Changes between Version 27 and Version 28 of u/erica/scratch

Timestamp:

02/08/16 12:50:18 (9 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/scratch

@@ -7 +7 @@
 In addition to being normalized (i.e. satisfying the first equation), this exponential function needs to go to zero at the boundary of the kernel. Thus, we have the following equation,
 
-[[latex($E_{acc} = \sum_{i=0}^{N} k(~e^{-dx*i} - e^{-Ndx} )~E_{acc}  $)]]
+[[latex($E_{acc} = \sum_{i=1}^{N} k(~e^{-dx*i} - e^{-Ndx} )~E_{acc}  $)]]
 
 which gives the normalization constant,
@@ -13 +13 @@
 [[latex($ \boxed{k = \frac{1}{~\Sigma ~e^{-dx*i}-e^{-Ndx}}} $)]]
 
-where N is the max number of cells in the kernel. Note that the sum runs over cells i=0 to 1, where the '0th' cell is the cell the sink is in, and the '4th' cell is the furthest cell from the sink in the kernel, that the exponential goes to zero at this boundary, and dx*i gives a position (which here assumes the sink is at the cell center -- the actual function is described below).
+where N is the max number of cells in the kernel. Note that i) the sum runs over cells i, where the '1st' cell is the cell the sink is in, and the 'Nth' cell is the furthest cell from the sink in the kernel, ii) that the exponential goes to zero at this boundary, iii) and dx*i gives a position (which here implicitly assumes the sink is at the cell center -- the actual function will be described below).
 
 
@@ -33 +33 @@
 [[Image(dx1kernel.png, 25%)]]
 
-In the most extreme case, as dx << 1 the function remains triangular, but as dx > 1, the function quickly drops to 0 (i.e. one cell removed from the origin). This is when the smoothing function would no longer serve its purpose (i.e. smoothing)... We could probably best avoid these issues by enforcing a larger kernel around the sink so that it most resembles an exponential, but then that would require changing the kernel of max refinement around a sink as well (which is currently set to 4 dx). We could also play with an exponential function that fell off faster. 
+We see that in this case, smaller dx resembles a triangle, but dx~1 gives an exponential. However, as dx << 1 the function remains triangular, but as dx > 1, the function quickly drops to 0 (i.e. one cell removed from the origin). This is when the smoothing function would no longer serve its purpose (i.e. smoothing)... We could probably best avoid these issues by enforcing a larger kernel around the sink so that it most resembles an exponential, but then that would require changing the kernel of max refinement around a sink as well (which is currently set to 4 dx).