This may happen on the first request due to CSS mimetype issues. Try clearing your browser cache and refreshing.

--- a/kwave/doc/interpolation.html +++ b/kwave/doc/interpolation.html @@ -1,58 +1,58 @@ -<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML//EN"> -<html> - <head> - <title>Types of Interpolation</title> - </head> - - <body> - <h1>Short introduction into Interpolation</h1> - -<CENTER> -<I>This part of help documentation was last updated on August. 26, 1998 for kwave 0.29 alpha</I> -</CENTER> -<p> - -To interpolate means, to find a rule to create points in between a given set of points based on a rule. Kwave supports different types of interpolation (rules). -<hr> -The most simple one is the linear Interpolation. -<p> -<img src="linear.gif"> -<p> -The linear interpolation connects all dots through a line. All points on the line define the points in between. This provides a statistical most errorfree interpolation, but does not provide a smooth function, that may be desirable for audio purposes ( )<p> -<hr> -A little bit more complex is the polynomial interpolation. -<p> -<img src="polynom1.gif"> -<p> -The rule is to find a polynom that goes though all the given points. Based on the function of the polynom the points in between are calculated. This has some -disadvantages.<p> -When changing one point, the whole interpolation function is affected, an effect that may not be desired. Here is an example that demonstrates this effect. -The red point was moved, and a new was function interpolated. -Both functions differ slightly at every point, but the given ones, only becauseone value was changed! -<p> -<img src="overlay.jpg"> -<p> -Furthermore the polynomial function is smooth, but does describe a rather long way. It oscillates around the given points. An interpolation rule (the cubic spline interpolation) that keeps closer to the points, while staying smooth will be introduced later<p> -<hr> -To be able to adjust one point, while leaving the rest of the interpolation untouched, one has take to a few points interpolate them, then take the next ones, and so on. This has been implemented in the piecewise polynomial interpolation, that takes 3 (5 or 7) points, interpolates them, but adds to the function only the interpolation between the first and the second point. Here follows an example for a 3rd order polynom. The red line is the unseen part of the interpolation starting at the point before the red one. -<p> -<img src="3polynom-mod.gif"> -<p> -You can also see, that this solution has a flaw. You don't get a smooth function, because the interpolated pieces don't fit together all the times. So perhaps there should be a found a rule, that allows to fit them together. -<hr> -The cubic spline interpolation does this. It calcuates a derrivative at the end points of the piecewise interpolation and assures the derrivative is equal for both pieces of interpolation. This minimizes the statistical function error while still obtaining a smooth function. The advantage of local editing still exists. -<p> -<img src="spline1.gif"> -<p> -One way to obtain derrivation of a point would be to take the two nearest point, draw a line through them, and then take the derrivation of this line. -<hr> -The next interpolation is again a rather simple one. It keeps the function on the value of the last point until a new point is given. This provides none of the above discussed properties, while still being useful for some effects or models. (for example a function of Pitch). -<p> -<img src="sample-hold.gif"> -<p> -<a href="index.html">Go back to Main Help Page</a> -</body> -</html> - - - +<!DOCTYPE HTML PUBLIC "-//IETF//DTD HTML//EN"> +<html> + <head> + <title>Types of Interpolation</title> + </head> + + <body> + <h1>Short introduction into Interpolation</h1> + +<CENTER> +<I>This part of help documentation was last updated on August. 26, 1998 for kwave 0.29 alpha</I> +</CENTER> +<p> + +To interpolate means, to find a rule to create points in between a given set of points based on a rule. Kwave supports different types of interpolation (rules). +<hr> +The most simple one is the linear Interpolation. +<p> +<img src="linear.gif"> +<p> +The linear interpolation connects all dots through a line. All points on the line define the points in between. This provides a statistical most errorfree interpolation, but does not provide a smooth function, that may be desirable for audio purposes ( )<p> +<hr> +A little bit more complex is the polynomial interpolation. +<p> +<img src="polynom1.gif"> +<p> +The rule is to find a polynom that goes though all the given points. Based on the function of the polynom the points in between are calculated. This has some +disadvantages.<p> +When changing one point, the whole interpolation function is affected, an effect that may not be desired. Here is an example that demonstrates this effect. +The red point was moved, and a new was function interpolated. +Both functions differ slightly at every point, but the given ones, only becauseone value was changed! +<p> +<img src="overlay.jpg"> +<p> +Furthermore the polynomial function is smooth, but does describe a rather long way. It oscillates around the given points. An interpolation rule (the cubic spline interpolation) that keeps closer to the points, while staying smooth will be introduced later<p> +<hr> +To be able to adjust one point, while leaving the rest of the interpolation untouched, one has take to a few points interpolate them, then take the next ones, and so on. This has been implemented in the piecewise polynomial interpolation, that takes 3 (5 or 7) points, interpolates them, but adds to the function only the interpolation between the first and the second point. Here follows an example for a 3rd order polynom. The red line is the unseen part of the interpolation starting at the point before the red one. +<p> +<img src="3polynom-mod.gif"> +<p> +You can also see, that this solution has a flaw. You don't get a smooth function, because the interpolated pieces don't fit together all the times. So perhaps there should be a found a rule, that allows to fit them together. +<hr> +The cubic spline interpolation does this. It calcuates a derrivative at the end points of the piecewise interpolation and assures the derrivative is equal for both pieces of interpolation. This minimizes the statistical function error while still obtaining a smooth function. The advantage of local editing still exists. +<p> +<img src="spline1.gif"> +<p> +One way to obtain derrivation of a point would be to take the two nearest point, draw a line through them, and then take the derrivation of this line. +<hr> +The next interpolation is again a rather simple one. It keeps the function on the value of the last point until a new point is given. This provides none of the above discussed properties, while still being useful for some effects or models. (for example a function of Pitch). +<p> +<img src="sample-hold.gif"> +<p> +<a href="index.html">Go back to Main Help Page</a> +</body> +</html> + + +

Powered by

Apache Allura™