Nick with a painting by Katie P. on his back. I did a touch of cleanup on this picture and it's agreed it would make a fine album cover.

Liz with her hawk standing tall

lucky for me I have found hairspray at my parents' house. Oh I shall have fun tomorrow in chruch getting eyed over by a bunch of shallow people cause of my hawk and clothing.

The first day of class for the spring semester. Although not attending classes, I showed up to being cookies for Altered Reality.

so you are a cookie then? if so what kind? I know you can't help your spelling at all but that turned out rather amusing.

Hey... that's pretty good for a guy who could barely breath all week!

Gosh, I look like I'm in that picture two times.

Rather abstract shot of one of the blue lights from last night's gathering

Gathering this evening which involved a lot of florescent paint

Last night Elizabeth got a hybrid moehawk

Interpolation is a method in which the values of unknown data are filled using known data. It is basically making an educated guessing--hypnotising. I've used interpolation a number of times in the field and today I'm going to write a little about it.

I've most often used linear interpolation. This is the easiest to implement and when filling in small gaps of data, quite sufficient. For larger gaps between data, one method is polynomial interpolation.

Here we have two singles, both missing 98% of the real data. The first chart uses linear interpolation to fill in the gaps. The second uses polynomial interpolation. The real signal is in green and the interpolated signal is in red. The black dots denote locations where data is present. Note that most of the green is covered because both systems of interpolation reconstruct the signal fairly well. However, aside from the ends, the polynomial interpolation is a much closer fit. This is evident when closely examining a segment of the chart.

Here is the equation for polynomial interpolation:

This expands to:

What this equation does is predict any new point with the existing point using all of the data from the existing points. This function creates a polynomial that intercepts each of the known points. Since this is a polynomial and all polynomials are continuous function, a value of y can be obtained for any value x.

It looks messy, but as source code turns into a couple of nested for-loops.

Because of the multiplication and division, the process is fairly CPU intensive. And the more data points, the longer this interpolation takes.

Polynomial interpolation can end up producing significantly worse results then linear interpolation when there are not enough data points.

Using a 5 Hz single

Using a 7 Hz single

Using a 7 Hz single

I've most often used linear interpolation. This is the easiest to implement and when filling in small gaps of data, quite sufficient. For larger gaps between data, one method is polynomial interpolation.

Here is the same function, but zoomed into a specific area. The linear interpolation's weak fit is much more apparent, where the polynomial interpolation is pretty much a perfect fit.

What this equation does is predict any new point with the existing point using all of the data from the existing points. This function creates a polynomial that intercepts each of the known points. Since this is a polynomial and all polynomials are continuous function, a value of y can be obtained for any value x.

It looks messy, but as source code turns into a couple of nested for-loops.

`//--------------------------------------------------------------------------- `

// Polynomial Interpolate

//---------------------------------------------------------------------------

function PolynomialInterpolate( $x , $Data )

{

$y = 0.0;

foreach ( $Data as $Y_Index => $Y_DataPoint )

{

$y_n = $Y_DataPoint[ "y" ];

$x_n = $Y_DataPoint[ "x" ];

$Numerator = 1.0;

$Denominator = 1.0;

foreach ( $Data as $X_Index => $X_DataPoint )

{

if ( $Y_Index == $X_Index )

continue;

$Numerator *= $x - $X_DataPoint[ "x" ];

$Denominator *= $x_n - $X_DataPoint[ "x" ];

}

$y += ( $Numerator / $Denominator ) * $y_n;

}

return $y;

}

Polynomial interpolation can end up producing significantly worse results then linear interpolation when there are not enough data points.

Using a 5 Hz single

Using a 7 Hz single

Using a 7 Hz single

The above charts show polynomial interpolation without enough sample points to discern the frequencies. Note how the interpolation still passes through each of the data points, but can become radically different between sample points. Linear interpolation actually ends up making a better fit under these conditions. Notice how the ends of the function are worse then the middle. This seems typical of polynomial interpolation.

There are other types of interpolation, such a bilinear and trilinear as well as the more popular Spline interpolation. I will probably write about these methods latter on.

There are other types of interpolation, such a bilinear and trilinear as well as the more popular Spline interpolation. I will probably write about these methods latter on.

I had a nice white-knuckled drive to Madison today for an interview. Turns out the snow storm last night replaced I-90 with a sheet of ice. Travel speeds were no more then 45 MPH with a lot of bumper to bumper. I noticed the ice shortly after getting onto 90. The off ramp was covered in a fair amount of snow, the the interstate looked clear. So after I reached what looked to be open road, I hit the accelerator to speed up. Shortly after I started accelerating, my back tires started spinning freely. I slowed down after that and pass no less then 10 cars in the ditch. Longest drive to Madison I've ever taken.

Pictured is Liz who fell in the snow walking around the driveway.

Pictured is Liz who fell in the snow walking around the driveway.

I'm fairly impressed. Last night I posted my resume on dice.com and all day I've been getting phone calls from staffing agencies--and that's not a bad thing :)

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