Linify: def. to turn something into a line.
I saw someone the other day using a variable density Hilbert curve to transform an image so as to be projectable using a laser and galvo mirror setup, this was on the facebook page Aesthetic Function Graphposting.
That kinda stayed in the back of my mind until I saw This Old Tony’s youtube video about making a (computer numeric controlled) CNC etch-a-sketch. The hardware looked great, but he was just using lame line sketches of outlines, the kind of limited images which you’d expect from an etch-a-sketch.
So, I put together a lazy easy way of turning a photo into a single line. Here is an example of my face. If you move quite far back from the image, it looks pretty good.
It uses a variable amplitude sinewave patches, which have a whole number of periods. Sine waves are used due to their simplicity and their periodicity. The same effect could be achieved with other periodic waves, in fact, square waves may be more efficient for CNC applications, as fewer commands would need to be issued to the orthogonal control axes.
The image is first downsampled to a resolution around 64 px on an edge. Then, for each pixel in the downsampled image, an approximate sinewave is found. Blocks of sine waves are added to a chain, which runs over the whole image. Currently, it raster scans, but it would be pretty easy to go R->L on even and L->R on odd. To improve the contrast, the downscaled image is contrast stretched to the 0-1 interval, and then each intensity value is raised to the (default) power of 1.5, this gives a little more contrast in the lighter regions and compensates somewhat for the low background level and the non-linear darkening with greater amplitude. This could be tuned better.
There are several factors which alter the contrast. The cell size, frequency, number of rendered points, linewidth, and also the gamma-like factor applied to the image before the linify. Images were then optimised by adjusting the gamma-like factor. Optimal values were found between ~0.75 and ~1.65. Higher values were better for darker images. Successive parabolic interpolation (SPI) was used, four values were selected at random, and their error with respect to the original image was found. These values were then used to fit a parabola, and the minima of the parabola were used as a new point. This process was iterated with the four best values being used for the fit. This process can be seen in the figures below. In the first figure, four points (the blue stars) are found. The first parabola is fitted, and the red point is the predicted best parameter value. In the second figure, the blue star shows the true value of the metric we are trying to minimise, it is slightly different than predicted. A new estimated best parameter value (green) is found. And so on. To ensure that the parabola is only used near the minima of the function, the points farthest from the minima are discarded. Typically, only three points are used, my implementation uses four for robustness, which is helpful as the curve is non-analytic and non-smooth.
A few starting locations were checked to ensure that the minimum found was global. This kind of root finding, SPI, is very simple and commonly used. It converges faster than a basic line search (about 33% faster) but does not always converge to the local extremum. Parabola are used, as the region around any minima or maxima of any function can be approximated by a parabola, which can be observed by the Taylor expansion about an extremum.
Whilst we have a non-linear intensity response and some artefacts from the process, it is much easier to get this kind of process to well represent real images than the skittliser process, as we have a wide range of possible intensity levels.
Of course, one of the issues with using this method on something like an etch-a-sketch is the extremely long paths taken without any kind of reference positions. Modifications could be made internally to an etch-a-sketch to have homing buttons which would click at certain x or y positions, thus giving the control system a method of resetting itself at least after every line. A much more difficult, but potentially interesting closed-loop control system would be using information from a video feed pointed at the etch-a-sketch. Taking the difference of successive frames would likely be a good indication of the tip location.
Finally, here is a landscape photograph of a body of water at sunrise. Just imagine it in dusky pink.