The Two-Dimensional Haar Transform


[IMAGE]

Figure 1. Input image

[IMAGE]

Figure 2. Haar Transform

Two-stage, two-dimensional Haar transform: The outputs of the first stage appear in the UR, LR and LL corners. The outputs of the second stage follow the same format in the UL quadrant. Refer to Fig. 14-26 in [2]. Click to download a double-size image in JPEG (31K) or GIF (150K) format.

[Flowchart]

Figure 3. Haar Transform Igraph

WiT igraph for the two-dimensional Haar transform: This igraph diagrams the algorithm used to compute one stage of the Haar transform. The 2x2 convolution kernels used are
[Equations]
The Haar transform, now 100 years old [1], is the simplest example of an orthonormal wavelet transform [2]. It decomposes an image into horizontal, vertical and diagonal edges at different (binary) scales. The WiT igraph shown here implements one stage of the forward and inverse discrete wavelet transform using the Haar basis functions and Mallat's fast wavelet transform algorithm [3]. It also compares the reconstructed image with the origonal. Click to download the WiT igraph file or the GIF image of the igraph (69K).
1. A. Haar, "Zur Theorie der Orthogonalen Funktionensysteme," Math. Ann. 69:331-371, 1910.
2. K. R. Castleman, Digital Image Processing, Prentice-Hall, 1996.
3. S. Mallat, "A Theory for Multiresolution Signal Decomposition: The Wavelet Representation," IEEE Trans. PAMI, 11:674-693, 1989.
Last updated 04 June, 2011.
Copyright © 1996 Kenneth R. Castleman
Permission granted for noncommercial educational use.
castleman@earthlink.net