Apparatus and method for unequal error protection in multiple-description coding using overcomplete expansions
First Claim
1. A computerized method of enhancing the recoverability of a set of data x having K dimensions for transmission on an erasure channel, the method comprising:
- generating an overcomplete expansion y of the set cf data x using an N×
K multiple description overcomplete expansion y=Fx, wherein N>
K and wherein at least some of the last N−
K rows of F are not duplicative of any of the first K rows; and
quantizing the coefficients of y to unequal resolution.
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Abstract
A projection onto convex sets (POCS)-based method for consistent reconstruction of a signal from a subset of quantized coefficients received from an N×K overcomplete transform. By choosing a frame operator F to be the concatenization of two or more K×K invertible transforms, the POCS projections are calculated in RK space using only the K×K transforms and their inverses, rather than the larger RN space using pseudo inverse transforms. Practical reconstructions are enabled based on, for example, wavelet, subband, or lapped transforms of an entire image. In one embodiment, unequal error protection for multiple description source coding is provided. In particular, given a bit-plane representation of the coefficients in an overcomplete representation of the source, one embodiment of the present invention provides coding the most significant bits with the highest redundancy and the least significant bits with the lowest redundancy. In one embodiment, this is accomplished by varying the quantization stepsize for the different coefficients. Then, the available received quantized coefficients are decoded using a method based on alternating projections onto convex sets.
67 Citations
23 Claims
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1. A computerized method of enhancing the recoverability of a set of data x having K dimensions for transmission on an erasure channel, the method comprising:
-
generating an overcomplete expansion y of the set cf data x using an N×
K multiple description overcomplete expansion y=Fx, wherein N>
K and wherein at least some of the last N−
K rows of F are not duplicative of any of the first K rows; and
quantizing the coefficients of y to unequal resolution. - View Dependent Claims (2, 3, 4, 5, 6, 7)
varying the quantization stepsize for the different coefficients.
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6. The method of claim 5, further comprising:
-
receiving at least some of the quantized coefficients; and
decoding the received quantized coefficients using a method based on alternating projections onto convex sets in order to generate a recovered set of data.
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7. A computer-readable medium having instructions stored thereon for causing a computer to perform the method of claim 1.
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8. A computerized system for enhancing the recoverability of a set of data for transmission on an erasure channel, the system comprising:
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a component for generating a multiple-description overcomplete expansion of the set of data wherein unequal encoding is employed. - View Dependent Claims (9, 10, 11, 12, 13, 14)
means for varying the quantization stepsize for the different coefficients. -
13. The system of claim 12, further comprising
means for receiving at least some of the quantized coefficients; - and
means for decoding the received quantized coefficients using a method based on alternating projections onto convex sets.
- and
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14. The system of claim 8, further comprising a computer-readable medium having instructions stored thereon for causing a computer to operatively control the means for generating the overcomplete expansion.
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15. A computerized method of reconstructing data after a transmission on an erasure channel, the method comprising:
-
providing an initial set of data x having K dimensions;
overcompletely transforming the set of data x to a multiple-description overcomplete expansion set of data y having N dimensions, where N>
K; and
quantizing the coefficients of y to unequal resolution. - View Dependent Claims (16, 17, 18, 19, 20, 21, 22, 23)
varying the quantization stepsize for the different coefficients.
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18. The method of claim 17, further comprising:
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transmitting the set of data y on the channel;
receiving a set of data y′
from the channel; and
iteratively transforming the received set of data y′
using projections onto convex sets that use at least two different K×
K transforms including;
(a) initializing t;
(b) setting an initial point in F1P;
(c) transforming p1(t) into the coordinate system of F2;
(d) projecting p2(t) onto F2Q;
(e) transforming q2(t+1) into the coordinate system of F1;
(f) projecting q1(t+1) onto F1P;
(g) checking for convergence and if not sufficiently converged then iterating (c) through (g); and
(h) reconstructing {circumflex over (x)}=Fn−
1pn(t+1) in order to generate a recovered set of data;
wherein;
F1 and F2 are each K×
K transforms, and n=1 or 2.
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19. The method of claim 15, further comprising:
-
transmitting the set of data y on the channel;
receiving a set of data y′
from the channel; and
iteratively transforming the received set of data y′
using projections onto convex sets that use at least two different K×
K transforms including;
(a) initializing t;
(b) setting an initial point in F1P;
(c) transforming p1(t) into the coordinate system of F2;
(d) projecting p2(t) onto F2Q;
(e) transforming q2(t+1) into the coordinate system of F1;
(f) projecting q1(t+1) onto F1P;
(g) checking for convergence and if not sufficiently converged then iterating (c) through (g); and
(h) reconstructing {circumflex over (x)}=Fn−
1pn(t+1) in order to generate a recovered set of data;
wherein;
F1 and F2 are each K×
K transforms, and n=1 or 2.
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20. The method of claim 19, wherein:
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(a) initializing further includes initializing t=0;
(b) starting further includes setting an initial point p1(0) such that for each index i, (c) transforming p1(t) further includes transforming p1(t) into the coordinate system of F2 using p2(t)=F2(F1−
1p1(t))=F12p1(t);
(d) projecting p2(t) further includes projecting p2(t) onto F2Q such that for each index i;
(e) transforming q2(t+1) further includes transforming q2(t+1) into the coordinate system of F1 using q1(t+1)=F1(F2−
1q2(t+1))=F21q2(t+1);
(f) projecting q1(t+1) further includes projecting q1(t+1) onto F1P such that for each index i;
(g) checking further includes checking for convergence by testing ∥
pn(t+1)−
pn(t)∥
2>
ε
, and if so then setting t←
t+1 and iterating (c) through (g); and
(h) reconstructing {circumflex over (x)}=Fn−
1pn(t+1), wherein n=1 or 2,and wherein; F1 and F2 are each K×
K transforms,F1−
1 is the inverse transform of F1, F2−
1 is the inverse transform of F2,ŷ
1 is a quantized version of y1, ŷ
2 is a quantized version of y2, such that ŷ
1 componentwise lies between respective lower and upper quantization cell boundary vectors l1≦
ŷ
1≦
u1 and ŷ
2 componentwise lies between respective lower and upper quantization cell boundary vectors l2≦
ŷ
2≦
u2,R1⊂
{1, . . . ,K} is the set of indices of the descriptions received of ŷ
1, andR2⊂
{1, . . . ,K} is the set of indices of the descriptions received of ŷ
2.
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21. The method of claim 20, wherein the (a) initializing t, (b) setting an initial point in F1P, (c) transforming p1(t) into the coordinate system of F2, (d) projecting p2(t) onto F2Q, (e) transforming q2(t+1) into the coordinate system of F1, (f) projecting q1(t+1) onto F1P, (g) checking for convergence and if not sufficiently converged then iterating (c) through (g);
- and (h) reconstructing {circumflex over (x)}=Fn−
1pn(t+1) are performed in the order listed.
- and (h) reconstructing {circumflex over (x)}=Fn−
-
22. The method of claim 19, wherein:
-
(a) initializing further includes initializing t=0;
(b) starting further includes setting an initial point p1(0) such that for each index i, (c) transforming p1(t) further includes transforming p1(t) into the coordinate system of F2 using p2(t)=F2(F1−
1p1(t))=F12p1(t);
(d) projecting p2(t) further includes projecting p2(t) onto F2Q such that for each index i;
(e) transforming q2(t+1) further includes transforming q2(t+1) into the coordinate system of F1 using q1(t+1)=F1(F2−
1q2(t+1))=F21q2(t+1);
(f) projecting q1(t+1) further includes projecting q1(t+1) onto F1P such that for each index i;
(g) checking further includes checking for convergence by testing ∥
p1(t+1)−
p1(t)∥
2>
ε
, and if so then setting t←
t+1 and iterating (c) through (g)and wherein; F1 and F2 are each K×
K transforms,F1−
1 is the inverse transform of F1, F2−
1 is the inverse transform of F2,ŷ
1 is a quantized version of y1, ŷ
2 is a quantized version of y2, such that ŷ
1 componentwise lies between respective lower and upper quantization cell boundary vectors l1≦
ŷ
1≦
u1 and ŷ
2 componentwise lies between respective lower and upper quantization cell boundary vectors l2≦
ŷ
2≦
u2,R1⊂
{1, . . . ,K} is the set of indices of the descriptions received of ŷ
1, andR2⊂
{1, . . . ,K} is the set of indices of the descriptions received of ŷ
2.
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23. The method of claim 22, wherein operations are performed in the order listed.
Specification