# The Steerable Pyramid Transform

### Figure 1. Input image

### Figure 2. Steerable Pyramid Transform

**Two-stage steerable pyramid transform for k=2:**
The output of the (single) highpass filter fills the LR quadrant.
The two bandpass filter outputs of the first stage appear in the UR and LL quadrants,
while the decimated lowpass output falls at the LR corner of the UL quadrant.
The three output images of the second stage follow the same format in the UL quadrant.
Click to download a double-size image in
JPEG (29K)
or GIF (154K) format.

### Figure 3. Steerable Pyramid Transform Igraph

**WiT igraph for the k=2 steerable pyramid transform:**
This igraph diagrams the algorithm used to compute one stage of the transform.
The five transfer functions used are

The steerable pyramid decomposition [1] is an outgrowth of the Laplacian pyramid [2].
It decomposes an image into oriented, bandpass filtered components
at different (binary) scales. It is overcomplete by the factor 4k/3,
but it avoids aliasing in the downsampling process, and it has useful shiftability
properties in both translation and rotation [3].

The WiT igraph shown here implements one stage of the forward and inverse
steerable pyramid transform using the algorithm of Simoncelli et. al. [1].
Click to download the WiT igraph file
or the GIF image of the igraph (40K).

For k = 1, the decomposition uses a nondirectional bandpass filter, similar to the Laplacian pyramid [2].
Click to download the convolution kernels
or the MathCAD file that generates them
[PDF file].

For k = 2, the decomposition uses two directional derivative (vertical and horizontal) bandpass filters.
Click to download the convolution kernels
or the MathCAD file that generates them
[PDF file].

For k = 3, the decomposition uses three directional derivative bandpass filters (0, 60, 120 degrees).
Click to download the convolution kernels
or the MathCAD file that generates them
[PDF file].

For k = 4, the decomposition uses four directional derivative bandpass filters (0, 45, 90, 135 degrees).
Click to download the convolution kernels
or the MathCAD file that generates them
[PDF file].

1. E. P. Simoncelli, and W. T. Freeman, "The Steerable Pyramid:
A Flexible Architecture for Multi-Scale Derivative Computation,"
*Proc. ICIP-95*: 444-447, 1995.

2. P. J. Burt, and E. H. Adelson, "The Laplacian Pyramid as a Compact Image Code,"
*IEEE Trans.* **C-31**:532-540, 1983.

3. E. P. Simoncelli, W. T. Freeman, E. H. Adelson, and D. J. Heeger,
"Shiftable Multiscale Transforms," *IEEE Trans.* **IT-38**(2):587-607, 1992.

Last updated 04 June, 2011.

*Copyright © 1996 Kenneth R. Castleman*

Permission granted for noncommercial educational use.

castleman@earthlink.net