Discrete Universal Denoiser
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In information theory and signal processing, the Discrete Universal Denoiser (DUDE) is a denoising scheme for recovering sequences over a finite alphabet, which have been corrupted by a discrete memoryless channel. The DUDE was proposed in 2005 by Tsachy Weissman, Erik Ordentlich, Gadiel Seroussi, Sergio Verdú and Marcelo J. Weinberger.
- 1 Overview
- 2 The discrete denoising problem
- 3 The DUDE scheme
- 4 Asymptotic optimality properties
- 5 Background
- 6 Extensions
- 7 Applications
- 8 References
The Discrete Universal Denoiser (DUDE) is a denoising scheme that estimates an unknown signal over a finite alphabet from a noisy version . While most denoising schemes in the signal processing and statistics literature deal with signals over an infinite alphabet (notably, real-valued signals), the DUDE addresses the finite alphabet case. The noisy version is assumed to be generated by transmitting through a known discrete memoryless channel.
For a fixed context length parameter , the DUDE counts of the occurrences of all the strings of length appearing in . The estimated value is determined based the two-sided length- context of , taking into account all the other tokens in with the same context, as well as the known channel matrix and the loss function being used.
The idea underlying the DUDE is best illustrated when is a realization of a random vector . If the conditional distribution , namely the distribution of the noiseless symbol conditional on its noisy context was available, the optimal estimator would be the Bayes Response to . Fortunately, when the channel matrix is known and non-degenerate, this conditional distribution can be expressed in terms of the conditional distribution , namely the distribution of the noisy symbol conditional on its noisy context. This conditional distribution, in turn, can be estimated from an individual observed noisy signal by virtue of the Law of Large Numbers, provided is “large enough”.
Applying the DUDE scheme with a context length to a sequence of length over a finite alphabet requires operations and space .
Under certain assumptions, the DUDE is a universal scheme in the sense of asymptotically performing as well as an optimal denoiser, which has oracle access to the unknown sequence. More specifically, assume that the denoising performance is measured using a given single-character fidelity criterion, and consider the regime where the sequence length tends to infinity and the context length tends to infinity “not too fast”. In the stochastic setting, where a doubly infinite sequence noiseless sequence is a realization of a stationary process , the DUDE asymptotically performs, in expectation, as well as the best denoiser, which has oracle access to the source distribution . In the single-sequence, or “semi-stochastic” setting with a fixed doubly infinite sequence , the DUDE asymptotically performs as well as the best “sliding window” denoiser, namely any denoiser that determines from the window , which has oracle access to .
The discrete denoising problem
Let be the finite alphabet of a fixed but unknown original “noiseless” sequence . The sequence is fed into a discrete memoryless channel (DMC). The DMC operates on each symbol independently, producing a corresponding random symbol in a finite alphabet . The DMC is known and given as a -by- Markov matrix , whose entries are . It is convenient to write for the -column of . The DMC produces a random noisy sequence . A specific realization of this random vector will be denoted by . A denoiser is a function that attempts to recover the noiseless sequence from a distorted version . A specific denoised sequence is denoted by . The problem of choosing the denoiser is known as signal estimation, filtering or smoothing. To compare candidate denoisers, we choose a single-symbol fidelity criterion (for example, the Hamming loss) and define the per-symbol loss of the denoiser at by
Ordering the elements of the alphabet by , the fidelity criterion can be given by a -by- matrix, with columns of the form
The DUDE scheme
Step 1: Calculating the empirical distribution in each context
The DUDE corrects symbols according to their context. The context length used is a tuning parameter of the scheme. For , define the left context of the -th symbol in by and the corresponding right context as . A two-sided context is a combination of a left and a right context.
The first step of the DUDE scheme is to calculate the empirical distribution of symbols in each possible two-sided context along the noisy sequence . Formally, a given two-sided context that appears once or more along determines an empirical probability distribution over , whose value at the symbol is
Thus, the first step of the DUDE scheme with context length is to scan the input noisy sequence once, and store the length- empirical distribution vector (or its non-normalized version, the count vector) for each two-sided context found along . Since there are at most possible two-sided contexts along , this step requires operations and storage .
Step 2: Calculating the Bayes response to each context
Denote the column of single-symbol fidelity criterion , corresponding to the symbol , by . We define the Bayes Response to any vector of length with non-negative entries as
This definition is motivated in the background below.
The second step of the DUDE scheme is to calculate, for each two-sided context observed in the previous step along , and for each symbol observed in each context (namely, any such that is a substring of ) the Bayes response to the vector , namely
Note that the sequence and the context length are implicit. Here, is the -column of and for vectors and , denotes their Schur (entrywise) product, defined by . Matrix multiplication is evaluated before the Schur product, so that stands for .
This formula assumed that the channel matrix is square () and invertible. When and is not invertible, under the reasonable assumption that it has full row rank, we replace above with its Moore-Penrose pseudo-inverse and calculate instead
By caching the inverse or pseudo-inverse , and the values for the relevant pairs , this step requires operations and storage.
Step 3: Estimating each symbol by the Bayes response to its context
The third and final step of the DUDE scheme is to scan again and compute the actual denoised sequence . The denoised symbol chosen to replace is the Bayes response to the two-sided context of the symbol, namely
This step requires operations and used the data structure constructed in the previous step.
In summary, the entire DUDE requires operations and storage.
Asymptotic optimality properties
The DUDE is designed to be universally optimal, namely optimal (is some sense, under some assumptions) regardless of the original sequence .
Let denote a sequence of DUDE schemes, as described above, where uses a context length that is implicit in the notation. We only require that and that .
For a stationary source
Denote by the set of all -block denoisers, namely all maps .
Let be an unknown stationary source and be the distribution of the corresponding noisy sequence. Then
and both limits exist. If, in addition the source is ergodic, then
For an individual sequence
Denote by the set of all -block -th order sliding window denoisers, namely all maps of the form with arbitrary.
Let be an unknown noiseless sequence stationary source and be the distribution of the corresponding noisy sequence. Then
Let denote the DUDE on with context length defined on -blocks. Then there exist explicit constants and that depend on alone, such that for any and any we have
where is the noisy sequence corresponding to (whose randomness is due to the channel alone)  .
In fact holds with the same constants as above for any -block denoiser . The lower bound proof requires that the channel matrix be square and the pair satisfies a certain technical condition.
To motivate the particular definition of the DUDE using the Bayes response to a particular vector, we now find the optimal denoiser in the non-universal case, where the unknown sequence is a realization of a random vector , whose distribution is known.
Consider first the case . Since the joint distribution of is known, given the observed noisy symbol , the unknown symbol is distributed according to the known distribution . By ordering the elements of , we can describe this conditional distribution on using a probability vector , indexed by , whose -entry is . Clearly the expected loss for the choice of estimated symbol is .
Define the Bayes Envelope of a probability vector , describing a probability distribution on , as the minimal expected loss , and the Bayes Response to as the prediction that achieves this minimum, . Observe that the Bayes response is scale invariant in the sense that for .
For the case , then, the optimal denoiser is . This optimal denoiser can be expressed using the marginal distribution of alone, as follows. When the channel matrix is invertible, we have where is the -th column of . This implies that the optimal denoiser is given equivalently by . When and is not invertible, under the reasonable assumption that it has full row rank, we can replace with its Moore-Penrose pseudo-inverse and obtain
Turning now to arbitrary , the optimal denoiser (with minimal expected loss) is therefore given by the Bayes response to
where is a vector indexed by , whose -entry is . The conditional probability vector is hard to compute. A derivation analogous to the case above shows that the optimal denoiser admits an alternative representation, namely , where is a given vector and is the probability vector indexed by whose -entry is Again, is replaced by a pseudo-inverse if is not square or not invertible.
When the distribution of (and therefore, of ) is not available, the DUDE replaces the unknown vector with an empirical estimate obtained along the noisy sequence itself, namely with . This leads to the above definition of the DUDE.
While the convergence arguments behind the optimality properties above are more subtle, we note that the above, combined with the Birkhoff Ergodic Theorem, is enough to prove that for a stationary ergodic source, the DUDE with context-length is asymptotically optimal all -th order sliding window denoisers.
The basic DUDE as described here assumes a signal with a one-dimensional index set over a finite alphabet, a known memoryless channel and a context length that is fixed in advance. Relaxations of each of these assumptions have been considered in turn. Specifically:
- Infinite alphabets
- Channels with memory
- Unknown channel matrix
- Variable context and adaptive choice of context length
- Two-dimensional signals
Application to image denoising
A DUDE-based framework for grayscale image denoising achieves state-of-the-art denoising for impulse-type noise channels (e.g., "salt and pepper" or "M-ary symmetric" noise), and good performance on the Gaussian channel (comparable to the Non-local means image denoising scheme on this channel). A different DUDE variant applicable to grayscale images is presented in.
Application to channel decoding of uncompressed sources
The DUDE has led to universal algorithms for channel decoding of uncompressed sources.
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