The goal is to do simple vector image editing by dragging the mouse across the screen like a polygon-shaped paintbrush, creating the Minkowski sum of the brush and the path of the mouse. The new polygon would be subtracted from any previously existing polygons of a different color and merged with any existing polygons of the same color.
The plan is to take each mouse movement as a line segment from the mouse's previous position to its current position, calculate the Minkowski sum on that line segment, then do use the Weiler–Atherton clipping algorithm to update the existing polygons to include that Minkowski sum.
Since it seems likely that Weiler–Atherton would cause UI delays if run for every mouse movement, we plan to delay that step by putting it into another thread that can take its time catching up to the latest mouse movements, or alternatively save all Weiler–Atherton calculations until the drawing is finished, then do it as a bulk operation when saving. We worry that this may lead to the accumulation of a very large number of overlapping polygons, to the point that UI would be delayed by the time required to render them all.
The question is: Is the above plan the way that Inkscape and other serious vector graphics editing software would perform this operation? It seems like a mad plan, both in the trickiness of the algorithm and its computational complexity. What would an expert do?
Another option under consideration: Do the painting using simple raster operations and then convert the raster into a vector image as the final step. Conversion from raster to vectors seems no less tricky than Weiler–Atherton, and quality of the final output might suffer, but could it be the better option?
While the user is holding down the mouse button and drawing, you can remember all the mouse movement line segments, and simultaneously render the brush*line Minkowski sums to a screen resolution bitmap.
You can use the bitmap for painting the screen until the user releases the button. At that time you can calculate the union of all the line segment Minkowski sums and add the resulting shape to your drawing.
To calculate the union of so many shapes simultaneously, some kind of sweep-line algorithm would be best. You should be able to do the job in O(N log N) or linear time, which will not cause any noticeable delay.
IMO the bottleneck in Weiler-Atherton is the detection of the intersections, which requires O(N²) operations when applied by brute-force. the rest of the processing is a reorganization of the links between the vertices and intersections, which should be limited to O(N), or even O(NI), where NI denotes the number of intersections.
In this particular case, you can probably speed-up the search for intersections by means of gridding or a hierarchy of bounding boxes. (Note that gridding parallels the idea of Matt to use auxiliary bitmap rendering.)
Unless you really have billion edges, I wouldn't fear the running time.
If you want to do this like professional vector graphics softwares do with arbitrary brush shapes (including features such as the brush shape can react to speed or stylus pressure), it can be quite involved unfortunately.
I was experimenting with the method described in "Ahn, Kim, Lim - Approximate General Sweep Boundary of a 2D Curved Object". It seems to handle a lot of cases that I would expect from a professional drawing application —especially the details where the brush shape can be dynamic while sweeping; adaptively generate the boundary curve for the resolution you want; variable width offsetting of 2d curves; etc..
It seems you can simplify this method if you don't need a generalised method. But I would like to add it here as a reference.
PS: I was searching for a non-paywalled link to include here in the answer and then this came up – Looking for an efficient algorithm to find the boundary of a swept 2d shape. Looks very similar to what you're trying to do :).
I'm working on a simple mapping application for fun, and one of the things I need to do is to find (and color) all of the points that are visible from the current location. In this case, points are pixels. My map is a raster image where transparent pixels are open space, and any other pixels are opaque. (There are no semi-transparent pixels; alpha is either 0 or 100%.) In this sense, it's sort of like a regular flood fill, with the constraint that each filled pixel has to have a clear line-of-sight to the origin point. The following image shows a couple of such areas colored in (the tiny crosshairs are the origin points, and white = transparent):
(http://tinyurl.com/nf3nqa4)
In addition, what I am ultimately interested in are the points that "border" other colors, i.e., I want the list of points that make up the edge of the visible region.
My current and very inefficient solution is the modified flood-fill I described above. This approach returns correct results, but due to the need to iterate every pixel on a line to the origin for every pixel in the flood fill, it's very slow. My images are downsized and quantized, but I still need about 1MP for acceptable accuracy, and typical LoS areas are at least 100,000 pixels each.
I may well be using the wrong search terms, but I haven't been able to find any discussion of algorithms that would solve this (rasterized) LoS case.
I suspect that this could be done more efficiently if your "walls" were represented as equations rather than simply pixels in a raster image. For example, polygons/triangles, circles, ellipses.
It would then be like raytracing (search for this term) in 2D. In other words, you could consider the ray/line from each pixel in the image to the point of interest and color the pixel only if it does not intersect with any object.
This method does require you to test the intersection for each pixel in the image with each object; however, if you look up raytracing you will find a number of efficient methods for testing these intersections. They will mostly be for the 3D case but it should be straightforward to convert them to 2D.
There are 3D raytracers that are very fast on MUCH larger images so this should be very doable.
You can try a delaunay triangulation on each color. I mean you can try to find the shape of each color with DT.
I'm doing a forward image warping with control points but as expected from any forward warping, target coordinates are non integers so it creates holes on the target image. Since the transform in warping is different for every pixel in the source image, i couldn't find how can I calculate inverse transform for every pixel in the destination image to do this as a backward warping and use bilinear interpolation.
So in short my questions are:
1 - Is this possible to calculate inverse transform for each destination pixel to do this as a backward warping?
2 - If I am forced to do it as a forward warping how I can take care of the holes in target image. Simply using nearest integers creates holes, distributing the same color to neighboring pixels take care of the problem but with heavy aliasing so i believe there is a better way to do this.
Any help is appreciated. Thanks.
You need to do bilinear filtering two times. First one to distribute the transform to pixels around the target pixel when the the target coordinate during forward warping is non integer. Then using these distributed transforms as inverse you'll do a backward warping and use bilinear interpolation again when sampling the pixels from the source
I am looking for an algorithm that takes vector image data (e.g. sets of edges) and interpolate another set of edges which is the "average" of the two (or more) sets.
To put it in another way, it is just like Adobe Flash where you "tween" two vector images and the software automatically computes the in-between images. Therefore you only specify the starting image and end image, then Flash takes care of all the in-between images.
Is there any established algorithm to do this? Especially in cases like different number of edges?
What exactly do you mean by edges? Are we talking about smooth vector graphics that use curves?
Well a basic strategy would be to simply do a linear interpolation on the points and directions of your control polygon.
Basically you could simply take two corresponding points (one of each curve/vector form) and interpolate them with:
x(t) = (1-t)*p1 + t*p2 with t in [0,1]
(t=0.5 would then of course give you the average between the two)
Since vector graphics usually use curves you'd need to do the same with the direction vector of each control point to get the direction vector of the averaged curve.
One big problem though is to match the right points of each control polygon, especially if both curves have a different degree. You could try doing a degree elevation on one to match the degree of the other and then one by one assign them to each other and interpolate.
Maybe that helps...
I've been working on a visualization project for 2-dimensional continuous data. It's the kind of thing you could use to study elevation data or temperature patterns on a 2D map. At its core, it's really a way of flattening 3-dimensions into two-dimensions-plus-color. In my particular field of study, I'm not actually working with geographical elevation data, but it's a good metaphor, so I'll stick with it throughout this post.
Anyhow, at this point, I have a "continuous color" renderer that I'm very pleased with:
The gradient is the standard color-wheel, where red pixels indicate coordinates with high values, and violet pixels indicate low values.
The underlying data structure uses some very clever (if I do say so myself) interpolation algorithms to enable arbitrarily deep zooming into the details of the map.
At this point, I want to draw some topographical contour lines (using quadratic bezier curves), but I haven't been able to find any good literature describing efficient algorithms for finding those curves.
To give you an idea for what I'm thinking about, here's a poor-man's implementation (where the renderer just uses a black RGB value whenever it encounters a pixel that intersects a contour line):
There are several problems with this approach, though:
Areas of the graph with a steeper slope result in thinner (and often broken) topo lines. Ideally, all topo lines should be continuous.
Areas of the graph with a flatter slope result in wider topo lines (and often entire regions of blackness, especially at the outer perimeter of the rendering region).
So I'm looking at a vector-drawing approach for getting those nice, perfect 1-pixel-thick curves. The basic structure of the algorithm will have to include these steps:
At each discrete elevation where I want to draw a topo line, find a set of coordinates where the elevation at that coordinate is extremely close (given an arbitrary epsilon value) to the desired elevation.
Eliminate redundant points. For example, if three points are in a perfectly-straight line, then the center point is redundant, since it can be eliminated without changing the shape of the curve. Likewise, with bezier curves, it is often possible to eliminate cetain anchor points by adjusting the position of adjacent control points.
Assemble the remaining points into a sequence, such that each segment between two points approximates an elevation-neutral trajectory, and such that no two line segments ever cross paths. Each point-sequence must either create a closed polygon, or must intersect the bounding box of the rendering region.
For each vertex, find a pair of control points such that the resultant curve exhibits a minimum error, with respect to the redundant points eliminated in step #2.
Ensure that all features of the topography visible at the current rendering scale are represented by appropriate topo lines. For example, if the data contains a spike with high altitude, but with extremely small diameter, the topo lines should still be drawn. Vertical features should only be ignored if their feature diameter is smaller than the overall rendering granularity of the image.
But even under those constraints, I can still think of several different heuristics for finding the lines:
Find the high-point within the rendering bounding-box. From that high point, travel downhill along several different trajectories. Any time the traversal line crossest an elevation threshold, add that point to an elevation-specific bucket. When the traversal path reaches a local minimum, change course and travel uphill.
Perform a high-resolution traversal along the rectangular bounding-box of the rendering region. At each elevation threshold (and at inflection points, wherever the slope reverses direction), add those points to an elevation-specific bucket. After finishing the boundary traversal, start tracing inward from the boundary points in those buckets.
Scan the entire rendering region, taking an elevation measurement at a sparse regular interval. For each measurement, use it's proximity to an elevation threshold as a mechanism to decide whether or not to take an interpolated measurement of its neighbors. Using this technique would provide better guarantees of coverage across the whole rendering region, but it'd be difficult to assemble the resultant points into a sensible order for constructing paths.
So, those are some of my thoughts...
Before diving deep into an implementation, I wanted to see whether anyone else on StackOverflow has experience with this sort of problem and could provide pointers for an accurate and efficient implementation.
Edit:
I'm especially interested in the "Gradient" suggestion made by ellisbben. And my core data structure (ignoring some of the optimizing interpolation shortcuts) can be represented as the summation of a set of 2D gaussian functions, which is totally differentiable.
I suppose I'll need a data structure to represent a three-dimensional slope, and a function for calculating that slope vector for at arbitrary point. Off the top of my head, I don't know how to do that (though it seems like it ought to be easy), but if you have a link explaining the math, I'd be much obliged!
UPDATE:
Thanks to the excellent contributions by ellisbben and Azim, I can now calculate the contour angle for any arbitrary point in the field. Drawing the real topo lines will follow shortly!
Here are updated renderings, with and without the ghetto raster-based topo-renderer that I've been using. Each image includes a thousand random sample points, represented by red dots. The angle-of-contour at that point is represented by a white line. In certain cases, no slope could be measured at the given point (based on the granularity of interpolation), so the red dot occurs without a corresponding angle-of-contour line.
Enjoy!
(NOTE: These renderings use a different surface topography than the previous renderings -- since I randomly generate the data structures on each iteration, while I'm prototyping -- but the core rendering method is the same, so I'm sure you get the idea.)
Here's a fun fact: over on the right-hand-side of these renderings, you'll see a bunch of weird contour lines at perfect horizontal and vertical angles. These are artifacts of the interpolation process, which uses a grid of interpolators to reduce the number of computations (by about 500%) necessary to perform the core rendering operations. All of those weird contour lines occur on the boundary between two interpolator grid cells.
Luckily, those artifacts don't actually matter. Although the artifacts are detectable during slope calculation, the final renderer won't notice them, since it operates at a different bit depth.
UPDATE AGAIN:
Aaaaaaaand, as one final indulgence before I go to sleep, here's another pair of renderings, one in the old-school "continuous color" style, and one with 20,000 gradient samples. In this set of renderings, I've eliminated the red dot for point-samples, since it unnecessarily clutters the image.
Here, you can really see those interpolation artifacts that I referred to earlier, thanks to the grid-structure of the interpolator collection. I should emphasize that those artifacts will be completely invisible on the final contour rendering (since the difference in magnitude between any two adjacent interpolator cells is less than the bit depth of the rendered image).
Bon appetit!!
The gradient is a mathematical operator that may help you.
If you can turn your interpolation into a differentiable function, the gradient of the height will always point in the direction of steepest ascent. All curves of equal height are perpendicular to the gradient of height evaluated at that point.
Your idea about starting from the highest point is sensible, but might miss features if there is more than one local maximum.
I'd suggest
pick height values at which you will draw lines
create a bunch of points on a fine, regularly spaced grid, then walk each point in small steps in the gradient direction towards the nearest height at which you want to draw a line
create curves by stepping each point perpendicular to the gradient; eliminate excess points by killing a point when another curve comes too close to it-- but to avoid destroying the center of hourglass like figures, you might need to check the angle between the oriented vector perpendicular to the gradient for both of the points. (When I say oriented, I mean make sure that the angle between the gradient and the perpendicular value you calculate is always 90 degrees in the same direction.)
In response to your comment to #erickson and to answer the point about calculating the gradient of your function. Instead of calculating the derivatives of your 300 term function you could do a numeric differentiation as follows.
Given a point [x,y] in your image you could calculate the gradient (direction of steepest decent)
g={ ( f(x+dx,y)-f(x-dx,y) )/(2*dx),
{ ( f(x,y+dy)-f(x,y-dy) )/(2*dy)
where dx and dy could be the spacing in your grid. The contour line will run perpendicular to the gradient. So, to get the contour direction, c, we can multiply g=[v,w] by matrix, A=[0 -1, 1 0] giving
c = [-w,v]
Alternately, there is the marching squares algorithm which seems appropriate to your problem, although you may want to smooth the results if you use a coarse grid.
The topo curves you want to draw are isosurfaces of a scalar field over 2 dimensions. For isosurfaces in 3 dimensions, there is the marching cubes algorithm.
I've wanted something like this myself, but haven't found a vector-based solution.
A raster-based solution isn't that bad, though, especially if your data is raster-based. If your data is vector-based too (in other words, you have a 3D model of your surface), you should be able to do some real math to find the intersection curves with horizontal planes at varying elevations.
For a raster-based approach, I look at each pair of neighboring pixels. If one is above a contour level, and one is below, obviously a contour line runs between them. The trick I used to anti-alias the contour line is to mix the contour line color into both pixels, proportional to their closeness to the idealized contour line.
Maybe some examples will help. Suppose that the current pixel is at an "elevation" of 12 ft, a neighbor is at an elevation of 8 ft, and contour lines are every 10 ft. Then, there is a contour line half way between; paint the current pixel with the contour line color at 50% opacity. Another pixel is at 11 feet and has a neighbor at 6 feet. Color the current pixel at 80% opacity.
alpha = (contour - neighbor) / (current - neighbor)
Unfortunately, I don't have the code handy, and there might have been a bit more to it (I vaguely recall looking at diagonal neighbors too, and adjusting by sqrt(2) / 2). I hope this enough to give you the gist.
It occurred to me that what you're trying to do would be pretty easy to do in MATLAB, using the contour function. Doing things like making low-density approximations to your contours can probably be done with some fairly simple post-processing of the contours.
Fortunately, GNU Octave, a MATLAB clone, has implementations of the various contour plotting functions. You could look at that code for an algorithm and implementation that's almost certainly mathematically sound. Or, you might just be able to offload the processing to Octave. Check out the page on interfacing with other languages to see if that would be easier.
Disclosure: I haven't used Octave very much, and I haven't actually tested it's contour plotting. However, from my experience with MATLAB, I can say that it will give you almost everything you're asking for in just a few lines of code, provided you get your data into MATLAB.
Also, congratulations on making a very VanGough-esque slopefield plot.
I always check places like http://mathworld.wolfram.com before going to deep on my own :)
Maybe their curves section would help? Or maybe the entry on maps.
compare what you have rendered with a real-world topo map - they look identical to me! i wouldn't change a thing...
Write the data out as an HGT file (very simple digital elevation data format used by USGS) and use the free and open-source gdal_contour tool to create contours. That works very well for terrestrial maps, the constraint being that the data points are signed 16-bit numbers, which fits the earthly range of heights in metres very well, but may not be enough for your data, which I assume not to be a map of actual terrain - although you do mention terrain maps.
I recommend the CONREC approach:
Create an empty line segment list
Split your data into regular grid squares
For each grid square, split the square into 4 component triangles:
For each triangle, handle the cases (a through j):
If a line segment crosses one of the cases:
Calculate its endpoints
Store the line segment in the list
Draw each line segment in the line segment list
If the lines are too jagged, use a smaller grid. If the lines are smooth enough and the algorithm is taking too long, use a larger grid.