IEEE Signal Processing - March 2018 - 119

tool crosses a boundary, the same change is visible in the
borehole image. This intersection is mapped to a sinusoid,
also known as a dip, where the amplitude and phase shift are
related to the angle at which the tool intersects with the formation layer. In essence, the sinusoids visible in the image
corresponds to the intersection of the cylindrical borehole
with a locally plane interface, and the amplitude of the sinusoid can be used to infer the dip angle. In real time, the
parameters of the apparent dip angles are used to infer the
tool orientation relative to the formation by well placement
engineers. Although this synthetic example represents a large
interval of several kilometers, the same patterns are visible on
a much smaller scale in the range of several meters. A sparse
representation and compression of borehole images is discussed next.

Total Vertical Depth

2-D Well Cross-Section

Drilled Well

(a)
Azimuthal Bin

Corresponding Borehole Image

Measured Depth
(b)

Sparse representation of borehole image features
An important step to efficiently compress an image is to first
find a sparse representation. Define an image by I ^ x, yh and a
dictionary set " z 1 ^ x, yh, z 2 ^ x, yh, f, z N ^ x, yh,. We assume
that this dictionary set spans the space of the image meaning
that the original image can be described as
I ^ x, yh =

/ a i z i ^ x, yh,

Figure 8. A synthetic example of a layered formation. The green curve
in (a) corresponds to the drilled well. The resulting (simulated) borehole
image is shown in (b). Each time the tool crosses a formation boundary,
the resulting borehole image shows a sinusoid. The amplitude and phase
shift of the sinusoid is related to the angle at which the tool intersects with
a formation layer.

(9)

i=1

where a i are the weights associated with each dictionary
atom. Clearly, the dictionary set " z i ^ x, yh,iN= 1 plays a vital
role in the sparsity. In general, there are two approaches
to design a dictionary: transform based and data driven.
Transform dictionaries are popular in today's prevailing standardized compression algorithms [i.e., discrete wavelet transform (DWT) in JPEG2000]. They are separable along the x-y
dimensions and thus have a low encoding complexity. This
structure means that the dictionaries are only efficient at representing vertical or horizontal features. Diagonal features are
described as a linear combination of horizontal/vertical atoms
and are more challenging to compress. This has been partly
addressed by using nonseparable directional transforms [21].
However, these methods are still designed for natural images
and require several encoding iterations to optimize the encoding parameters.
On the other hand, data-driven dictionaries are learned from
training data. They inherently represent directional features;
however, they have a higher encoding complexity and do not
generalize well to images outside of the training set. Recent
innovation in this field can impose additional orthogonal constraints on the dictionary set [22], and they can be applied in a
separable manner to reduce the complexity.
In our work, we have focused on specializing both types
of dictionaries for borehole images. The dictionaries are
designed for compression of sinusoidal features shown in Figure 8 and for longitudinal fractures. For both transform-based
and data-driven dictionaries, we first run a detection algorithm that outputs a parametric representation of the borehole
image features. The features are then used to design a set of

(a)

(b)

Figure 9. An example of six dictionary atoms from (a) standard 9-7 DWT
and (b) proposed directional DWT.

dictionary atoms " z i ^ x, yh,iN= 1 that produce a sparse representation of the data. In the transform-based algorithm, a DWT
is modified by steering the wavelet atoms in the direction of
the features. We use the lifting implementation, which allows
the directional information to be easily added to the transform [23]. An example of the Daubechies 9-7 atoms and our
proposed directional atoms is shown in Figure 9. A complete
coding algorithm, including quantization, rate allocation,
and entropy coding has been implemented [24]. An example
of the results obtained from a synthetic borehole image at
0.1 bits/pixel is shown in Figure 10. In comparison, we also
show compression results using JPEG2000. Note that the dips
can be clearly identified in the proposed method, whereas in
JPEG2000, the DWT basis functions are visible as artifacts
around diagonal lines.

IEEE Signal Processing Magazine

March 2018

119

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