|The helical coordinate|
In this book section only, I use abnormal notation for bold letters. Here , are signals, while and are images, being neither matrices or vectors. Recall from Chapter that a filter is a signal packed into a matrix to make a filter operator.
Let the time reversed version of be denoted . This notation is consistent with an idea from Chapter that the adjoint of a filter matrix is another filter matrix with a reversed filter. In engineering, it is conventional to use the asterisk symbol `` '' to denote convolution. Thus, the idea that the autocorrelation of a signal is a convolution of the signal with its time reverse (adjoint) can be written as .
Wind the signal around a vertical-axis helix to see its 2-dimensional shape :
Physics on a helix can be viewed through the eyes of matrices and numerical analysis. This presentation is not easy, because the matrices are so huge. Discretize the -plane to an array, and pack the array into a vector of components. Likewise, pack minus the Laplacian operator into a matrix. For a plane, that matrix is shown in equation (14).
The 2-dimensional matrix of coefficients for the Laplacian operator is shown in (14), where on a Cartesian space, , and in the helix geometry, . (A similar partitioned matrix arises from packing a cylindrical surface into a array.) Notice that the partitioning becomes transparent for the helix, . With the partitioning thus invisible, the matrix simply represents 1-dimensional convolution and we have an alternative analytical approach, 1-dimensional Fourier transform. We often need to solve sets of simultaneous equations with a matrix similar to (14). The method we use is triangular factorization.
Although the autocorrelation has mostly zero values, the factored autocorrelation has a great number of nonzero terms. Fortunately, the coefficients seem to be shrinking rapidly towards a gap in the middle, so truncation (of those middle coefficients) seems reasonable. I wish I could show you a larger matrix, but all I can do is to pack the signal into shifted columns of a lower triangular matrix like this:
Spectral factorization produces not merely a causal wavelet with the required autocorrelation. It produces one that is stable in deconvolution. Using in 1-dimensional polynomial division, we can solve many formerly difficult problems very rapidly. Consider the Laplace equation with sources (Poisson's equation). Polynomial division and its reverse (adjoint) gives us , which means we have solved by using polynomial division on a helix. Using the 7 coefficients shown, the cost is 14 multiplications (because we need to run both ways) per mesh point. An example is shown in Figure 10.
Figure 10. Deconvolution by a filter with autocorrelation being the 2-dimensional Laplacian operator. Amounts to solving the Poisson equation. Left is ; Middle is ; Right is .
Figure contains both the helix derivative and its inverse. Contrast those filters to the - or -derivatives (doublets) and their inverses (axis-parallel lines in the -plane). Simple derivatives are highly directional, whereas, the helix derivative is only slightly directional achieving its meagre directionality entirely from its phase spectrum.
|The helical coordinate|