Vectorization of binary grayscale image

Question

I have binary grayscale bitmap images (black and white) that contain lines, curves and some simple shapes (ellipses, and polygones), my goal is to describe these elements as formulas.

One of the options is to apply vectorization on the images, but I am not expert in this domain so I need your help in suggesting what can I do. is there any tool or library that is able to provide the formulas that describe these objects?

Thank you

Answer 1

Perhaps cubic bezier is what you need:

This is a project I've done, (1) I use Ramer-Douglas-Peucker to remove noise and (2) represent the curves as cubic bezier I obtain by using least square fitting:

This is original drawing:

Vectorized image:

Since it's already converted to mathematical formula, It can be zoomed infinitely.

Sorry I can't share the code since it's quite enormous but I hope you get the idea.

If you want tracing library you can use this: http://potrace.sourceforge.net/

Also If you're interested to only remove noise you can try CSS: http://www.morethantechnical.com/2012/12/07/resampling-smoothing-and-interest-points-of-curves-via-css-in-opencv-w-code/

Answer 2

If you have nice proper uninterrupted shapes you can just trace their contours using something like findContours().. But if your input (that you did not describe properly) is noisy and sketchy, the approach should rely on a Hough transform, see below. By the same coin, in fitting curves a lot will depend on the level of noise and the presence of outliers (e.g. background elements that aren't shapes or are inaccurate shapes that only approximate, say a proper ellipse). It is hard to imagine proper clean lines and proper shapes in a typical task unless it is a homework.

Hough lines and Hough circles are the most widely used functions in openCV library. Note that fitting ellipses is non-trivial since they have 5 parameters (lines have 2 and circles have 3) and Hough space grows too much. Rectangles can be found either with Hough lines or a special rectangle Hough. Other shapes can be detected using generalized non-parametric Hough.

Fitting curves should use RANSAC to get rid of outliers, and geometric (least square in terms of point distances) fit to minimize pixel noise. The latter procedure typically involves non-linear optimization that should be initialized by a simpler algebraic fit. Luckily, for simple geometric primitives fitting functions have already been written, see fitLine().

The bottom line, given that your shapes are a bit noisy, your task is non-trivial (to the degree you probably don't realize) and thus should be split on several sub-projects like finding shapes, fitting curves, etc.