The Steerable Pyramid

What is a steerable pyramid?

The Steerable Pyramid is a linear multi-scale, multi-orientation image decomposition that provides a useful front-end for image-processing and computer vision applications. We developed this representation in 1990, in order to overcome the limitations of orthogonal separable wavelet decompositions that were then becoming popular for image processing (specifically, those representations are heavily aliased, and do not represent oblique orientations well). Once the orthogonality constraint is dropped, it makes sense to completely reconsider the filter design problem (as opposed to just re-using orthogonal wavelet filters in a redundant representation, as is done in cycle-spinning or undecimated wavelet transforms!). Detailed information may be found in the references listed below.
The basis functions of the steerable pyramid are Kth-order directional derivative operators (for any choice of K), that come in different sizes and K+1 orientations. As directional derivatives, they span a rotation-invariant subspace (i.e., they are equi-variant) and they are designed and sampled such that the whole transform forms a tight frame. An example decomposition of an image of a white disk on a black background is shown to the right. This particular steerable pyramid contains 4 orientation subbands, at 2 scales. The smallest subband is the residual lowpass information. The residual highpass subband is not shown.
The block diagram for the decomposition (both analysis and synthesis) is shown to the right. Initially, the image is separated into low and highpass subbands, using filters L0 and H0. The lowpass subband is then divided into a set of oriented bandpass subbands and a low(er)-pass subband. This low(er)-pass subband is subsampled by a factor of 2 in the X and Y directions. The recursive (pyramid) construction of a pyramid is achieved by inserting a copy of the shaded portion of the diagram at the location of the solid circle (i.e., the lowpass branch). More detailed descriptions may be found in the references (below).

What advantages does it have over separable orthonormal wavelets?

More importantly, the representation is translation-invariant (i.e., the subbands are aliasing-free, or equivariant with respect to translation) and rotation-invariant (i.e., the subbands are steerable, or equivariant with respect to rotation). This can make a big difference in applications that involve representation of position or orientation of image structure.

The primary drawback is that the representation is overcomplete by a factor of 4k/3, where k is the number of orientation bands. Also, the filter design problem is messy, and so space-domain implementation is not perfect-reconstruction (although errors are small enough for most applications). We typically use a frequency-domain implementation, which provides perfect reconstruction, but the resulting filters exhibit more spatial "ringing".

Here is a table comparing properties to other well-known transforms (more information about these transforms may be found in this book chapter):

	Steerable Pyramid	Separable Orthog. Wavelet	Laplacian Pyramid	Gabor (octave)	Block DCT
jointly-localized (space/frequency)	yes	yes (can be)	yes	not inverse	not in frequency
translation-equivariant (no aliasing)	yes (approx)	no	yes (approx)	no	no
oriented kernels	yes	no (not diagonals)	N/A	yes	no
rotation-equivariant (steerable)	yes (approx)	no	N/A	no	no
tight frame (self-inverting)	yes (approx)	yes	no	no	yes
overcompleteness	4k/3	1	4/3	1	1

For what applications is the steerable pyramid useful?