Changes between Version 12 and Version 13 of ppSub_Issues

Timestamp:

Jan 12, 2010, 1:22:26 PM (16 years ago)

Author:

Paul Price

Comment:

Adding information about dual convolution performance and choice of method to keep kernels compact.

ppSub_Issues

@@ -42 +42 @@
 ||[[Image(source:/trunk/doc/ppsub/pics/diff.gp/diff.gp.04.03.dual.p0.isis.jpg)]] ||[[Image(source:/trunk/doc/ppsub/pics/diff.gp/diff.gp.04.03.dual.p0.herm.jpg)]] ||[[Image(source:/trunk/doc/ppsub/pics/diff.gp/diff.gp.04.03.dual.p0.deco.jpg)]] || 5.2pix, A/R = 1.2, Theta = +30 vs 5.2pix, A/R = 1.2, Theta = -30, dual penalty = 0.0 || 
 ||[[Image(source:/trunk/doc/ppsub/pics/diff.gp/diff.gp.04.03.dual.p1.isis.jpg)]] ||[[Image(source:/trunk/doc/ppsub/pics/diff.gp/diff.gp.04.03.dual.p1.herm.jpg)]] ||[[Image(source:/trunk/doc/ppsub/pics/diff.gp/diff.gp.04.03.dual.p1.deco.jpg)]] || 5.2pix, A/R = 1.2, Theta = +30 vs 5.2pix, A/R = 1.2, Theta = -30, dual penalty = 1.0 || 
+
+
+=== Dual convolution following fix (r26562) ===
+
+Following simplification of the math (and corresponding simplification of the code), dual convolution is now working.
+
+In the presence of noise, it is difficult for the least-squares solution to provide kernels that are compact (since convolution of one image by a large kernel can be matched with the convolution of the other image by a similarly large kernel).  One solution is to add a penalty term into the least-squares problem (c.f. Yuan & Akerlof, 2008ApJ...677..808Y).  Another solution is to solve the equation once, and then compare the corresponding terms in each of the kernels, masking the smaller (or both if small).  Below we test these solutions using the cross-directed Gaussian and PS1_V1 PSFs (5.2pix, A/R = 1.2, Theta = +30 vs 5.2pix, A/R = 1.2, Theta = -30).
+
+ * NONE: No attempt was made to force the kernels to be compact.
+ * MASK: We compared corresponding terms in each of the kernels, masking the smaller; we also masked any with a value less than 10^-3^ of the derived normalisation.
+ * PENALTY: We added a penalty function following Yuan & Akerlof, scaled to match the background diagonal term in the least-squares matrix.
+ * BOTH: We applied both the MASK and PENALTY methods.
+
+The below images are a montage of the residuals and the convolution kernels for the above methods, laid out:
+
+|| NONE || MASK ||
+|| PENALTY || BOTH ||
+
+The conclusion is that the PENALTY method works well.  NONE blows up the kernel too much (as expected).  MASK does not allow sufficient flexibility in the kernel to provide cross-directed kernels for both images.  BOTH suffers from the weaknesses of the MASK method.
+
+
+I also tried using Singular Value Decomposition to solve the least-squares equation, with masking of low-significance singular values, but found that the quality of the subtraction is strongly dependent on the choice of the threshold, so am putting this off further evaluation of this until later.
+
+==== Gaussian PSFs ====
+
+|| Method || Normalisation || Mean dev. || Peak-peak residuals ||
+|| NONE   || 1.000113      || 0.002095  || No apparent residuals ||
+|| MASK   || 1.006291      || 0.427949  || +90,-92 ||
+|| PENALTY|| 1.000060      || 0.015114  || +9,-11 ||
+|| BOTH   || 0.995888      || 0.206492  || +90,-74 ||
+
+Residuals:
+[[Image(source:/trunk/doc/ppsub/pics/dual/gaussians_resid.jpg)]]
+
+Kernels:
+[[Image(source:/trunk/doc/ppsub/pics/dual/gaussians_kernels.jpg)]]
+
+
+==== PS1_V1 PSFs ====
+
+|| Method || Normalisation || Mean dev. || Peak-peak residuals ||
+|| NONE   || 1.000116      || 0.042257  || +10,-9 ||
+|| MASK   || 1.008349      || 0.169106  || +38,-53 ||
+|| PENALTY|| 1.000726      || 0.081741  || +18,-17 ||
+|| BOTH   || 1.003305      || 0.557793  || +104,-110 ||
+
+Residuals:
+[[Image(source:/trunk/doc/ppsub/pics/dual/ps1v1_resid.jpg)]]
+
+Kernels:
+[[Image(source:/trunk/doc/ppsub/pics/dual/ps1v1_kernels.jpg)]]