## Graph filters

I've put together two graph filters that can be applied to the VD.

The first one detects the interior or exterior of a polygon. When the VD is constructed the polygon boundary must be input in CW order, and any islands inside the polygon in CCW order (or vice versa). This allows running other downstream algorithms only on the parts of the VD that pass the filter. Like these exterior and interior offsets:

The other filter looks at the interior VD and tries to produce an approximate medial axis. We can start with the complete interior VD, such as this "J":

By definition the medial axis consists of "the set of all points having more than one closest point on the object's boundary". The separator edges shown in purple above can clearly be eliminated, since their adjacent/defining sites are an open line-segment and the segment's endpoint. Removing separators gives us this:

Now we can either finish here, or try to filter out some more edges to make it look better. Since we approximated smooth curves with line-segments we should try to detect which parts of the boundary are really distinct curves, and which are merely many consecutive line-segments approximating a single smooth curve. I've compared the dot-product (angle) between two consecutive segments, and applied an arbitrary threshold:

For the whole alphabet it looks like this.

The choice of threshold value for the angle-filtering is arbitrary. In many cases such as "x" and "m" it results in small or large left-over branches. This could probably be avoided by (1) tuning the angle-threshold, (2) approximating smooth curves with a larger number of line-segments, (3) eliminating branches below a certain length, or (4) choosing a font that's made for v-carving (are there any?).

Although it's probably not right to call it a "medial axis" , the same filter applied to the exterior VD also looks interesting. It divides the plane into organic looking shapes around each letter. It could probably be used for a lot of shape analysis. For example in a smart pocketing routine to find large areas that can be cleared with a large cutter, before a smaller cutter is required for the details. Note that in addition to the geometric shape of all the blue-ish edges the diagram also holds distance-information at each point of an edge. The distance stored is a clearance-disk radius, i.e. we can draw a circle at any point of an edge with this radius, and no input geometry (in yellow) will intersect the circle.

## 8-channel 4th-order 60 kHz anti-alias low-pass filter

I used this Sallen-Key design to build an 8-channel 4th order low-pass anti-alias filter for a 16-bit 200 kS/s +/- 10 V AD-Converter. I calculated the components for the 60 kHz low-pass Butterworth design with this on-line calculator. Previously I've used the MAX274, but that component is limited to +/- 5 V signals. Here I really need the +/- 10 V voltage swing. The exact design calls for 2872 pF, 2452 pF, 6935 pF, and 1016 pF capacitors, but I looked at the transfer function with what values were available in 1% tolerance from Farnell, and the response looked fine with (R= 1 k, C1=C2= 2700 pF for the first stage and C1=6800 pF, C2=1000 pF for the second stage). Both the resistors and capacitors (~1.5 eur/pcs!) have a tolerance of 1 %, which according to a monte-carlo simulation should not affect the response that much. I'm using OP42 op-amps with a unity-gain bandwidth of 10 MHz, which should be adequate (100x the cut-off frequency was recommended in a guide I read, that would be 6 MHz in this case).

For testing I hooked up a signal generator and an oscilloscope and wrote a LabVIEW program to loop trough around 250 different frequencies while recording the peak-to-peak value of the filter input and output signals. The oscilloscope only has an 8-bit AD converter, but I adjusted the analogue gain between 5 V/div and 2 mV/div to achieve effectively around 16-bit dynamic range.

This is the result of testing all channels with a 20 Vpp sine wave between 100 Hz and 10 MHz. The blue curve shows the design response and the red and green curves show the maximum and minimum expected response from the monte-carlo simulation (I drew all component values from normal distributions with 1 % standard deviations). Pretty nice agreement until ~500 kHz. Here's another view of the data:

This figure shows the deviation of the real filters from the design response, again confirming that everything works as it should up to 500 kHz.

Log-log plots can be confusing, so here's a semilog plot and a linear plot of the same data:

Here are the source files for this design:

The box actually looks like this.