IEEE Signal Processing - March 2018 - 53
computationally intensive and are generally slow to converge.
Moreover, they would incur large latency in data gathering
since they require availability of the entire data prior to learning the dictionary. More importantly, learning one dictionary
for the entire data is suboptimum since seismic signals are
nonstationary and the characteristics of traces from different
shots may vary significantly. As an alternative approach, ODL
methods can process streaming data. When receiving a new
data batch, the algorithm updates the previous dictionary by
applying only a few iterations of a simple optimization algorithm. Although this reduces both the latency in data gathering and the required computational power at the sensor, it
suffers from two major drawbacks for seismic compression:
1) The dictionary update is performed by processing all the
data received thus far, which requires a considerable processing resource.
2) Since seismic traces are nonstationary, the dictionary
updates never converge to zero. Hence, the encoder needs
to send the dictionary updates of each batch to the receiver,
imposing a significant overhead to the overall compression.
We propose the SIODL to address both issues. Due to the
nonstationarity of seismic traces, SIODL applies a limited buffer and a sliding window mechanism to hold only a few past
traces that is used to update the dictionary. To mitigate the
overhead of sending the dictionary updates, SIODL enforces
the dictionary updates to be sparse in a fixed domain.
Let X (k) ! R M # I denote the kth batch of seismic data,
where I is the number of data points in each batch and D (k) is
the dictionary used for encoding X (k) . The dictionary update
for the kth batch would be DD (k) = D (k) - D (k - 1) . SIODL
relies on representing the dictionary update as DD (k) = UU (k),
where U ! R M # P is an appropriate off-the-shelf fixed dic(k) (k)
(k)
(k)
tionary and U (k) = 6u 1 , u 2 , f, u N @, u i ! R P is the sparse
coefficients (of the dictionary updates) to be transmitted to
the decoder. SIODL uses the data corresponding to the last
L batches to update the dictionary at the current batch. Let
X = 6X (k - L + 1), f, X (k)@ and W = 6W (k - L + 1), f, W (k)@ be the
data and coefficients of the last L batches, respectively. This
setup leads to the following nonconvex optimization problem
for each column of U (k) [21]:
" u (jk), W j, J , = argmin Q j - Uuv T
u, v
F
subject to u
0
# t D,
(2)
for j = 1, f, N,
where t D is the desired sparsity for the dictionary updates
and J is the index group of the signals whose representations used d j (the jth atom of the dictionary). J can be found
by locating nonzero elements in the jth row of W, and
Q j = X J - D (k - 1) W J - / i ! j Uu i W i, J .
We use an alternative optimization to solve the problem for
each j, i.e., we fix W Tj, J and solve for u j . According to [22,
Lemma 1], if we assume ` v = W Tj, J W j, J 2 j we have
u j = argmin Q j v - Uu
u
for j = 1, f, N.
2
subject to u
0
Using the updated value of u j, W Tj, J can be found via
W Tj, J = ` Q Tj Uu j Uu j 2 j .
Rate-optimized DL
The objective of sparse DL techniques is to represent the data
by few nonzero coefficients. However, the sparse representation
is not necessarily optimal for data compression whose objective
is to minimize the overall bit rate in encoding the coefficients.
Therefore, we consider the rate function rather than optimizing
for sparsity, i.e.,
t,W
t = argmin ` X - DW
D
D,W
2
F
+ mR (W) j,
(4)
where R (W) measures the bit rate in encoding and transmitting W.
Computing the rate requires quantizing and estimating the
probability distribution of the coefficients. Our simulations with
real seismic data suggest that optimizing for the quantizer yields
only a subtle compression gain despite its computational complexity. Thus, we use a scalar uniform quantizer that has asymptotic rate-distortion optimality [23]. Probability distributions can
be estimated using either nonparametric models such as kernel
methods or a family of parametric models. Due to the transmission overhead of the nonparametric approaches, a parametric
method is adopted here. In our proposed approach, we select the
Gaussian mixture model (GMM) as the family of distributions
for the coefficients, i.e., p (w) = / s r (s) p ^w s h, where r (s) is
the weight of the source, indexed by s, and p ^. sh ~N ^ n s, v 2s h
is normal distribution with mean n s and variance v 2s . Using
the GMM makes the derivations of the solution of the optimization problem more tractable. Moreover, by arbitrarily increasing
the number of components in the GMM, almost any continuous probability distribution function can be well-approximated.
Note that, by adding a source with n 0 = 0 and v 0 = 0, we can
further promote the sparsity of the coefficients, as each coefficient would be zero with probability r (0). Since each data
sample is encoded separately, p (w) = % i p (w i). Moreover,
R (w) . - / i log 2 p (w i), which can be upper bounded by
- log 2 p (w i) # - log 2 r (st i) - log 2 p ^w i st i h, wher e st i is t he
maximum a posteriori estimation of the source index s given
w i . The resulting iterative DL for compression can be summarized as [24].
1) Given the current coefficients, update the dictionary to mint = argmin D X - DW 2 . Usually a
imize the error, i.e., D
F
dictionary norm constraint (e.g., unit column norm) is
added to the optimization for stability. The resulting optimization can then be solved using techniques such as
MOD, K-SVD, or via closed-form solutions for orthonormal dictionaries.
2) Fixing the dictionary and the probability distributions of
coefficients for the data sample x, the best sources (indexed
by s ) ) that generated the coefficients and w ) (the optimum
value for the w) are found by minimizing (4). This results in
# t D,
min
x - Dw
w, s
(3)
IEEE Signal Processing Magazine
|
March 2018
|
2
2
- m / ^log 2 p ^w i s i h + log 2 r ^s i hh.
i
53
Table of Contents for the Digital Edition of IEEE Signal Processing - March 2018
Contents
IEEE Signal Processing - March 2018 - Cover1
IEEE Signal Processing - March 2018 - Cover2
IEEE Signal Processing - March 2018 - Contents
IEEE Signal Processing - March 2018 - 2
IEEE Signal Processing - March 2018 - 3
IEEE Signal Processing - March 2018 - 4
IEEE Signal Processing - March 2018 - 5
IEEE Signal Processing - March 2018 - 6
IEEE Signal Processing - March 2018 - 7
IEEE Signal Processing - March 2018 - 8
IEEE Signal Processing - March 2018 - 9
IEEE Signal Processing - March 2018 - 10
IEEE Signal Processing - March 2018 - 11
IEEE Signal Processing - March 2018 - 12
IEEE Signal Processing - March 2018 - 13
IEEE Signal Processing - March 2018 - 14
IEEE Signal Processing - March 2018 - 15
IEEE Signal Processing - March 2018 - 16
IEEE Signal Processing - March 2018 - 17
IEEE Signal Processing - March 2018 - 18
IEEE Signal Processing - March 2018 - 19
IEEE Signal Processing - March 2018 - 20
IEEE Signal Processing - March 2018 - 21
IEEE Signal Processing - March 2018 - 22
IEEE Signal Processing - March 2018 - 23
IEEE Signal Processing - March 2018 - 24
IEEE Signal Processing - March 2018 - 25
IEEE Signal Processing - March 2018 - 26
IEEE Signal Processing - March 2018 - 27
IEEE Signal Processing - March 2018 - 28
IEEE Signal Processing - March 2018 - 29
IEEE Signal Processing - March 2018 - 30
IEEE Signal Processing - March 2018 - 31
IEEE Signal Processing - March 2018 - 32
IEEE Signal Processing - March 2018 - 33
IEEE Signal Processing - March 2018 - 34
IEEE Signal Processing - March 2018 - 35
IEEE Signal Processing - March 2018 - 36
IEEE Signal Processing - March 2018 - 37
IEEE Signal Processing - March 2018 - 38
IEEE Signal Processing - March 2018 - 39
IEEE Signal Processing - March 2018 - 40
IEEE Signal Processing - March 2018 - 41
IEEE Signal Processing - March 2018 - 42
IEEE Signal Processing - March 2018 - 43
IEEE Signal Processing - March 2018 - 44
IEEE Signal Processing - March 2018 - 45
IEEE Signal Processing - March 2018 - 46
IEEE Signal Processing - March 2018 - 47
IEEE Signal Processing - March 2018 - 48
IEEE Signal Processing - March 2018 - 49
IEEE Signal Processing - March 2018 - 50
IEEE Signal Processing - March 2018 - 51
IEEE Signal Processing - March 2018 - 52
IEEE Signal Processing - March 2018 - 53
IEEE Signal Processing - March 2018 - 54
IEEE Signal Processing - March 2018 - 55
IEEE Signal Processing - March 2018 - 56
IEEE Signal Processing - March 2018 - 57
IEEE Signal Processing - March 2018 - 58
IEEE Signal Processing - March 2018 - 59
IEEE Signal Processing - March 2018 - 60
IEEE Signal Processing - March 2018 - 61
IEEE Signal Processing - March 2018 - 62
IEEE Signal Processing - March 2018 - 63
IEEE Signal Processing - March 2018 - 64
IEEE Signal Processing - March 2018 - 65
IEEE Signal Processing - March 2018 - 66
IEEE Signal Processing - March 2018 - 67
IEEE Signal Processing - March 2018 - 68
IEEE Signal Processing - March 2018 - 69
IEEE Signal Processing - March 2018 - 70
IEEE Signal Processing - March 2018 - 71
IEEE Signal Processing - March 2018 - 72
IEEE Signal Processing - March 2018 - 73
IEEE Signal Processing - March 2018 - 74
IEEE Signal Processing - March 2018 - 75
IEEE Signal Processing - March 2018 - 76
IEEE Signal Processing - March 2018 - 77
IEEE Signal Processing - March 2018 - 78
IEEE Signal Processing - March 2018 - 79
IEEE Signal Processing - March 2018 - 80
IEEE Signal Processing - March 2018 - 81
IEEE Signal Processing - March 2018 - 82
IEEE Signal Processing - March 2018 - 83
IEEE Signal Processing - March 2018 - 84
IEEE Signal Processing - March 2018 - 85
IEEE Signal Processing - March 2018 - 86
IEEE Signal Processing - March 2018 - 87
IEEE Signal Processing - March 2018 - 88
IEEE Signal Processing - March 2018 - 89
IEEE Signal Processing - March 2018 - 90
IEEE Signal Processing - March 2018 - 91
IEEE Signal Processing - March 2018 - 92
IEEE Signal Processing - March 2018 - 93
IEEE Signal Processing - March 2018 - 94
IEEE Signal Processing - March 2018 - 95
IEEE Signal Processing - March 2018 - 96
IEEE Signal Processing - March 2018 - 97
IEEE Signal Processing - March 2018 - 98
IEEE Signal Processing - March 2018 - 99
IEEE Signal Processing - March 2018 - 100
IEEE Signal Processing - March 2018 - 101
IEEE Signal Processing - March 2018 - 102
IEEE Signal Processing - March 2018 - 103
IEEE Signal Processing - March 2018 - 104
IEEE Signal Processing - March 2018 - 105
IEEE Signal Processing - March 2018 - 106
IEEE Signal Processing - March 2018 - 107
IEEE Signal Processing - March 2018 - 108
IEEE Signal Processing - March 2018 - 109
IEEE Signal Processing - March 2018 - 110
IEEE Signal Processing - March 2018 - 111
IEEE Signal Processing - March 2018 - 112
IEEE Signal Processing - March 2018 - 113
IEEE Signal Processing - March 2018 - 114
IEEE Signal Processing - March 2018 - 115
IEEE Signal Processing - March 2018 - 116
IEEE Signal Processing - March 2018 - 117
IEEE Signal Processing - March 2018 - 118
IEEE Signal Processing - March 2018 - 119
IEEE Signal Processing - March 2018 - 120
IEEE Signal Processing - March 2018 - 121
IEEE Signal Processing - March 2018 - 122
IEEE Signal Processing - March 2018 - 123
IEEE Signal Processing - March 2018 - 124
IEEE Signal Processing - March 2018 - 125
IEEE Signal Processing - March 2018 - 126
IEEE Signal Processing - March 2018 - 127
IEEE Signal Processing - March 2018 - 128
IEEE Signal Processing - March 2018 - 129
IEEE Signal Processing - March 2018 - 130
IEEE Signal Processing - March 2018 - 131
IEEE Signal Processing - March 2018 - 132
IEEE Signal Processing - March 2018 - 133
IEEE Signal Processing - March 2018 - 134
IEEE Signal Processing - March 2018 - 135
IEEE Signal Processing - March 2018 - 136
IEEE Signal Processing - March 2018 - 137
IEEE Signal Processing - March 2018 - 138
IEEE Signal Processing - March 2018 - 139
IEEE Signal Processing - March 2018 - 140
IEEE Signal Processing - March 2018 - 141
IEEE Signal Processing - March 2018 - 142
IEEE Signal Processing - March 2018 - 143
IEEE Signal Processing - March 2018 - 144
IEEE Signal Processing - March 2018 - 145
IEEE Signal Processing - March 2018 - 146
IEEE Signal Processing - March 2018 - 147
IEEE Signal Processing - March 2018 - 148
IEEE Signal Processing - March 2018 - 149
IEEE Signal Processing - March 2018 - 150
IEEE Signal Processing - March 2018 - 151
IEEE Signal Processing - March 2018 - 152
IEEE Signal Processing - March 2018 - 153
IEEE Signal Processing - March 2018 - 154
IEEE Signal Processing - March 2018 - 155
IEEE Signal Processing - March 2018 - 156
IEEE Signal Processing - March 2018 - 157
IEEE Signal Processing - March 2018 - 158
IEEE Signal Processing - March 2018 - 159
IEEE Signal Processing - March 2018 - 160
IEEE Signal Processing - March 2018 - 161
IEEE Signal Processing - March 2018 - 162
IEEE Signal Processing - March 2018 - 163
IEEE Signal Processing - March 2018 - 164
IEEE Signal Processing - March 2018 - 165
IEEE Signal Processing - March 2018 - 166
IEEE Signal Processing - March 2018 - 167
IEEE Signal Processing - March 2018 - 168
IEEE Signal Processing - March 2018 - 169
IEEE Signal Processing - March 2018 - 170
IEEE Signal Processing - March 2018 - 171
IEEE Signal Processing - March 2018 - 172
IEEE Signal Processing - March 2018 - 173
IEEE Signal Processing - March 2018 - 174
IEEE Signal Processing - March 2018 - 175
IEEE Signal Processing - March 2018 - 176
IEEE Signal Processing - March 2018 - Cover3
IEEE Signal Processing - March 2018 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201809
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201807
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201805
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201803
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_201801
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1117
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0917
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0717
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0517
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0317
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0117
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1116
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0916
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0716
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0516
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0316
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0116
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1115
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0915
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0715
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0515
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0315
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0115
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1114
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0914
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0714
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0514
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0314
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0114
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1113
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0913
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0713
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0513
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0313
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0113
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1112
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0912
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0712
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0512
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0312
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0112
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1111
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0911
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0711
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0511
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0311
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0111
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1110
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0910
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0710
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0510
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0310
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0110
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1109
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0909
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0709
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0509
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0309
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0109
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_1108
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0908
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0708
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0508
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0308
https://www.nxtbook.com/nxtbooks/ieee/signalprocessing_0108
https://www.nxtbookmedia.com