IEEE Signal Processing - March 2018 - 93

of images that have the same subsurface structure to generate
u . The retrieved images are then
our augmented training data X
assigned the same image-level label as the query image. While
these image-level annotations can be used to train machine-learning models, recent work on weakly supervised labeling such as
that proposed in [64] show that these image-level labels can be
mapped into pixel-level labels that can be much more useful in
training powerful fully supervised deep-learning models such as
fully convolutional networks [44]. Figure 9 shows several examples of this mapping for various subsurface seismic structures.
The approach in [64] is based on nonnegative matrix factorization (NMF) [65]. NMF is a commonly used matrix factorization technique that is closely related to many unsupervised
machine-learning techniques such as k-means and spectral clusu ! R +N p # N s
tering [66]. NMF decomposes a nonnegative matrix X
into the product of two lower-rank matrices W ! R +N p # N f , and
H ! R +N f # N s such that both W and H are nonnegative. In
other words,
u . WH
X

s.t. W, H $ 0.

The regular NMF problem does not have an analytical solution, and is typically formulated as the following nonconvex optimization problem:
arg min X - WH
W, H

s.t. WH $ 0,

(7)

where, · F is the Frobenius norm. Lee et al. [67] showed
that NMF can be used to learn a "parts-based" representation, where each feature would represent a localized "part" of
the data. In practice, this is rarely achieved using the formulation in (7). To remedy this, the feature matrix W is initialized
u sepusing k-means applied on each class in the data matrix X
arately. This initialization simplifies the feature learning and
u . Then, a sparsity
makes W robust to mislabeled images in X
constraint is imposed on these initial features using the following sparsity measure:
t (w) =

(6)

u . In
Here, N f is the number of components (or the rank) of X
u represents the augmented data matrix from the secour work, X
tion "Similarity-Based Retrieval," where each column is a single
u has N s such
seismic image in vector form. The data matrix X
images, each of which is a vector of length N p . NMF factorizes
this data matrix into two nonnegative matrices: a basis matrix W
and a coefficient matrix H.

2
F

Np - w

1

Np - 1

w

2

,

(8)

where · 1 and · 2 are the l 1 and l 2 norms, respectively, and
t (·) indicates the sparsity of a vector. To enforce this constraint,
we follow the algorithm proposed by [68]. Additionally, to make
sure that each feature w i in the matrix W represents a single class
only, we impose an orthogonality constraint on the coefficients
matrix H. We also add two regularization terms on W and H to
avoid overfitting. The problem then becomes

(a) Original

(b) Labeled

(c) Original

(d) Labeled

Figure 9. Results of the weakly supervised pixel-level annotation approach in [64] for various subsurface structures. (a) shows images containing
chaotic and fault structures and (b) shows their corresponding pixel-level labels in blue and green respectively. (c) and (d) show various images
that contain salt-dome bodies or boundaries, and their corresponding pixel-level labels in red, respectively.
IEEE Signal Processing Magazine

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March 2018

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93
Table of Contents for the Digital Edition of IEEE Signal Processing - March 2018
