Anonymous user
Talk:Deconvolution/2D+: Difference between revisions
m
link fixed
m (→general: signed) |
m (link fixed) |
||
Line 1:
I got interested in higher dimensional deconvolution when contacted by someone about using it as a way of looking for trading indicators in financial time series. I set this task as an example of something that I'm speculating can be done well with functional and array processing languages, but only with difficulty otherwise, which I hope someone will weigh in to confirm or refute. In case anyone wants to know, consistent [[http://www.filedropper.com/tdat test data]] were generated partly by this higher dimensional convolution function,
<lang Ursala>conv = +^|(~&x+,*DrlDSNiCK9xxSNiCK9K7iFS+ *)=>times+ **+ *K7|\x+ iota; * ! plus:-0</lang>
invoked as <code>(conv d)(h,f)</code> with dimension <code>d > 0</code> and conforming <code>h</code> and <code>f</code>. (This function essentially subsumes the [[Image convolution]] task as a special case with <code>d = 2</code> and <code>|h| = 3</code>.) I suggest a development methodology based on warming up with [[Deconvolution/1D]], then hand coding the solutions for the next few dimensions, and then looking for the pattern.
Line 43:
odd number). If the solver precludes over determined systems
and the language also doesn't let you delete rows from matrices, you might
have to write the code in the <code>band</code> function to build it that way in
the first place.
Line 52:
the <code>band</code> function above to construct a matrix <math>A</math> of vectors following
the same pattern, with which each <math>i,j</math>-th element of <math>A</math> being either a
vector from <math>f</math>, or a zero vector of conforming length.
I'm not saying it can't be done, but I'm not sure how pruning the matrix generalizes to higher dimensions, so the rest of this description assumes the solver copes with over determined systems. Such a solver is readily available in the [[http://www.netlib.org/lapack Lapack]] library, which is callable from many languages.
Next, pair up each row of the matrix-of-vectors <math>A</math> with the corresponding member of
<math>g</math>. Since <math>A</math> is a matrix of vectors, each row of it can be considered a matrix
of scalars on its own. For each <math>i</math> ranging from 0 to <math>|g|-1</math>, the pair <math>(g_i,A_i)</math> of the <math>i</math>-th member of <math>g</math> and the <math>i</math>-th row of <math>A</math> will consist of a vector <math>g_i</math> and a matrix <math>A_{i}</math>. Pair up a copy of <math>g_i</math> with each row of <math>A_{i}</math> and apply the 1 dimensional <code>band</code> function to each member of the list of all such pairs. This step will yield a list of matrices (of scalars) each with <math>|g_i|</math> rows. Flatten this list of matrices into a single matrix with <math>|g_i|</math> rows by concatenating them row-wise (i.e., horizontally). Having done so for all <math>(g_i,A_i)</math> pairs, flatten the resulting list of <math>|g|</math> matrices into a one large matrix having <math>|g|*|g_0|</math> rows by concatenating them column-wise (i.e., vertically). This matrix should then be fed to the linear equation solver along with a flattened copy of <math>g</math>.
The result returned by the linear equation solver will be a list of
| |||