Changes between Initial Version and Version 1 of Stack_Algorithm

Timestamp:

Oct 22, 2013, 3:44:02 PM (13 years ago)

Author:

watersc1

Comment:

--

Stack_Algorithm

@@ +1 @@
+= Intro =
+
+== Input selection ==
+
+== PSF Matching ==
+
+== Pixel Rejection ==
+
+The following conventions in the remainder of this discussion.  Lower case values are for a single
+pixel on image "i", upper case values are values that are constant for
+a given image "i" (i.e., all pixels on image i have that parameter),
+greek values are combined values (the stacked values).
+
+# Set up
+For a given pixel, define four parameters:
+f_i   pixel value
+v_i   pixel variance value
+m_i   input mask value, 0 = bad, 1 = good
+s_i   suspect mask value, 0 = not suspect, 1 = suspect
+W_i   image weight
+
+There are also a set of configurable options:
+NMIN     = 5       Minimum number of values for KMM test
+MIN_UNI  = 0.05    Minimum probability to accept a bimodal solution
+SYS_FRAC = 0.1     Systematic variance fraction
+NSIG_REJ = 3.5     N-sigma rejection threshold
+DISCARD  = 0.2     Fraction of values to exclude in Olympic mean
+ITER     = 0.5     Number of rejection iterations per input
+
+After convolution to the target PSF, we add an additional variance
+(ADDV_i) term equal to the chi^2 value found during the PSF matching.
+For 1d Gaussian convolution, we have no real concept of this value, so
+all chi^2 (and ADDV_i) are defined to be 1.0;
+
+v_i = v_i + ADDV_i
+
+The image weights are calculated by taking the inverse of the robust
+median of the variance image:
+
+W_i ~ 1 / sum_{x,y} v_i_{x,y}
+
+# Calulate initial stack values
+
+- Calculate the weighted mean:
+
+mu    = sum_i( f_i * W_i) / sum_i(W_i)
+sigma = 1 / sum_i(1 / W_i)
+nu    = sum_i(m_i)
+
+# Rejection loop
+
+This loop is repeated ITER * Ninput times (minimum of once).
+
+- Determine distribution values:
+
+if (nu > NMIN) do Mixture Model test:
+ mu_KMM      mean of most popular gaussian mode
+ sigma_KMM   sigma of most popular gaussian mode
+ pi_KMM      fraction of inputs in most popular gaussian mode
+ rho_UNI     Probability that inputs are unimodal
+
+- Set rejection limits for each input:
+
+if (nu > NMIN) sys_var = sigma_KMM^2
+else           sys_var = SYS_FRAC * f_i
+
+limit_i = (NSIG_REJ)^2 * (v_i + sys_var)
+
+- Determine distribution median
+
+if (nu > NMIN) mu_median = mu_KMM
+else           mu_median = olympic_mean
+where:
+olympic_mean = sum_i(f_i * W_i) / sum_i(W_i)
+from int((DISCARD * nu + 0.5)/2.0)
+to int((DISCARD * nu + 0.5)/2.0) + int(nu) - int(DISCARD * nu + 0.5)
+
+- Determine most discrepant point
+
+if ( (f_i - mu_median)^2 > limit_i ) AND
+   ( (f_i - mu_median)^2 / limit_i > delta ) THEN
+ delta = (f_i - mu_median)^2 / limit_i
+ delta_index = i
+
+- Do rejection.  If we have suspect pixels, throw those out first (so
+  they'll be skipped in the next iteration).  Otherwise, remove the
+  most discrepant point
+
+if DEFINED(dev_i)
+   if (s_i != 0) m_i = 0
+   else if (i = delta_index) m_i = 0
+
+- End rejection loop.
+
+
+This completes the initial stack stage of ppStack.  We then use the
+set of rejected pixels to identify the source pixels in the
+unconvolved input images.  These are masked out, and the images are
+reconvolved and a final stack is constructed using the same algorithm,
+however the rejection loop is skipped (so the weighted mean is all
+that is performed).
+
+
+== Output Products ==
+