So I came upon an interesting algorithms problem and I'm trying to think of the most efficient solution. So there is a 2d array, and each square in the array is either red, white, or blue. We want to count the number of subrectangles of the array which contain at least one red square, no blue squares, and white squares don't matter. What is the best way to go about this?
It seems like a simple problem statement. Clearly we can iterate through all possible subgrids and test if they work, but this is very inefficient. Any help? Thanks.
You need a way to avoid duplicates; for example, you can identify a rectangle using x0,y0,x1,y1.
open, valid = empty sets of rectangles
add all red squares to a set `open`
while not `open` empty,
take the first `open` rectangle
for each direction,
try to grow it 1 square in that direction
if growth possible (not outside matrix, not blue),
and result not in `valid` or `open`,
add to `open`
add it to `valid`
The size of valid is the count you seek. Could probably be more optimal, but at least it seems correct, and should be substantially faster than looking at all possible rectangles because it a) never looks a rectangles without a red, and b) never keeps looking once a rectangle would involve a blue.
A "look at all possible rectangles" algorithm on a size-n grid would have to look at ~n^4 rectangles (back-of-the envelope, as there are n^2 x0,y0 pairs & n^2 x1,y1 pairs); and, for each rectangle, check that there is a blue and a red (so, O(n^5) is a low-ball estimate; O(n^6) may be closer to the mark). Importantly, it will have that cost regardless of distribution of red & blue. My proposal has better complexity if there are less valid results: O(n^2) if there are no reds, or if they are all surrounded by blues (checkerboard pattern). It is also of lower complexity in the worst case, as it only grows when valid -- probably around O(n^4) worst-case, defined as all-red map.
Related
I have a set of pages that look like this:
I have the content in grids with * Heights and Widths so the grid correctly scales when the entire window resizes. I would like the text to resize with the grid. Basically I would like the user to resize from this:
To this:
(preserving white space)
One way to do this would be to wrap the TextBlock in a ViewBox with margins on the right and bottom (for Grid.Row="3") to account for white space. But because I have several pages with different lengths and line counts I would have to set the margin specifically for each page otherwise the text sizes would differ on each page. Is there a better way to do this??
I don't think there is a better way to do this. There are different ways. But, I think it isn't just a matter of opinion that they would not be better.
Ways I can think of.
Render your text offscreen, rendertargetbitmap that so you've got a picture. Change your textblocks on screen to images and stretch them.
Or
Work out the size your text wants to be. Then do some calculation comes up with a different fontsize which is "better". This is a lot easier to write a description of than do.
In my opinion.
A viewbox is easier to implement. Way less error prone than calculations. Will give at least as good results as rendering to a picture.
I just want to add one more solution to the ones suggested by Andy, which is more of a scientific approach and takes a bit of practice to master.
Suppose you have to find a function F, which maps one or more variables to a desired single value. In your case that would be a function F, which takes aspect ratio of the window as input and outputs an appropriate font size.
How can you find such a function?
Well... you don't need to do any math yourself!
First, you need some data to begin with:
1. Resize the window randomly
2. Calculate aspect ration (X)
3. Pick an appropriate font size that looks good enough (Y)
4. Repeat the measurement 7 to 10 times (sorry data scientists)
5. Enter the data in Excel - one column for X and another one for Y
6. Insert a scatter chart
7. Choose the best trendline for your data, but avoid the polynomial one
8. Display the trendline equation and use the expression in your code
Now I should mention the pros and cons of this regression technique.
Pros:
1. It can solve a wide range of tricky problems:
"I use this 3rd party control, but when the text is too long it overlaps the title bar. How to trim it so it doesn't go beyond the top border?. Deadline is coming!"
2. Even if it doesn't solve the problem perfectly, the results are often acceptable
3. It takes minutes to try out unlike spending a day refreshing your math skills
Cons:
1. The biggest problem is that to keep it simple, you often lower the number of
variables by assuming some of them to be constant. In this post I've assumed that
the font family won't change for example, neither the font weight.
2. If any of the assumptions does not hold the final result could be even worse
This technique is fragile, but powerful. Use it as your last weapon and never leave magic expression like
fontSize = (int)(0.76 + 1.2 * aspectRation) without documenting how it came to be.
I'm given a 2d binary array. Some of the dots are on, some are off (1 for on, 0 for off).
I know that the "on" dots were created before by putting circles on the 2d array.
The circles are of the same radius, and each time a circle was put, the dots inside it changed to 1 instead of 0.
All the circles are within the edges of the array and dot touching the edge of the circle is lit.
An illustration can be seen below. The circles are ordered randomly and may touch.
Notice that the dots inside the circles are 1 and all other are 0.
Can you find how many circles were there just by looking at the 2d array without the circles after I had put them? Is this problem solvable?
My attempt at solving this problem was:
First, I assumed that my circles can contain dots as in the figure (radius big enough to contain 4 to 7 dots.
Then I tried to categorize what possible orientation can the circles have, however there are just a lot.
I would like to find these two circles. Notice that they can cannot overlap but can be just one near the other.
If your circles don't overlap, you can use connected component labeling algorithm and get number of circles:
NCircles = (NComponents - 1) / 2
(if inner empty regions of circles and outer empty place form separate components)
Edit: with these dots it is worth to select only connected conponents with size in some range to exclude dots and other false regions.
Simple kind of CCL suitable for this picture:
scan image until black pixel is met
do flood fill while possible, keep bounding box of scanned black pixels
if box corresponds to circle size, count it
scan further from any unmarked pixel
One more possible approach: you can try Hough algorithm for circles of predefined radius.
For example, OpenCV library contains labeling function that works with images and arrays (and Hough transform too)
Why not just generate randomly generate circles and count them?
When you insert a new circle, just check if they do not overlap.
And stop inserting new circles after you tried a certain times and failed to insert a new circle. With this last value you probably need to play a bit.
You can probably repeat this a couple of times and average the result like that.
So, to start off, I'm not very good at computer graphics. I'm trying to implement a GUI toolkit where one of the features is being able to apply 3D transformations to 2D "layers". (a layer only has one Z coordinate, as pre-transform, it's a two dimensional axis aligned rectangle)
Now, this is pretty straightforward, until you come to 3D transformations that would push the layer back, requiring splitting the layer into several polygons in order to render it correctly, as illustrated here. And because we can have transparency, layers may not get completely occluded, while still requiring getting split.
So here is an illustration depicting the issue and the desired outcome. In this scenario, the blue layer (call it B) is on top of the red layer (R), while having the same Z position (but B was added after R). In this scenario, if we rotate B, its top two points will get a Z index lower than 0 while the bottom points will get an index higher than 0 (with the anchor point being the only point/line left as 0).
Can somebody suggest a good way of doing this on the CPU? I've struggled to find a suitable algorithm implementation (in C++ or C) that would be appropriate to this scenario.
Edit: To clarify myself, at this stage in the pipeline, there is no rendering yet. We just need to produce a set of polygons for each layer that would then represent the layer's transformed and occluded geometry. Then, if required, rendering (either software or hardware) is done if required, which is not always the case (for example, when doing hit testing).
Edit 2: I looked at binary space partitioning as an option of achieving this but I have only been able to find one implementation (in GL2PS), which I'm not sure how to use. I do have a vague understanding of how BSPs work, but I'm not sure how they can be used for occlusion culling.
Edit 3: I'm not trying to do colour and transparency blending at this stage. Just pure geometry. Transparency can be handled by the renderer, and overdraw is okay. In this case, the blue polygon can just be drawn under the red one, but with more complicated cases, depth sorting or even splitting up the polygons may be required (example of a scary case like that below). Although the viewport is fixed, because all layers can be transformed in 3D, creating a shape shown below is possible.
So what I'm really looking for is an algorithm that would geometrically split layer B into two blue shapes, one of which would be drawn "above" and one of which would be drawn below R. The part "below" would get overdraw, yes, but it's not a major issue. So B just need to be split into two polygons so it would appear to cut through R when those polygons are drawn in order. No need to worry about blending.
Edit 4: For the purpose of this, we cannot render anything at all. This all has to be done purely geometrically (producing 2D polygons). This is what I was originally getting at.
Edit 5: I should note that the overall number of quads per subscene is around 30 (average). Definitely won't go above 100. Unless the layers are 3D transformed (which is where this problem arises), they are just radix sorted by Z positions before being drawn. Layers with the same Z position are drawn in order in which they were added (first in, first out).
Sorry if I didn't make it clear in the original question.
If you "aren't good with computer graphics", Doing it on CPU (software rendering) will be extremely difficult for you, if polygons can be transparent.
The easiest way to do it is to use GPU rendering (OpenGL/Direct3D) with Depth Peeling technique.
Cpu solutions:
Soltuion #1 (extremely difficult):
(I forgot the name of this algorithm).
You need to split polygon B into two, - for example, using polygon A as clip plane, then render result using painter's algorithm.
To do that you'll need to change your rendering routines so they'll no longer use quads, but textured polygons, plus you'll have to write/debug clipping routines that'll split triangles present in scene in such way that they'll no longer break paitner's algorithm.
Big Problem: If you have many polygons, this solution can potentially split scene into infinite number of triangles. Also, writing texture rendering code yourself isn't much fun, so it is advised to use OpenGL/Direct3D.
This can be extremely difficult to get right. I think this method was discussed in "Computer Graphics Using OpenGL 2nd edition" by "Francis S. Hill" - somewhere in one of their excercises.
Also check wikipedia article on Hidden Surface Removal.
Solution #2 (simpler):
You need to implement multi-layered z-buffer that stores up to N transparent pixels and their depth.
Solution #3 (computationally expensive):
Just use ray-tracing. You'll get perfect rendering result (no limitations of depth peeling and cpu solution #2), but it'll be computationally expensive, so you'll need to optimize rendering routines a lot.
Bottom line:
If you're performing software rendering, use Solution #2 or #3. If you're rendering on hardware, use technique similar to depth-peeling, or implement raytracing on hardware.
--edit-1--
required knowledge for implementing #1 and #2 is "line-plane intersection". If you understand how to split line (in 3d space) into two using a plane, you can implement raytracing or clipping easily.
Required knowledge for #2 is "textured 3d triangle rendering" (algorithm). It is a fairly complex topic.
In order to implement GPU solution, you need to be able to find few OpenGL tutorials that deal with shaders.
--edit-2--
Transparency is relevant, because in order to get transparency right, you need to draw polygons from back to front (from farthest to closest) using painter's algorithms. Sorting polygons properly is impossible in certain situation, so they must be split, or you should use one of the listed techniques, otherwise in certain situations there will be artifacts/incorrectly rendered images.
If there's no transparency, you can implement standard zbuffer or draw using hardware OpenGL, which is a very trivial task.
--edit-3--
I should note that the overall number of quads per subscene is around 30 (average). Definitely won't go above 100.
If you will split polygons, it can easily go way above 100.
It might be possible to position polygons in such way that each polygon will split all others polygon.
Now, 2^29 is 536870912, however, it is not possible to split one surface with a plane in such way that during each split number of polygons would double. If one polygon is split 29 timse, you'll get 30 polygons in the best-case scenario, and probably several thousands in the worst case if splitting planes aren't parallel.
Here's rough algorithm outline that should work:
Prepare list of all triangles in scene.
Remove back-facing triangles.
Find all triangles that intersect each other in 3d space, and split them using line of intersection.
compute screen-space coordinates for all vertices of all triangles.
Sort by depth for painter's algorithm.
Prepare extra list for new primitives.
Find triangles that overlap in 2D (post projection) screen space.
For all overlapping triangles check their rendering order. Basically a triangle that is going to be rendered "below" another triangles should have no part that is above another triangle.
8.1. To do that, use camera origin point and triangle edges to split original triangles into several sub-regions, then check if regions conform to established sort order (prepared for painter's algorithm). Regions are created by splitting existing pair of triangles using 6 clip planes created by camera origin points and triangle edges.
8.2. If all regions conform to rendering order, leave triangles be. If they don't, remove triangles from list, and add them to the "new primitives" list.
IF there are any primitives in new primitives list, merge the list with triangle list, and go to #5.
By looking at that algorithm, you can easily understand why everybody uses Z-buffer nowadays.
Come to think about it, that's a good training exercise for universities that specialize in CG. The kind of exercise that might make your students hate you.
I am going to come out say give the simpler solution, which may not fit your problem. Why not just change your artwork to prevent this problem from occuring.
In problem 1, just divide the polys in Maya or whatever beforehand. For the 3 lines problem, again, divide your polys at the intersections to prevent fighting. Pre-computed solutions will always run faster than on the fly ones - especially on limited hardware. From profesional experience, I can say that it also does scale, well it scales ok. It just requires some tweaking from the art side and technical reviews to make sure nothing is created "ilegally." You may end up getting more polys doing it this way than rendering on the fly, but at least you won't have to do a ton of math on CPUs that may not be up to the task.
If you do not have control over the artwork pipeline, this won't work as writing some sort of a converter would take longer than getting a BSP sub-division routine up and running. Still, KISS is often the best solution.
Suppose we have an image (pixel buffer) that is in black and white, so each pixel is either black or white (not gray scale).
Now somewhere in the middle of that images, place a green dot. It may have a radius of n for rendering purposed, but it is really a just point. Give the dot a randomly selected direction and speed, and start it moving. If the image is all white pixels, the dot will bounce off the edges of the image, infinitely wandering around the picture. This is quite easy... just reverse either the rise or run of the dot's vector.
Next, suppose the image has some globs of black pixels. As the dot encounters these globs of black pixels, the angle of reflection needs to be calculated. This is also quite easy of the the black pixels have a fixed slope, as in my sketch (blue X represents black pixels). You can find the slope of the blue Xs and easily calculate the new vector.
But how about the case where the black pixels form really unfriendly surfaces? What are some approaches to figuring out this angle?
This is the subject that I am interested in.
There must be some algorithms that exist for this kind of purpose, but I never ran across any in school. I am not asking how to code this, rather approaches to writing the algorithm to do this. I have a few ideas that I'll try, but if there are some standard ways to do this that exist, I'd like to learn about them.
Obviously I'd like to start with Black and White then move into RGBA.
I am looking for any reference material on just this sort of subject. Websites, books, or other references are very very welcome.
Also, if there are different StackOverflow tags that could be good, let me know.
Thanks much!
Edit********** More pics and information
Maybe I wasn't clear what I meant by "unfriendly surfaces". In the previous picture, our blue X's happened to just be a line. Picture a case where it is not a line, rather a wierd shape.
We start with our green pixel traveling at a slope of 2. Suppose it's vector is that of 12 pixels per frame. It would have a projected path like this:
But suppose instead of a nice friendly line, we have this:
In my mind I can kinda of see what is likely to happen if this were a ball and some walls.
Look for edge detection algorithms used in image processing. Some edge detectors also approximate the direction of edges.
You can think of the pixel neighborhood of the green dot, maybe somewhere between 3x3 and 7x7, as a small edge direction detection problem. One approach would take two passes at the pixels:
In the first pass, smooth the sharp black/white pixels using a Gaussian filter.
In the second pass, apply an edge detection operator, such as Sobel, Prewitt or Roberts to produce the X and Y derivatives of the pixels' intensity. You can then approximate the direction as:
angle = arctan(dx/dy)
The motivation for the smoothing pass is to give the edge detection operator information from farther-away pixels.
The Wikipedia page on the Canny edge detector has a good discussion on obtaining the direction (the "gradient") of an edge, including an example of a particular Gaussian filter you can use for smoothing.
Am doing something similar with a ball and randomly generated backgrounds.
The filter and edge detection is highly technical but all other processes using a 5*5 or 3*3 grid seem similarly difficult.
However, I think I may have a cheap way around this. Assuming a ball travelling in any direction, scan all leading edges of the ball - a semicircle. The further to the edge of the ball the collision occurs the closer to vertical is the collision. Again, I think, this should allow you to easily infer the background normal and from there the answer is fairly simple
I'm in the process of writing a C program using OpenCV to detect some rectangles made with tape, which are hollow on the inside. Problem is, each physical rectangle gives two digital rectangles: one for the inner perimeter, one for the outer perimeter. The outer rectangle in all cases completely encloses the inner rectangle.
I need some way to remove the inner rectangles, and in a reasonably efficient manner, as this is being run on a video feed and must not drop framerate considerably (approx. 15fps, on a BeagleBoard xM, which is not terribly powerful).
There are always four physical rectangles, and somewhere between four to eight digital rectangles depending on the cleanliness of the processing operations. The outer rectangle is detected reliably; the inner rectangle is not. The image is thresholded, eroded, and dilated such that the image is clean and detection is reliable in general.
I feel that this problem is separate from OpenCV and is really just working with rectangles and could probably be solved by me with some time, but the project is on a crunch deadline, so I'm also throwing this question in. Thanks in advance, guys.
there is a function called grouprectangle in opencv.
The function can remove multiple rectangles...
Have a happy coding.
Since you only have at most 8 digital rectangles, I think it would be fine to use the natural, brute force, algorithm to figure out which rectangles are inside other rectanges. It's OK to do O(N^2) algorithms when N is small, and 8 is small.
Here is the pseudo code:
for each rectangle i {
for each rectangle j {
if i != j and rectangle i is inside rectangle j {
disregard rectangle i
}
}
}
Solved - the speedy solution is to take the distance to one of the corners from the center point of the rectangle, and compare that distance between rectangles whose centers are very close together. The one with the shorter distance must be the inner rectangle.
Code-wise you'd want to calculate the center, then find, say, the bottom right point, which is just the point with both min x and min y. Calculate the distance between them and store it somehow. For each rectangle, iterate over the other ones and check if their centers are very close (a constant of ~30px works fine for this in my case). Compare the distances calculated earlier; the rectangle with the shorter distance should be deleted from the list of rectangles.