CWP-591

Detection of channels in seismic images using thesteerable pyramid

John Mathewson and Dave Hale

Center for Wave Phenomena, Colorado School of Mines, Golden CO 80401, USA

ABSTRACTChannels have always been important geologic features in the exploration foroil and gas. With 3-D seismic data they can often be mapped easily on time ordepth slices. In other situations they can be difficult to detect, due to structuralcomplexity or other factors. There are a number of image-processing algorithmsthat can be used to enhance linear features such as channels in 3-D seismicvolumes.One way involves the use of steerable pyramid filters to partition a seismic imagein terms of scale and orientation. Features can then be characterized accordingto dimensionality and direction using the partitioned image. Here, we explainour implementation of the steerable pyramid in 2-D and 3-D, and show how itcan be used to enhance image features. Examples of channel enhancement onsynthetic seismic images demonstrate the efficacy of this processing.

Key words: seismic image processing, interpretation

1 INTRODUCTION

Sands associated with channels often make good reser-voirs for hydrocarbons. For this reason, the detection ofchannels is an important part of seismic interpretationfor oil and gas exploration. ¿From 3-D seismic data itis possible, for horizontal layering, to perform detailedmapping of channel systems directly on time slices. Forexample, Figure 1 is a view of a synthetic 3-D seismicdataset with two channel systems, which are clearly visi-ble on the displayed time slice. However, in the majorityof cases layers are not horizontal, and horizon flatteningneeds to be done first.

In theory, it should be possible to automatically de-tect channels whether or not they are horizontal. Theyhave unique characteristics that differentiate them fromother types of features that we encounter in 3-D seismicdata. Channels are long, sinuous objects with arbitraryorientation. They are locally linear in that their extentis much greater in one direction than in any other direc-tion, at a given point. Other features in a 3-D seismicvolume are either locally planar (e.g. bedding planes,faults) or locally isotropic (e.g. pinnacle reefs, salt bod-ies). This difference in dimensionality between different

Figure 1. Synthetic 3-D seismic volume with channels

features in the input image can be used to detect andenhance (or attenuate) particular types of features.

Previous study of the dimensionality and orienta-tion of image features has mostly involved eigenvalue de-composition of the structure tensor. This analysis can

56 J. Mathewson & D. Haleky

kx

B00

B01

B02

B00

B02

B01

B10

B11

B12

B10

B12

B11

L

Figure 2. Illustration of decomposition into scale and

orientation subbands for a 3-level, 3-direction 2-D steer-

able pyramid, shown in frequency space. Spatial frequencies

(wavenumbers) kx and ky correspond to x- and y-directions.

also be made using the steerable pyramid, which is amethod of image decomposition into scale and orienta-tion subbands first introduced in the early 1990’s (Free-man and Adelson 1991, Simoncelli et al 1992).

The steerable pyramid has some nice features thatmay be important in the detection and enhancement ofchannels. For example, it is inherently a multi-scale pro-cess. Given the diversity of shapes and sizes of channelsthis is potentially an advantage over other approaches.Also, the steerable pyramid may have an advantage ininteractive work, as the precomputed filters can be usedfor rapid analysis and enhancement of image features.Finally, little work has been done on dimensionality es-timation using the steerable pyramid, and we believethat it has potential for this type of analysis.

2 THE 2-D STEERABLE PYRAMID

The steerable pyramid is a transform in which steerablefilters are used in a multi-scale recursive scheme, result-ing in a decomposition in terms of scale and orientation.Figure 2 shows an illustration of this decomposition inthe wavenumber domain, for a 2-D steerable pyramidwith three scale levels and three basis orientations.

2.1 Review of 2-D steerable filters

To illustrate the concept of a steerable filter we canconsider the directional derivative of a Gaussian. A two-dimensional symmetric filter g with Gaussian impulseresponse is

g(x, y) = e−(x2+y2)/2. (1)

Figure 3. Example of steering for a first derivative. Di-

rectional 1st-derivative-of-Gaussian filters with orientations

θ = 0 and θ = π/2 are applied to a test image. A weighted

sum of these filtered images yields a directionally filtered im-

age with an orientation θ = π/6. (modified from Freeman

and Adelson 1991)

Differentiation of this function with respect to x and ygives us 1st-derivative-of-Gaussian filters with impulseresponses

h(x, y; θ = 0) =∂

∂xg(x, y) = −xg(x, y) (2)

and

h(x, y; θ = π/2) =∂

∂yg(x, y) = −yg(x, y) (3)

A directional 1st-derivative-of-Gaussian filter witharbitrary orientation θ can be made by performing aweighted sum of these filters,

h(x, y; θ) = cos(θ)h(x, y; θ = 0)+sin(θ)h(x, y; θ = π/2),(4)

which is the concept of steering in its simplest form. Ifwe then convolve the filter h(x, y; θ) with an input im-age p(x, y), we get a directionally filtered output imageq(x, y; θ):

q(x, y; θ) =

∫dr

∫ds h(r, s; θ)p(x− r, y − s), (5)

where r and s are dummy integration variables.The problem is that steering of the filters is not

useful when θ varies spatially. For this case we muststeer the filtered images. Let us define two basis filtersbj(x, y) and their Fourier transforms Bj(kx, ky):

b0(x, y) ≡ h(x, y; θ = 0) ⇐⇒ B0(kx, ky) (6)

b1(x, y) ≡ h(x, y; θ = π/2) ⇐⇒ B1(kx, ky), (7)

When convolved with input image p(x, y), these produceoutput images

qj(x, y) =

∫dr

∫ds bj(r, s; θ)p(x− r, y − s), (8)

such that

q(x, y; θ) = cos(θ)q0(x, y) + sin(θ)q1(x, y). (9)

For the case where θ varies spatially, i.e. θ = θ(x, y),

Steerable pyramid detection of channels 57

q(x, y) = cos θ(x, y)q0(x, y) + sin θ(x, y)q1(x, y). (10)

Using this result, the output q(x, y) can be steered forany θ(x, y) from the precomputed qj(x, y).

The concept of steering extends to any order of di-rectional derivative and to other types of functions aswell. For the 1st-derivative-of-Gaussian case there wereonly two basis filters b0 and b1, scaled by weighting func-tions cos θ and sin θ. A general expression for a steerablefilter in terms of basis filters bj and weighting functionswj is

h(x, y; θ) =

M−1∑j=0

wj(θ)bj(x, y), (11)

where M is the number of basis filters. We can also writethe general form of equation 10:

q(x, y) =

M−1∑j=0

wj(x, y)qj(x, y). (12)

In order to extend steering to functions otherthan the directional first-derivative-of-Gaussian filter,we need answers to the following questions:

• For what sorts of basis filters bj does steering work?• How many basis filters do we need? (What is M?)• What directions θj should we use?• What are the weights wj(θ)?

These questions are answered in steering theorems de-veloped by Freeman and Adelson (1991), which can befound in Appendix A of this paper.

Let’s consider the extension of steerable filters tothe directional 2nd-derivative-of-Gaussian filter. First,we wish to determine the required number of basis di-rections M . For θ = 0 this filter is

b0(x, y) = h(x, y; θ = 0) = (x2 − 1)e−(x2+y2). (13)

This can be written in polar coordinates as

b0(r, φ) = (r2cos2φ− 1)e−r2, (14)

where r =√

x2 + y2 and φ = arg(x, y), or

b0(r, φ) = (r2e−i2φ + 2(r2 − 1)e0 + r2ei2φ)e−r2. (15)

Therefore, the number of non-zero Fourier coefficientsis three. By Theorem 1 at least three basis filters are re-quired, so M = 3. In fact it turns out that the numberof basis directions needed to steer a directional deriva-tive in 2-D is in general one greater than the order ofthe derivative. In this case, equation 11 then becomes

h(x, y; θ) =

2∑j=0

wj(θ)bj(x, y). (16)

The directions of the M basis filters should be cho-sen such that

θj = jπ/M, j = 0, 1, · · · , M − 1, (17)

Figure 4. Example of steering for the 2nd-derivative-of-

Gaussian filter. Directional filters with orientations θ =

0, pi/3, and 2π/3 are applied to the same test image that

was used in Figure 3. A weighted sum of these filtered im-

ages yields a directionally filtered image with an orientation

of θ = π/6.

which gives uniformly spaced basis directions. This isnot the only set of θj that can be used, but the steering ismore accurate than for other choices, and works for anyangle of rotation. In this case, we choose θj = jπ/3, j =0, 1, 2, consistent with equation (17), and the basis filtersare

bj(x, y) = h(x, y; θ = jπ/3). (18)

Using Theorem 1, we can now solve for the weight-ing functions wj . After removing all but the non-zeroterms, equation A3 becomes

[1

ei2θ

]=

[1 1 1

ei2θ1 ei2θ2 ei2θ3

]w1(θ)

w2(θ)

w3(θ)

. (19)

When we solve this system of equations, with θ0 = 0,θ1 = π/3, and θ2 = 2π/3, we find that

wj(θ) =1

3[1 + 2 cos(2(θ − jπ/3))]. (20)

The basis filters bj need not be directional deriva-tives. Other filters can be steered (Freeman and Adel-son 1991). In particular, we may choose bj(x, y), withFourier transforms Bj(kx, ky), such that∣∣∣∣∣M−1∑j=0

Bj(kx, ky)

∣∣∣∣∣ = C (21)

for all wavenumbers (kx, ky), where C is a constant in-dependent of (kx, ky). For example, for M = 3 we mightchoose (Castleman, Schulze, and Wu, 1998)

Bj(kx, ky) = cos2(θ − θj), (22)

where θj = jπ/3, j = 0, 1, 2. For this filter, the constantC = 3/2. The equation for the filter may also be writtenusing direction cosines αj and βj :

Bj(kx, ky) =(αjkx + βjky)2

k2x + k2

y, (23)

58 J. Mathewson & D. Hale

Figure 5. Schematic showing the process flow for creation of

a 2-D steerable pyramid with M = 3 with three levels. Steps

that are recursed to produce each pyramid level are shown

with dotted outline.

where αj = cos θj , βj = sin θj , and θj = jπ/3.

2.2 2-D steerable pyramid

To create a steerable pyramid, we combine steerablefilters described in the previous section with multi-scale decomposition by recursively high-cut filtering anddownsampling the image. Figure 5 shows the steps in-volved in creation of a 2-D steerable pyramid with threedirections and three levels, like the one depicted in Fig-ure 2.

The input image enters on the left and low-passfilters L0 and L1 are applied. These symmetric filters areapplied by multiplication in the wavenumber domain.They are both of the form (from Castleman, Schulze,and Wu, 1998)

L(ka, kb,k) =

1 ; |k| ≤ ka

12[1 + cos[π( |k|−ka

kb−ka)]] ; ka < |k| < kb

0 ; |k| ≥ kb

(24)

where ka and kb are the beginning and end of the taperzone of the filter, and |k| =

√k2

x + k2y.

The purpose of filter L0 is to attenuate high-wavenumber energy in the corners of the wavenumberdomain. The part of the energy attenuated by L0 can bepassed to the output, but it doesn’t contribute anythinguseful to the output image and we generally discard it.Low-pass filter L1 removes any data that will be aliasedin downsampling. For L0 we have been using ka = 0.75π

Figure 6. Fourier domain impulse response displays of 2-

D steerable pyramid filters for a single pyramid level. The

amplitude ranges from zero (dark blue) to one (dark red).

and kb = π. Filter cutoff parameters for L1 are halfthose used for L0, so ka = 0.375π and kb = 0.5π.

The results of filtering with L0 and L1 are sub-tracted to produce a bandpass-filtered image to whichthe basis directional filters Bj are applied, also bymultiplication in the wavenumber domain, producingbandpass-filtered images q00, q01 and q02. This makesup one level of the pyramid.

The output of filter L1 is then downsampled byselecting every other sample, giving us an image onequarter the size of the input image. The L1 filter is thenapplied, the output subtracted from the downsampledinput, and the basis filters Bj are applied exactly as de-scribed above, producing bandpass-filtered images q10,q11 and q12. This makes up the next level of the pyramid.

The process can be applied recursively as manytimes as we like, until we run out of samples. At eachlevel the size of the image is one quarter that of the pre-vious one. The final pyramid level, l contains only verylow wavenumbers. The basis filters Bj are not applied tothis final level. Figure 7 is a display of a time slice fromthe 3-D synthetic dataset from Figure 1 which has beenprocessed with a three level, three directions steerablepyramid transform.

Reconstruction of the image is straightforward.First, the weighting functions w10, w11 and w12 are ap-plied to the directionally filtered images q10, q11 andq12, and the low-wavenumber image, l, is upsampled. Inthe next section we will describe how we determine wj .For the moment let’s just say it is a spatially-variantscaling function designed to optimally steer the direc-tionally filtered images.

The four resulting intermediate images aresummed. Filtered images q00, q01 and q02 are multi-plied by their respective weighting functions w00, w01

and w02. Note that different levels of the pyramid have

Steerable pyramid detection of channels 59

Figure 7. The three-level, three-direction 2-D steerable

pyramid applied to a time slice from the synthetic 3-D vol-

ume.

Figure 8. Schematic showing the process flow for recon-

struction of an image from a 2-D steerable pyramid withthree directions and three levels. Steps that are recursed for

each pyramid level are shown with dotted outline.

different weighting functions, so they can be steeredin different directions. These outputs are added to thesummed result from the previous level of the pyramid.The output is the reconstructed image.

If the steering weights wj are all equal to 1, theoutput image is identical to the input, less the high-wavenumber energy that is removed by filter L0. How-ever, if weighting functions have been applied to en-hance features in the image, the output may look quitedifferent than the input. For example, figure 9 showsthe time slice seen in figure 7 before and after steerablepyramid processing with steering weights designed to

(a)

(b)

Figure 9. Time slice from a 3-D synthetic dataset before (a)

and after (b) processing with a 5-level, 3-directions steerable

pyramid designed to enhance coherent features in the input

image.

enhance coherent features in the image. The two chan-nel systems are enhanced while noise is generally re-duced. The question is, how do we determine the opti-mum weights?

2.3 2-D Estimation of feature orientation

There are a number of effective ways to determine thedirectionality of features in an an image. Previous workin this area has been done with the structure tensor (van

60 J. Mathewson & D. Hale

(a) (b) (c)

Figure 10. Steerable filter result for a simple image of tilted

lines. Input (a) and steered versions with weights calculated

for the correct orientation(b) and perpendicular to the cor-

rect orientation(c).

Figure 11. Plot of absolute amplitude vs. θ for a single

steered output sample from the input image shown in fig-

ure 10.

Vliet and Verbeek, 1995), plane-wave destruction filters(Fomel, 2002), and other techniques (van Spaendonck,2000).

The steerable pyramid can give very accurate esti-mates of direction. To see how this is done, consider thesimple image of tilted lines in figure 10. Three basis fil-ters as given in equation 22 were applied to this imagefor directions θ = 0, π/3, and 2π/3. From the orienta-tion of the planes in the input image and of the basisfilters, we can calculate the proper weighting functionswj by using equation 20. If we use weights calculatedfor some different direction to steer the basis images,the amplitude of the reconstructed image is reduced.For an error of π/2 the output amplitude is near zero,as in figure 10(c). If the weights are correct, the am-plitude of the reconstructed image is the same as theinput, as in figure 10(b).

Figure 11 shows the amplitude of a single recon-structed output sample given the input image shownin figure 10(a). For this image, wj were calculated for0 ≤ θ ≤ 180. The amplitude maximum occurs at 138,which is consistent with the direction of the lines in theinput image.

The direction that gives us the largest output am-plitude can be found analytically. From equation 12, forthe case where the number of basis filters M = 3,

Figure 12. Line segments indicating the spatially-variant

dominant direction calculated using steerable filters. The in-

put is one level of a steerable pyramid representation of a

time slice, on which the lines are superimposed.

q(x, y) =

2∑j=0

wj(x, y)qj(x, y). (25)

We also have the equation for wj as a function of θ(equation 20). Combining these gives us

q(x, y; θ) =

2∑j=0

1

3[1 + 2 cos(2(θ(x, y)− θj))]qj(x, y).(26)

We find an extremum of this function by setting∂q/∂θ = 0, and solve for θ in terms of the basis imageamplitudes, giving us

θ(x, y) =1

2arctan

( √3(q1(x, y)− q2(x, y))

2q0(x, y)− q1(x, y)− q2(x, y)

).(27)

As we can see in figure 11, this function has twoextrema separated by 90, and we select the one withthe larger absolute amplitude. Using this method we areable to calculate the local dominant direction at everysample location in the image. This estimate is somewhatunstable for sample locations where the input amplitudeis small and in the presence of noise, so we apply asmoothing filter for stability.

To illustrate the estimation of direction, we haveapplied a steerable pyramid to the time slice from fig-ure 1, and calculated orientation estimates at every sam-ple location in the second pyramid level image. Thisresult is shown in figure 12. The line segments superim-posed on the seismic image depict the dominant direc-tion calculated in the middle of the line segment. Linesare displayed for every fifth sample, though they arecalculated for all samples.

Steerable pyramid detection of channels 61

(a) (b) (c)

(d) (e) (f)

Figure 13. Wireframe representations of a regular icosahe-

dron (a-c) and a cuboctahedron (d-f), illustrating how lines

between opposite vertices define three perpendicular rect-

angles. The red lines indicate the directions of the axes of

symmetry of six basis filters.

3 THE 3-D STEERABLE PYRAMID

3.1 3-D steerable filters

When we extend the concept of steerable filters to threedimensions, we need to take another look at the ques-tions that we answered for the 2-D case. For example,how many directional filters will we need in three di-mensions? What directions should these filters take?

We will continue with an approximation to the sec-ond derivative, as we did for the two-dimensional case.According to Theorem 3 (Freeman and Adelson 1991),we will need six basis filters.

The determination of which directions to use for ourfilters is more complicated for three dimensions than itwas for two. For the 2-D case we stated that filters whichequally divide the angle range are most accurate, andcan be steered to any orientation. This holds for the 3-Dcase also, though we note again that there is more thanone set of angles that will work.

To find directions that will equally partition the fullrange of angles, a good approach is to consider the reg-ular polyhedra (Delle Luche et al, 2004). The regularicosahedron has twenty equilateral triangles for faces,and twelve vertices. Lines between opposite vertices ofthe regular icosahedron define six directions that evenlypartition the range of 3-D angles. The angle differencebetween any two of these directions is 63.43495. Also,these directions define three perpendicular rectanglesfor which the ratio of the length of the sides is the goldenratio 1 : (1 +

√5)/2. The fact that our filter directions

can lie on these perpendicular planes simplifies the matha good deal, if we set the directions of the planes parallelto our coordinate axes.

We have found an alternative set of directions thatis every bit as good as the scheme described above,

Figure 14. Diagram of the coordinate references and quan-

tities used in the filter design, in 3-D wavenumber domain.

based on the vertices of the cuboctahedron. The cuboc-tahedron can be formed by shrinking the three perpen-dicular rectangles that define the vertices of the regularicosahedron, so that they become three squares. For thisset of filter directions, the angle difference between onedirection and the four others that are not coplanar is60. The other direction in the same plane is perpen-dicular. Figure 13 shows how the filter directions areconstructed using the two polyhedra.

We have found that the two geometries give essen-tially identical results. The system based on the icosa-hedron has the disadvantage that the mathematical ex-pressions are more complex. For this reason we haveused filter directions based on the vertices of the cuboc-tahedron in our research.

The three-dimensional directional filters are appliedby multiplication in the wavenumber domain, and theyare just a 3-D extension of the filters that we appliedin 2-D. In our discussion of 2-D steerable filters, thedirectional filters are defined in equation 22. In 3-D wecan define the six directional filters in the same way.

In figure 14, the vector ~OC is parallel to the axis ofsymmetry of one of the six basis filters Bj . Its directioncan be described using angles θj and φj . The directionof vector ~OM is from the origin (kx = ky = kz = 0)to some point M , and its direction can be described byangles θ and φ. Let Ω be the angle between vectors ~OCand ~OM .

Bj(θ, φ) = cos2 Ω(θ, φ; θj , φj), j = 0, 1, · · · , 5, (28)

The calculation of cosΩ is straightforward using the dotproduct ~OM · ~OC (Delle Luche et al, 2004):

62 J. Mathewson & D. Hale

j αj βj γj

0 1√2

1√2

0

1 1√2

− 1√2

0

2 1√2

0 1√2

3 − 1√2

0 1√2

4 0 1√2

1√2

5 0 1√2

− 1√2

Table 1. Direction cosines for the axes of symmetry of six

basis filters Bj with geometry based on the vertices of the

cuboctahedron.

Bj(θ, φ) =

(~OM · ~OC

| ~OM || ~OC|

)2

. (29)

A simpler way to define the directions for the 3-Dcase is using the direction cosines:

α = sin θ cos φ β = sin θ sin φ γ = cos θ (30)

The directions of the axes of symmetry of the six basisfilters Bj are given in terms of αj , βj , and γj in Table1. Similarly to the 2-D case, the equation for the basisfilters may also be written using direction cosines:

Bj(kx, ky, kz) =(αjkx + βjky + γjkz)

2

k2x + k2

y + k2z

. (31)

So far we have discussed cosine-squared filters, butwe can also use 1 − cos2 = sin2 filters. In 2-D the onlydifference between these is a 90 rotation, but they arevery different in 3-D. The cos2 filters are conic in shapeand the sin2 filters have an annular shape. These areapplied by multiplication in the wavenumber domain,followed by inverse 3-D Fourier transform to producethe image in the space domain. Figure 16a shows a sin-gle cos2 filter in yellow, and the equivalent sin2 filterin blue, in the wavenumber domain. Figure 16b showsthe identical filters after inverse Fourier transform, inthe space domain. The conic cos2 filter in wavenumberdomain becomes an annulus in space domain, and theannular sin2 filter becomes conic. Amplitudes in thesedisplays have been clipped so that we can see the shapesof the filters.

The effect of the conic cos2 filters applied in thewavenumber domain is to enhance planar features in thespace domain, whose normal direction is parallel to theaxis of symmetry of the filter. The sin2 filters enhancelinear features aligned with the filter axis of symmetry.

Along with directional filters Bj , radial low-passfilters L0 and L1 are also applied as for the 2-D steer-able pyramid. These are identical to the filters in 2-D, applied using equation 24 by multiplication in thewavenumber domain. The only difference is that for 3D,|k| =

√k2

x + k2y + k2

z .With the basis filters Bj and radial filters defined,

the steerable pyramid transform and subsequent im-

(a)

(b) (c)

Figure 15. Image in 3-D wavenumber domain of six direc-

tional filters based on the cuboctahedron. Amplitudes less

than 92% have been zeroed so that we can see the conic

shape of the filters. Three views (a-c) have been displayed,

allowing us to appreciate the symmetry of the directional

filters.

(a) (b)

Figure 16. Conic cos2 filter (yellow) transforms to annular

filter and annular sin2 filter (blue) transforms to conic filter,

going from wavenumber domain (a) to space domain (b).

Steerable pyramid detection of channels 63

age reconstruction proceed according to the 2-D processflows in figures 5 and 8, except that there are now sixbasis images at every pyramid level.

To find the weighting functions for the 3-D steerablefilters, we use equation A5:

α2

αβ

αγ

β2

βγ

γ2

=

α20 α2

1 α22 α2

3 α24 α2

5

α0β0 α1β1 α2β2 α3β3 α4β4 α5β5

α0γ0 α1γ1 α2γ2 α3γ3 α4γ4 α5γ5

β20 β2

1 β22 β2

3 β24 β2

5

β0γ0 β1γ1 β2γ2 β3γ3 β4γ4 β5γ5

γ20 γ2

1 γ22 γ2

3 γ24 γ2

5

w0

w1

w2

w3

w4

w5

.(32)

The direction cosines αj , βj , and γj for our filtersbased on the cuboctahedron are listed in Table 1. Thesevalues are inserted into equation 32 and we can thensolve for wj(α, β, γ), giving the following set of equa-tions:

w0 =1

2[(α + β)2 − γ2]

w1 =1

2[(α− β)2 − γ2]

w2 =1

2[(γ + α)2 − β2] (33)

w3 =1

2[(γ − α)2 − β2]

w4 =1

2[(β + γ)2 − α2]

w5 =1

2[(β − γ)2 − α2]

As in the 2-D case, we multiply the filtered images qj

by the weights wj to produce a steered image:

q(x, y, z) =

5∑j=0

wj(x, y, z)qj(x, y, z). (34)

3.2 3-D Estimation of orientation

The 3-D steerable pyramid can be used to estimate theorientation of image features exactly as we did in twodimensions, but it is more complicated. We are now cal-culating a three-dimensional angle. Also, the type ofsteerable filter, cos2 or sin2, will depend on whether weare looking for planar or linear features.

For an example of orientation analysis in three di-mensions, let’s consider a synthetic 3-D data volumecomprising parallel planes of known orientation, shownin figure 17. The orientation of the planes is given bythe direction of the normal vector, with θ = 72 andφ = 33, as we defined them in figure 14. We calcu-lated the absolute amplitude of a single steered outputsample for all values of θ and φ. This was done by calcu-lating and applying the appropriate wj , then summing

Figure 17. Three orthogonal slices through a synthetic 3-

D data volume consisting of parallel planes. The diagram at

top right indicates the location and orientation of the three

slices.

Figure 18. Amplitude as a function of 3-D steering angles

θ and φ for a single sample in the synthetic volume fromfigure 17. Large amplitudes are shown in red; dark blue is an

amplitude near zero.

the basis images, iterating through all angles. The resultis shown in figure 18.

In the display, we note that there is a maximum atθ = 72, φ = 33, which is consistent with the knownorientation of the planes. We also see maxima at 180

from this, and the minimum amplitude is essentially

64 J. Mathewson & D. Hale

zero, at 90 from the correct direction. Obviously, wecan’t do this kind of search for every sample in a 3-D image. In two dimensions we were able to calculatethe optimum steering angle analytically from the am-plitudes of the basis images, and this should be possiblein three dimensions as well.

4 DETECTION AND ENHANCEMENT OFCHANNELS

We noted earlier in this paper that steerable cos2 fil-ters enhance planar features, while sin2 filters shouldenhance linear features. This gives us an idea of how touse these filters in channel detection.

In most seismic datasets the dominant features areplanar reflectors. In many situations this can make de-tection of channels difficult, as they can be obscured bythe reflections. The steerable pyramid can be made toenhance planar events, using cos2 filters. The enhancedreflections can then be subtracted from the seismic data,revealing the channels. Figure 19 shows an example ofthis approach. Strong planar events are greatly atten-uated by the process, but amplitudes of channels aresimilar to the input.

Another approach that we have been studying isto use the 3-D steerable pyramid with sin2 filters to en-hance the channels. Early trials using the 3-D sin2 filterswere unsuccessful, and we have been testing variationsof this method on a time slice from the 3-D syntheticdataset using 2-D filters. Figure 20 shows one result ofthis testing. Four cascaded iterations of 2-D steerablepyramid adaptive smoothing were applied to the im-age. The amount of smoothing applied is varied overthe image using a linearity attribute that is calculatedfor every sample. The result of this variable smooth-ing is that the continuity of channels is improved whilesurrounding areas are essentially unchanged.

5 CONCLUSION

The steerable pyramid transform is potentially a use-ful tool for analysis of orientation and dimensionality,though it has seen little use in geophysical applicationsuntil now. During our research we have established thatit can give us accurate and detailed estimates of the ori-entation of image features. We have also found linearityattributes in 2-D using steerable pyramid filters.

One thing that we noted regarding the steerablepyramid was the lack of research into 3-D implemen-tation. The 3-D steerable pyramid concept was intro-duced by Freeman and Adelson in 1991, but there areonly a couple of technical papers in this area. We havedone quite a lot of research in 3-D and have discovereda number of new developments. We have found a setof filter directions based on the vertices of the cuboc-tahedron that simplify the math while giving accurate

(a)

(b)

Figure 19. Example of use of 3-D steerable pyramid for

attenuation of planar events. Strong horizontal reflections on

3-D synthetic input (a) are enhanced using the 3-D steerablepyramid, then subtracted (b).

results. We have also found analytic expressions for fea-ture orientation in terms of image amplitudes.

Quite a lot of work remains to be done. We arecurrently working to improve our orientation and di-mensionality estimates for 3-D data, which is absolutelynecessary for detection of channels. We are also workingto improve the program efficiency so that we can workwith larger datasets. We also plan to test the applicabil-

Steerable pyramid detection of channels 65

(a) (b)

Figure 20. Example of use of 2-D steerable pyramid for

enhancement of channel features. Four cascades of steerable

pyramid adaptive smoothing are applied to input time slice

(a), resulting in more continuous channels on the output im-

age (b)

ity of the steerable pyramid to other seismic processingsuch as attenuation of coherent noise, and interpolation.

REFERENCES

Freeman, W.T. and E. Adelson, 1991, The design and use of

steerable filters: IEEE Trans. Patt. Anal. Machine Intell.,

13, 891–906.

Simoncelli, E.P., W.T. Freeman, E.H. Adelson, and

D.J. Heeger, 1992, Shiftable multi-scale transforms: IEEE

Trans. Information Theory, 38, 587–607.Simoncelli, E.P. and W.T. Freeman, 1995, The steerable

pyramid: A flexible architecture for multi-scale derivative

computation: 2nd IEEE Int’l Conf. Image Processing, volIII, 444–447.

Castleman K., M. Schulze and Q. Wu, 1998, Simplified De-

sign of Steerable pyramid Filters: To appear in Proceed-

ings of ISCAS’98

Wu Q., M. Schulze and K. Castleman, 1998, Steerable pyra-mid filters for selective image enhancement applications:

Proc. IEEE ISCAS, Paper 1838, Session WAA2-2

Denis F.and A. Baskurt, 2004, Multidirectional curvilinearstructures detection using steerable pyramid: Journal of

Electronic Imaging, 13, 756–765.

Delle Luche C., Denis F. and A. Baskurt, 2004, 3-D steer-

able pyramid based on conic filters: Wavelet Applications

in Industrial Processing. Edited by Truchetet, Frederic.Proceedings of the SPIE, 5266, 260–268.

Van Spaendonck,R.L.C., F.M. Hindriks, F.C.A. Fernandes,

and G.G. Drijkoningen, 2000, Three-dimensional at-

tributes for seismic interpretation: Annual Meeting Ab-

stracts, 70th Annual Int’l Meeting:SEG, 2059–2062.

van Vliet, L.J., and P.W. Verbeek, 1995, Estimators for ori-

entation and anisotropy in digitized images: Proceedings

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APPENDIX A: STEERING THEOREMS

Below are steering theorems from Freeman and Adelson(1991) that were used in this research:

Given a function fθ rotated by an angle θ, and fθj ,

which is the same function rotated by a number of anglesθj , j = 0, 1, ..., M . The general steering constraint is

fθ(x, y, z) =

M−1∑j=0

wj(θ)fθj (x, y, z), (A1)

where M is the number of basis functions required forsteering and wj(θ) are the weighting functions.

Let f be any function which can be expanded ina Fourier series in polar angle, φ, with r =

√x2 + y2

and φ = arg(x, y),

f(r, φ) =

N∑n=−N

an(r)einφ. (A2)

Theorem 1: The steering condition, equation A1, holdsfor functions expandable in the form of equation A2 ifand only if the weighting functions wj(θ) are solutionsof:

1

eiθ

· · ·

eiNθ

=

1 1 · · · 1

eiθ1 eiθ2 · · · eiθM

......

......

eiNθ1 eiNθ2 · · · eiNθM

w1(θ)

w2(θ)

...

wM (θ)

.(A3)

If, for any n, an(r) = 0, then the corresponding nthrow of the left side and of the matrix of the right side ofequation (A3) should be removed.

Theorem 2: Let T be the number of positive ornegative frequencies −N ≤ n ≤ N for which f(r, φ) hasnon-zero coefficients an(r). Then the minimum numberof basis functions which are sufficient to steer f(r, φ)by Eq. (A1) is T (i.e., M >= T ).

Theorem 3: Given a three dimensional axiallysymmetric function f(x, y, z) = A(r)PN (x), wherePN (x) is an even or odd symmetry N th order poly-nomial in x. Let α, β and γ be the direction cosines

66 J. Mathewson & D. Hale

of the axis of symmetry of fR(x, y, z) and αj, βj andγj be the direction cosines of the axis of symmetry offRj (x, y, z). Then the steering equation,

fR(x, y, z) =

M−1∑j=0

wj(α, β, γ)fRj (x, y, z), (A4)

holds if and only if(a) M ≥ (N + 1)(N + 2)/2 and(b) the wj(α, β, γ) satisfy

αN

αN−1β

αN−1γ

αN−2β2

...

γN

=

αN1 αN

2 · · · αNM

αN−11 β1 αN−1

2 β2 · · · αN−1M βM

αN−11 γ1 αN−1

2 γ2 · · · αN−1M γM

......

......

γN1 γN

2 · · · γNM

×

w1(α, β, γ)

w2(α, β, γ)

w3(α, β, γ)

...

wM (α, β, γ)

. (A5)