E-Splines) Unser
• Functions with rational Fourier transforms
• Our Focus
• Sampling extensions in 2-D using above mentioned
  kernels, in particular, for
• Sets of 2-D Diracs ? Local Global schemes
  Local kernels Complex moments (AFM)
• Bilevel polygons ? Global scheme Complex
  moments Annihilating filter method (AFM)
• Planar polygons ? Local scheme Directional
  derivatives Directional kernels

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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION2. SAMPLING FRI SIGNALS in 2-D

• Sampling setup
• Properties of sampling kernels
• In current context, any kernel that reproduce
  polynomials ?
• of degrees ? 0,1,2??-1 such that
• Partition of unity
• Reproduction of polynomials along x-axis
• Reproduction of polynomials along y-axis

Set of samples in 2-D
Input signal
Sampling kernel
e.g., B-Splines (biorthogonal) and Daubechies
scaling functions (orthogonal) are valid kernels
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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION3. SETS OF 2-D DIRACS

• Sets of 2-D Diracs Local reconstruction
• Consider and with
  support such that
• there is at most one Dirac
  in an area of size .
  Assume .
• From the partition of unity
• (reproducing of polynomial of degree 0),it
  follows that

The amplitude
This is derived as follows,
B-Splines of order one
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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION2. SAMPLING FRI SIGNALS in 2-D

• Properties of sampling kernels
• In current context, any kernel that reproduce
  polynomials ?
• of degrees ? 0,1,2??-1 such that
• Partition of unity
• Reproduction of polynomials along x-axis
• Reproduction of polynomials along y-axis

e.g., B-Splines (biorthogonal) and Daubechies
scaling functions (orthogonal) are valid kernels
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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION3. SETS OF 2-D DIRACS contd.

• Sets of 2-D Diracs Local reconstruction contd.
• and using polynomial reproduction properties
  along x and y directions,
• the coordinate positions are given by

Above relations are derived as,
As long as any two Diracs are sufficiently apart,
we can accurately reconstruct a set of Diracs,
considering one Dirac per time.
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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION4. BILEVEL POLYGONS DIRACS
Complex-moments

• Complex-moments for polygonal shapes earlier
  works
• Since decades, moments are used to characterize
  unspecified objects. Shohat and Tamarkin 1943,
  Elad et al. 2004. Here, we present a sampling
  perspective to the results of Davis 1964,
  Milanfar et al. 1995, Elad et al. 2004 on
  reconstruction of polygonal shapes using
  complex-moments.

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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION4. BILEVEL POLYGONS DIRACS
Complex-moments

• Complex-moments for polygonal shapes a modern
  connection

Now, we will briefly review the annihilating
filter method due to its relevance in finding
weights and positions of the corner points zi
from the observed complex moments.
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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION4. BILEVEL POLYGONS DIRACS
Complex-moments

• Annihilating filter method (we refer to
  Vetterli, Stoica and Moses for more details)
• This method is well known in the field of
  Error-correcting codes and Spectral estimation.
  Especially, in second application, it is employed
  to determine the weights and locations
  of the spectral components, generally
  observed in form of

The annihilating filter method consists of the
following steps
2. Locations The convolution condition is solved
by the following Yule-Walker system
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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION4. BILEVEL POLYGONS DIRACS
Complex-moments

• A sampling perspective (using Complex-moments
  Annihilating filter method)

Consider g(x,y) as a non-degenerate, simply
connected, and convex bilevel polygon with N
corner points
Consider g(x,y) as a set of N 2-D Diracs
Then from the complex-moments formulation of
Milanfar et al.
Because of the polynomial reproduction property
of the kernel, we derive that
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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION4. BILEVEL POLYGONS DIRACS
Complex-moments

• Simulation results

A set of N3 Diracs
Bilevel polygon with N3 corner points
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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION4. BILEVEL POLYGONS DIRACS
Complex-moments

• Summary
• A polygon with N corner points is uniquely
  determined from its samples using a kernel that
  reproduce polynomials up to degree 2N-1 along
  both x and y directions.
• A set of Diracs is uniquely determined from its
  samples using a kernel that reproduces
  polynomials up to degree 2N-1 along both x and y
  directions and that there are at most N Diracs in
  any distinct area of size .
• Global reconstruction algorithm
• Complexity ? Complexity of the signal
• Numerical instabilities in algorithmic
  implementations for very close corner points

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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION5. PLANAR POLYGONS
Directional-derivatives

• Problem formulation Intuitively, for a planar
  polygon, two successive directional derivatives
  along two adjacent sides of the polygon result
  into a 2-D Dirac at the corner point formed by
  the respective sides.

Discrete challenge

• Lattice theory Directional derivatives ?
  Discrete differences
• Subsampling over integer lattices and
• Local directional kernels in the framework of
  2-D Dirac sampling (local reconstruction)

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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION5. PLANAR POLYGONS
Directional-derivatives

• Lattice theory We refer to Cassels, Convey and
  Sloan for more detail.

Base lattice is a subset of points of Z2
(integer lattice)
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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION5. PLANAR POLYGONS
Directional-derivatives

• Proposed sampling scheme

By using Parsevals identities and after certain
manipulations, we have
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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION5. PLANAR POLYGONS
Directional-derivatives

• Directional kernels

Modified kernel is a directional kernel. For
each corner point ? independent directional
kernel.
For example,
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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION5. PLANAR POLYGONS
Directional-derivatives

• Local reconstruction of the corner point

The directional kernel can
reproduce polynomials of degrees 0 and 1 in both
x and y directions.
Assume that there is only one corner point ?
support of its associated directional kernel.
Then from the local reconstruction scheme of
Diracs, we can reconstruct the amplitude and the
position of an equivalent Dirac at a given corner
point (e.g. at point A ) as
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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION5. PLANAR POLYGONS
Directional-derivatives

• Simulation result

Local reconstruction
Pair of directional differences
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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION5. PLANAR POLYGONS
Directional-derivatives

• Summary reconstruction algorithm

Initial intuition
Final realization

• Advantage
• Local reconstruction.
• Only local reconstruction complexity,
  irrespective of the number of corner points in a
  polygon.

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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION6. CONCLUSION FUTURE WORK

• Conclusion
• We have proposed several sampling schemes for the
  classes of 2-D non-bandlimited signals.
• In particular, sets of Diracs and (bilevel and
  planar) polygons can be reconstructed from their
  samples by using kernels that reproduce
  polynomials.
• Combining the tools like annihilating filter
  method, complex-moments, and directional
  derivatives, we provide local and global sampling
  choices with varying degrees of complexity.

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SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE
OF INNOVATION6. CONCLUSION FUTURE WORK

• Future work
• From March 2005 to October 2005
• Exploring a different class of kernels, namely,
  exponential splines (E-Splines).
• Extending the sampling schemes in higher
  dimension. For instance, using the notion of
  complex numbers in 4-D (quaternion).
• Considering more intricate cases such as
  piecewise polynomials inside the polygons, and
  planar shapes with piecewise polynomial
  boundaries.
• We plan to submit a paper for IEEE Transactions
  on Image Processing by summer 2005.
• From November 2005 to June 2006
• Studying the wavelet footprints Dragotti 2003
  and then extending them in 2-D
• Integrating the proposed sampling schemes with
  the footprints in 2-D
• Investigating the sampling situations when the
  signals are perturbed with the noise
• Developing resolution enhancement algorithms for
  satellite images.

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Questions?