Weighted overcomplete de-noising

A method for obtaining an estimate {circumflex over (x)} of a noise-free portion x of a noise-containing signal y, comprising the steps of:

• obtaining an estimate {circumflex over (x)}(n) for each element n of {circumflex over (x)} according to the following equation:
• x
• ^
• ⁡
• (
• n
• )
• =
• ∑
• i
• =
• 1
• M
• ⁢
• ⁢
• α
• i
• ⁡
• (
• n
• )
• ⁢
• x
• ^
• i
• ⁡
• (
• n
• )
• ,
• n
• =
• 1
• ,
• …
• ⁢
• ,
• N
• ,
• wherein weight factors αi(n), i=1, . . . , M, n=1, . . . , N are optimally determined by formulating a combination thereof as a linear estimation problem;
• and
• wherein M is the number weight factors αi(n) in the summation used to obtain the estimate {circumflex over (x)}(n);
• and
• combining the N {circumflex over (x)}(n) estimates to obtain {circumflex over (x)}.

An apparatus for obtaining an estimate of a noise-free portion of a noise-containing digital signal, the apparatus comprising:

• one or more components configured to:
• (a) apply a set of M linear transforms to the noise-containing digital signal;
• (b) determine M initial de-noised estimates of each digital element of the digital signal;
• (c) derive a combination of weight factors for the M initial de-noised estimates of each digital element by formulating the combination as a linear estimation problem and solving it for the individual weight factors;
• and
• (d) formulate a final de-noised estimate of each digital element based on the corresponding M initial de-noised estimates and the combination of weight factors determined in steps (b) and (c) respectively.

A computer-readable medium embodying a program of instructions for directing a machine to perform a process of obtaining an estimate of a noise-free portion of a noise-containing digital signal, the program comprising:

• (a) instructions for applying a set of M linear transforms to the noise-containing digital signal;
• (b) instructions for determining M initial de-noised estimates of each digital element of the digital signal;
• (c) instructions for deriving a combination of weight factors for the M initial de-noised estimates of each digital element by formulating the combination as a linear estimation problem and solving it for the individual weight factors;
• and
• (d) instructions for formulating a final de-noised estimate of each digital element based on the corresponding M initial de-noised estimates and the combination of weight factors determined in steps (b) and (c) respectively.

A computer-readable medium embodying a program of instructions for directing a machine to perform a process of obtaining an estimate {circumflex over (x)} of a noise-free portion x of a noise-containing signal y, the program comprising:

• instructions for obtaining an estimate {circumflex over (x)}(n) for each element n of {circumflex over (x)} according to the following equation:
• x
• ^
• ⁡
• (
• n
• )
• =
• ∑
• i
• =
• 1
• M
• ⁢
• ⁢
• α
• i
• ⁡
• (
• n
• )
• ⁢
• x
• ^
• i
• ⁡
• (
• n
• )
• ,
• n
• =
• 1
• ,
• …
• ⁢
• ,
• N
• ,
• wherein weight factors αi(n), i=1, . . . , M, n=1, . . . , N are optimally determined by formulating a combination thereof as a linear estimation problem;
• and
• wherein M is the number weight factors αi(n) in the summation used to obtain the estimate {circumflex over (x)}(n);
• and
• instructions for combining the N {circumflex over (x)}(n) estimates to obtain {circumflex over (x)}.