1. INTRODUCTION AND MOTIVATION 2. SAMPLING OF SIGNALS WITH FINITE RATE OF INNOVATION (FRI) IN 2-D • SETS OF 2-D DIRACS (LOCAL RECONSTRUCTION) • BILEVEL POLYGONS & DIRACS using COMPLEX-MOMENTS (GLOBAL APPROACH) • PLANAR POLYGONS based on DIRECTIONAL DERIVATIVES (LOCAL APPROACH) 3. CONCLUSION AND ONGOING WORK 4. KEY REFERENCES SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE OF INNOVATION USING KERNELS THAT REPRODUCE POLYNOMIALS IEEE International Conference on Image Processing (ICIP) 2005, Genova, Italy. Pancham Shukla and Pier Luigi Dragotti Communications and Signal Processing Group, Electrical and Electronic Engineering Department, Imperial College London, UK Email: { P.Shukla, P.Dragotti } @ imperial.ac.uk Consider and with support such that there is at most one Dirac in an area of size j k k j xy k j y y x x a y x g ) , ( ) , ( , ) , ( y x xy y x L L ) , ( , q p xy q p y y x x a . y y x x T L T L From the partition of unity the amplitude is determined as: x y L j L k k j q p S a 1 1 , , And using polynomial reproduction properties along x-axis and y- axis, the coordinate position is determined as: q p L j L k k j x j p a S C x x y , 1 1 , , 1 (simple complex- moment) (weighted complex- moment) where are complex weights and are corner point coordinates of the bilevel polygon i i z ), , ( y x g . 1 2 ,... 3 , 2 N n Moments are used to characterize unspecified objects. [Shohat et al 1943, Elad et al. 2004]. For a convex, bilevel polygon with N corner points, and an analytic function in closure , the complex-moments of the polygon follow [Milanfar et al.][3]: ) , ( y x g The can be retrieved from using annihilating filter (Prony’s like method) such that i z w n ) ( z A . 0 ] [ w n i A Now for both bilevel polygon and set of Diracs, using complex- moments and annihilating filter method, it is straightforward to see that i z w n ) ( z A ) ( z A i z i a s n Theorem [Milanfar et al.] [3]: For a given non-degenerate, simply connected, and convex polygon in the complex Cartesian plane, all its N corner points are uniquely determined by its weighted complex-moments up to order 2N-1. w n A sampling perspective to above theorem follows… As the kernel can reproduce polynomials up to degree 2N-1 ( = 0,1...2N-1), all 2N moments are determined from weighted sums of the samples For instance, ) , ( y x xy w n . , k j S Complex-moments and annihilating filter Original image with three bilevel polygons Sampled version of the image Samples around pentagon Original pentagon and reconstructed corner points (marked with +) Sampling kernel that can reproduce polynomials up to degree 2N-1=9 ) , ( 9 y x xy Conclusion: Local and global sampling choices for the classes of 2-D FRI signals with varying degrees of complexity. Current investigations: Sampling of more general shapes such as circles, ellipses, and polygons containing polygonal holes inside. Future plans: Extension of our 2-D sampling results in higher dimensions and effect of noise. Integration of sampling results with wavelet footprints for image resolution enhancement and super-resolution photogrammetry. 1. M Vetterli, P Marziliano, and T Blu, “Sampling signals with finite rate of innovation,” IEEE Trans. on Signal Processing, 50(6): 1417-1428, June 2002. 2. P L Dragotti, M Vetterli, and T Blu, “Exact sampling results for signals with finite rate of innovation using Strang-Fix conditions and local kernels,” Proc. IEEE ICASSP, Philadelphia, USA, March 2005. 3. P Milanfar, G Verghese, W Karl, and A Willsky, “Reconstructing polygons from moments with connections to array processing,” IEEE Trans. on Signal Processing, 43(2): 432-443, February 1995. 4. V Velisavljevic, B Beferull-Lozano, M Vetterli, and P L Dragotti, “Discrete multi-directional wavelet bases,” Proc. IEEE ICIP, Barcelona, Spain, Intuitively, for a planar polygon, two successive directional derivatives along two adjacent sides of the polygon results into a 2-D Dirac at the corner point formed by the respective sides. Continuous model Discrete challenge Lattice theory [4] and [Convey and Sloan] Link directional derivatives discrete differences. Subsampling over integer lattices. Local directional kernels in the framework of 2-D Dirac sampling (local reconstruction). Sampling matrix 2 , 2 1 , 2 2 , 1 1 , 1 2 1 v v v v v v V 1 , 1 2 , 1 1 1 tan v v 1 , 2 2 , 2 1 2 tan v v wher e ) , ( , ) , ( ) det( 2 1 2 1 1 2 , y x y x g V S D D k j ) sin( ) , ( * ) , ( ) , ( 1 2 0 2 1 2 1 y x y x y x xy Dirac at A Modified kernel ). , ( 2 1 y x At each corner point Independent modified (directional) kernel Assume only one corner point in support of directional kernel. Using local reconstruction scheme, the amplitude and the position of Dirac at any corner point (e.g. at point A ) follows: ) ( det , , 1 2 V S D D a j k k j q p ) ( det , , , 1 1 2 V a S D D C x q p j k k j x j p ) ( det , , , 1 1 2 V a S D D C y q p j k k j y k q The directional kernel can reproduce polynomials of degrees 0 and 1 in both x and y directions. ) , ( 2 1 y x ) , ( y x xy ) , ( 0 2 1 y x support : y x L v v L v v 2 , 2 2 , 1 1 , 2 1 , 1 support : y x L L ) , ( 2 1 y x Samples of the polygon using Haar kernel A planar triangle needs three pairs of directional differences to get decomposed in three 2-D Diracs. Original polygon with three corner points After first pair of directional difference on samples After third pair of directional difference After second pair of directional difference Pair of directional differences Local reconstruc tion A 2 1 ) tan( 2 ) tan( 2 1 PR ? * Motivation Image resolution enhancement and super-resolution photogrammetry. Samplin g ) , ( y x xy ) , ( y x g j k y x xy kT y jT x ) , ( L L L L L L S S S S S S S S S , 1 , 0 , , 1 1 , 1 0 , 1 , 0 1 , 0 0 , 0 Set of samples in 2-D Acquisition device Sampling kernel Input signal ) / , / ( ), , ( , k T y j T x y x g S y x xy k j A generic sampling setup in 2-D Z k xy Z j x j x k y j x C ) , ( , j x C , Z k xy Z j y k y k y j x C ) , ( , Z j Z k xy k y j x 1 ) , ( 0 Partition of unity: Reproduction of polynomials along y- axis: Reproduction of polynomials along x- axis: 1 e.g., B-splines and Daubechies scaling functions are valid kernels. What signals? Non-bandlimited signals but with a finite number of degrees of freedom (rate of innovation), and thus known as signals with Finite Rate of Innovation (FRI) [Vetterli et al] [1,2]. E.g., Streams of Diracs, non-uniform splines, and piecewise polynomials. Present Focus: Sets of 2-D Diracs, bilevel and planar polygons. Reconstruction algorithms? Polynomial reproduction, Complex- moments, Annihilating filter, and Directional derivatives. Sampling kernel and properties? Any kernel (x,y) that is of compact support and can reproduce polynomials of degrees =0,1,2… -1 such that
1. INTRODUCTION AND MOTIVATION 2. SAMPLING OF SIGNALS WITH FINITE RATE OF INNOVATION (FRI) IN 2-D
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1. INTRODUCTION AND MOTIVATION
2. SAMPLING OF SIGNALS WITH FINITE RATE OF INNOVATION (FRI) IN 2-D
• SETS OF 2-D DIRACS (LOCAL RECONSTRUCTION)
• BILEVEL POLYGONS & DIRACS using COMPLEX-MOMENTS (GLOBAL APPROACH)
• PLANAR POLYGONS based on DIRECTIONAL DERIVATIVES (LOCAL APPROACH)
3. CONCLUSION AND ONGOING WORK 4. KEY REFERENCES
SAMPLING SCHEMES FOR 2-D SIGNALS WITH FINITE RATE OF INNOVATION
USING KERNELS THAT REPRODUCE POLYNOMIALS
IEEE International Conference on Image Processing (ICIP) 2005, Genova, Italy.
Pancham Shukla and Pier Luigi Dragotti
Communications and Signal Processing Group, Electrical and Electronic Engineering Department, Imperial College London, UK
Email: { P.Shukla, P.Dragotti } @ imperial.ac.uk
Consider and with support such that there is at most one Dirac in an area of size j k kjxykj yyxxayxg ),(),( , ),( yxxy yx LL ),(, qpxyqp yyxxa .yyxx TLTL
From the partition of unity the amplitude is determined as:
x yL
j
L
kkjqp Sa
1 1,,
And using polynomial reproduction properties along x-axis and y-axis, the coordinate position is determined as:
qp
L
j
L
kkj
xj
p a
SC
x
x y
,
1 1,,1
(simple complex-moment)
(weighted complex-moment)
where are complex weights and are corner point coordinates of the bilevel polygon
i iz
),,( yxg .12,...3,2 Nn
Moments are used to characterize unspecified objects. [Shohat et al 1943, Elad et al. 2004]. For a convex, bilevel polygon with N corner points, and an analytic function in closure , the complex-moments of the polygon follow [Milanfar et al.][3]:
),( yxg
The can be retrieved from using annihilating filter (Prony’s like method) such that
izwn
)(zA
.0][ wniA
Now for both bilevel polygon and set of Diracs, using complex-moments and annihilating filter method, it is straightforward to see that
izwn
)(zA )(zA iz
iasn
Theorem [Milanfar et al.] [3]: For a given non-degenerate, simply connected, and convex polygon in the complex Cartesian plane, all its N corner points are uniquely determined by its weighted complex-moments up to order 2N-1.w
n
A sampling perspective to above theorem follows…
As the kernel can reproduce polynomials up to degree 2N-1 ( = 0,1...2N-1), all 2N moments are determined from weighted sums of the samples For instance,
),( yxxywn
.,kjS
Complex-moments and annihilating filter
Original image with three bilevel polygons
Sampled version of the image
Samples around pentagonOriginal pentagon and reconstructed
corner points (marked with +)
Sampling kernel that can reproduce polynomials up to degree 2N-1=9
),(9 yxxy
Conclusion: Local and global sampling choices for the classes of 2-D FRI signals with varying degrees of complexity.
Current investigations: Sampling of more general shapes such as circles, ellipses, and polygons containing polygonal holes inside.
Future plans: Extension of our 2-D sampling results in higher dimensions and effect of noise. Integration of sampling results with wavelet footprints for image resolution enhancement and super-resolution photogrammetry.
1. M Vetterli, P Marziliano, and T Blu, “Sampling signals with finite rate of innovation,” IEEE Trans. on Signal Processing, 50(6): 1417-1428, June 2002.
2. P L Dragotti, M Vetterli, and T Blu, “Exact sampling results for signals with finite rate of innovation using Strang-Fix conditions and local kernels,” Proc. IEEE ICASSP, Philadelphia, USA, March 2005.
3. P Milanfar, G Verghese, W Karl, and A Willsky, “Reconstructing polygons from moments with connections to array processing,” IEEE Trans. on Signal Processing, 43(2): 432-443, February 1995.
4. V Velisavljevic, B Beferull-Lozano, M Vetterli, and P L Dragotti, “Discrete multi-directional wavelet bases,” Proc. IEEE ICIP, Barcelona, Spain, September 2003.
Intuitively, for a planar polygon, two successive directional derivatives along two adjacent sides of the polygon results into a 2-D Dirac at the corner point formed by the respective sides.
Continuous model Discrete challenge
Lattice theory [4] and [Convey and Sloan]
Link directional derivatives discrete differences.
Subsampling over integer lattices.
Local directional kernels in the framework of 2-D Dirac sampling (local reconstruction).
Sampling matrix
2,21,2
2,11,1
2
1
vv
vv
v
vV
1,1
2,111 tan
v
v
1,2
2,212 tan
v
v
where
),(,),()det( 21
21
12
,yxyxg
V
SDD kj
)sin(
),(*),(),(
12
021
21
yxyxyx
xy
Dirac at A Modified kernel
).,(21
yxAt each corner point Independent modified (directional) kernel
Assume only one corner point in support of directional kernel. Using local reconstruction scheme, the amplitude and the position of Dirac at any corner point (e.g. at point A ) follows:
)(det
,
,
12
V
SDD
aj k
kj
qp
)(det,
,,1 12
Va
SDDC
xqp
j kkj
xj
p
)(det,
,,1 12
Va
SDDC
yqp
j kkj
yk
q
The directional kernel can reproduce polynomials of degrees 0 and 1 in both x and y directions.
),(21
yx
),( yxxy ),(021
yx
support: yx LvvLvv 2,22,11,21,1support: yx LL
),(21
yx
Samples of the polygon using Haar kernel
A planar triangle needs three pairs of directional differences to get decomposed in three 2-D Diracs.
Original polygon with three corner points
After first pair of directional difference on samples
After third pair of directional difference
After second pair of directional difference
Pair of directional
differences
Local reconstruction
A2
1)tan(
2)tan(
2
1
PR ?
*
Motivation Image resolution enhancement and super-resolution photogrammetry.
Sampling
),( yxxy ),( yxg
j k
yxxy kTyjTx ),( LLLL
L
L
SSS
SSS
SSS
,1,0,
,11,10,1
,01,00,0
Set of samples in 2-DAcquisition device
Sampling kernelInput signal )/,/(),,(, kTyjTxyxgS yxxykj
A generic sampling setup in 2-D
Zkxy
Zj
xj xkyjxC ),(,
jxC ,
Zkxy
Zj
yk ykyjxC ),(,
Zj Zk
xy kyjx 1),(0
Partition of unity:
Reproduction of polynomials along y-axis:
Reproduction of polynomials along x-axis:
1
e.g., B-splines and Daubechies scaling functions are valid kernels.
What signals? Non-bandlimited signals but with a finite number of degrees of freedom (rate of innovation), and thus known as signals with Finite Rate of Innovation (FRI) [Vetterli et al] [1,2]. E.g., Streams of Diracs, non-uniform splines, and piecewise polynomials. Present Focus: Sets of 2-D Diracs, bilevel and planar polygons.
Reconstruction algorithms? Polynomial reproduction, Complex-moments, Annihilating filter, and Directional derivatives.
Sampling kernel and properties? Any kernel (x,y) that is of compact support and can reproduce polynomials of degrees =0,1,2…-1 such that
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