{"id":277,"date":"2012-01-01T10:12:33","date_gmt":"2012-01-01T10:12:33","guid":{"rendered":"http:\/\/ahay.org\/blog\/?p=277"},"modified":"2015-09-04T12:13:16","modified_gmt":"2015-09-04T12:13:16","slug":"program-of-the-month-sfsmooth","status":"publish","type":"post","link":"https:\/\/ahay.org\/blog\/2012\/01\/01\/program-of-the-month-sfsmooth\/","title":{"rendered":"Program of the month: sfsmooth"},"content":{"rendered":"

sfsmooth<\/a>, one of the most useful Madagascar programs, implements smoothing by triangle filtering. <\/p>\n

The idea of triangle filtering is explained in Jon Claerbout’s books<\/a> Processing versus Inversion<\/em> and Image Estimation by Example<\/em>. You can find the explanation in the chapter Smoothing with box and triangle<\/a>. The triangle filter of radius is a correlation of two box filters. In the Z<\/em>-transform notation,
\n$$T_k(Z) = B_k(1\/Z)\\,B_k(Z)$$
\nwhere
\n$$B_k(Z) = \\displaystyle \\frac{1}{k}\\,\\left(1+Z+Z^2+\\cdots+Z^k\\right) = \\frac{1}{k}\\,\\frac{1-Z^{k+1}}{1-Z}$$
\nand $k$ is the smoothing radius. Triangle filtering is a very efficient operation, because it requires only $N$ multiplications for the input of size $N$. <\/p>\n

Triangle filtering serves as a good approximation to more expensive Gaussian filtering. Repeated triangle filtering rapidly approaches Gaussian filtering (a consequence of the central limit theorem). The following example from jsg\/shape\/smoo<\/a> demonstrates how repeated application of triangle smoothing turns a triangle filter into a Gaussian filter <\/p>\n

\"\" <\/p>\n

The following example from gee\/ajt\/triangle<\/a> demonstrated how sfsmooth<\/strong> handles boundary conditions: the energy is reflected from the side in order to preserve the zero-frequency (DC) component of the input signal. Repeated smoothing or smoothing with a very large radius turns every signal into a constant. <\/p>\n

\"\" <\/p>\n

In multiple dimensions, triangle smoothing is applied sequentially in different directions, possible with different radii (controlled by rect#=<\/strong> ) parameter. The following example from gee\/ajt\/mound<\/a> shows shapes created by 2-D triangle smoothing after one pass and two passes (repeat=2<\/strong>). <\/p>\n

\"\" \"\"<\/p>\n

Triangle smoothing is theoretically a self-adjoint operator. Numerically, the adjoint and forward operators are different. The action is controlled by the adj=<\/strong> parameter. <\/p>\n

For smoothing with a box instead of a triangle, use sfboxsmooth<\/a>. <\/p>\n

For a different approach to smoothing, try sfgaussmooth<\/a>, which implements smoothing by recursive Gaussian filtering. <\/p>\n

Previous programs of the month<\/h3>\n