I have a math question in order to make some objective-c code for my app.
I have n images with fixed dimensions. I would like to divide these images over a view in rows and columns. The view has dimensions x / y. I want the images to have the maximum size (surface) although their dimensions are fixed.
I think this is a math problem I probably learned in school but I forgot.
I contemplated writing a bit of trial and error code but there must be a more elegant solution.
If the surface is square the solution is simply the square root. But this is not always the case.
Cheers
Nick
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There's this problem that my professor showed me that I thought was interesting.
The problem is as follows:
If you have a 2^n by 2^n checkerboard and remove one square from it, will you be able to fill it with L-shaped trominos?
The answer to this is yes, the method of which I thought was interesting.
Example
In this example, X is the removed square and the numbers represent the shape of the L-shaped trominos. In every possible permutation of checkerboards, there should be a solution to fill up every square with trominos.
Does anyone know if this kind of problem is named in a field of math? I'd love to learn more about these.
I'm also going to attempt to program this, anyone have any ideas that could help?
There's a broad family of problems called tiling problems that ask when, whether, and how to tile different shapes using a fixed collection of figures.
There are other questions about figure subdivisions, where the question is when, whether, or how to subdivide some larger figure apart into smaller figures of various shapes.
I am not sure how to put this problem in a single sentence, sorry if the title is misleading.
I am currently developing a simple terrain editor with a circle-shaped brush size. The image below shows a few cases that represent my problem.
additional info: the square size is fixed and uniform and in the current version, my concern is only to find which one is hit and which one is not (the amount of region covered is important for weighting the hit, but probably not right now)
My current solution (which is not even correct for a certain condition) is: given a hit in a position (x, y) with radius r, loop through all square from (x-radius, y-radius) to (x+radius, y+radius) and apply 2-D box to circle collision detection. But I don't think this is optimal (or even correct IMO).
Can anyone help me with this one? Thank you
Since i can't add a simple comment due to bureaucracy on this website i have to type it out here.
Anyway you're in luck since i was trying to do this recently as well! The way i did it is i iterated through the vertex array and check if the current vertex falls inside the radius of the circle. But perhaps what you want is to check it against each quad center and if that center falls inside the radius then add the whole quad as it's being collided.
Of course depending on the size of your grid the performance will vary so it's good to try to iterate through as few quads as needed. Though accessing these quads from the array is something you have to figure out yourself.
I have successfully calculated Rotation, Translation with the intrinsic camera matrix of two cameras.
I also got rectified images from the left and right cameras. Now, I wonder how I calculate the 3D coordinate of a point, just one point in an image. Here, please see the green points. I have a look at the equation, but it requires baseline which I don't know how to calculate. Could you show me the process of calculating the 3d coordinate of the green point with the given information (R, T, and intrinsic matrix)?
FYI
1. I also have a Fundamental matrix and Essential matrix, just in case we need them.
2. Original image size is 960 x 720. Rectified ones are 925 x 669
3. The green point from the left image: (562, 185), from the right image: (542, 185)
The term "baseline" usually just means translation. Since you already have your rotation, translation and intrinsics matrices (let's not them R, T and K). you can triangulate and don't need either the Fundamental or Essential matrices (they could be used to extract R, T etc but you already have them). You don't really need your images to be rectified either, since it doesn't change the triangulation process that much. There are many ways to triangulate, each with their pros and cons, and many libraries that implement them. So, all I can do here is give you and overview of the problem and potential solutions, as well as pointers to resources that you can either use as their are or as a source of inspiration to write your own code.
Formalization and solution outlines. Let's formalize what we are after here. You have a 3d point X, with two observations x_1 and x_2 respectively in the left and right images. If you backproject them, you obtain two rays:
ray_1=K^{1}x_1
rat_2=R*K^{-1}x_2+T //I'm assuming that [R|T] is the pose of the second camera expressed in the referential of the first camera
Ideally, you'd want those two rays to meet at point X. Since in practice we always have some noise (discretization noise, rounding errors and so on) the two rays wont meet at X, so the best answer would be a point Q such that
Q=argmin_X {d(X,ray_1)^2+d(X,ray_2)^2}
where d(.) denotes the Euclidian distance between a line and a point. You can solve this problem as a regular least squares problem, or you can just take the geometric approach (called midpoint) of considering the line segment l that is perpendicular to both ray_1 and ray_2, and take its middle as your solution. Another quick and dirty way is to use the DLT. Basically, you re-write the constrains (i.e. X should be as close as possible to both rays) as a linear system AX=0 and solve it with SVD.
Usually, the geometric (midpoint) method is the less precise. The DLT based one, while not the most stable numerically, usually produces acceptable results.
Ressources that present in depth formalization
Hartley-Zisserman's book of course! Chapter 12. A simple DLT-based method, which is the one used in opencv (both in the calibration and sfm modules) is explained on page 312. It is very easy to implement, it shouldn't take more that 10 minutes in any language.
Szeliski'st book. It has an intersting discussion on triangulation in the chapter on SFM, but is not as straight-forward or in depth as Hartley-Zisserman's.
Code. You can use the triangulation methods from opencv, either from the calib3d module, or from the contribs/sfm module. Both use the DLT, but the code from the SFM module is more easily understandable (the calib3d code has a lot of old-school C code which is not very pleasant to read). There is also another lib, called openGV, which has a few interesting methods for triangulation.
cv::triangulatePoints
cv::sfm::triangulatePoints
OpenGV
The openGV git repo doesn't seem very active, and I'm not a big fan of the design of the library, but if I remember correctly (feel free to tell me otherwise) it offers methods other that the DLT for triangulations.
Naturally, those are all written in C++, but if you use other languages, finding wrappers or similar libraries wont be difficult (with python you still have opencv wrappers, and MATLAB has a bundle module, etc.).
I'm not sure this is the right place but here I go:
I have a database of 300 picture in high-resolution. I want to compute the PCA on this database and so far here is what I do: - reshape every image as a single column vector - create a matrix of all my data (500x300) - compute the average column and substract it to my matrix, this gives me X - compute the correlation C = X'X (300x300) - find the eigenvectors V and Eigen Values D of C. - the PCA matrix is given by XV*D^-1/2, where each column is a Principal Component
This is great and gives me correct component.
Now what I'm doing is doing the same PCA on the same database, except that the images have a lower resolution.
Here are my results, low-res on the left and high-res on the right. Has you can see most of them are similar but SOME images are not the same (the ones I circled)
Is there any way to explain this? I need for my algorithm to have the same images, but one set in high-res and the other one in low-res, how can I make this happen?
thanks
It is very possible that the filter you used could have done a thing or two to some of the components. After all, lower resolution images don't contain higher frequencies that, too, contribute to which components you're going to get. If component weights (lambdas) at those images are small, there's also a good possibility of errors.
I'm guessing your component images are sorted by weight. If they are, I would try to use a different pre-downsampling filter and see if it gives different results (essentially obtain lower resolution images by different means). It is possible that the components that come out differently have lots of frequency content in the transition band of that filter. It looks like images circled with red are nearly perfect inversions of each other. Filters can cause such things.
If your images are not sorted by weight, I wouldn't be surprised if the ones you circled have very little weight and that could simply be a computational precision error or something of that sort. In any case, we would probably need a little more information about how you downsample, how you sort the images before displaying them. Also, I wouldn't expect all images to be extremely similar because you're essentially getting rid of quite a few frequency components. I'm pretty sure it wouldn't have anything to do with the fact that you're stretching out images into vectors to compute PCA, but try to stretch them out in a different direction (take columns instead of rows or vice versa) and try that. If it changes the result, then perhaps you might want to try to perform PCA somewhat differently, not sure how.
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I'm interested in creating poster-sized images that contain repeating patterns, similar to the two (public domain) images below, the Flower of Life and a Penrose tiling:
My questions:
How do people usually create images like these on a computer? I'm hoping the answer isn't, "Open Adobe Illustrator and guess at intersection points," since such points can be defined mathematically. But I also imagine that not everyone with an interest in geometric patterns is also familiar with programming.
What is the best environment for creating such images? In particular, what's the best way to get high-resolution images out of Java, Python, Processing, etc? Or, is Mathematica the best tool?
Actually calculating the points and doing the math isn't the hard part, in my mind (at least, it's not the focus of this question). I'm interested in the best way to get a high-quality visual product out of a program.
The best way to create images like these is to learn to write PostScript. It's a clean language, easy to learn, and quite powerful once you know it well.
Bill Casselman's manuals are by far the best reference for high quality mathematical illustration.
Use a vector image format like SVG. This will scale perfectly to any resolution.
Inkscape is a great tool for creating these.
Once you have a vector image format, there are many options for using it in programming languages, depending on your language of choice.
For example -
.NET - SvgNet
ActionScript - svgweb
C++ - LeadTools
XAML for WPF/Silverlight - ViewerSVG (Converts SVG to XAML)
I don't know how those images were created, I would guess they were scanned from a book, but, in my work with fractals, I tend to start with just using the <canvas> tag, mainly so that I can change the size of the element and see it drawn more iterations, so I can get the highest resolution.
That is the problem with something like SVG is that you would need to pick a resolution and then create it, and it will scale well up and down, but if you developed it at one resolution, then you go to a higher resolution to demo, you may see more gaps than you would like.
If you want to just do it and save it as a static image then any GUI will work, as you are saving a GIF at that point, but if you want it, for example, on a web page, and have it look as good as it can on that browser then you may want to look at using javascript.
The math part isn't hard, and so drawing the image is fairly easy, once you derive the recursive algorithm that is needed. I tend to go to the next iteration until the size is below a threshold, for example, a radius of < 3, then it exits.
Well, #2 is going to be kind of a holy war so I'll address #1. :)
The key to images of this nature is recursion. Basically they are the same image repeated over and over in a controlled way to get an intersting result. Take the flower of life for example. You repeat the center petal six times (the method to do the petals is up to you). Then you create six more flowers using the petals tip as the center and overlapping one of the petals. You then recursively move outward. After a few "rounds" you stop and draw the containing circle. Basically the recusion simulates the stamp, move and rotate that would be required if you were doing it by hand.
When I have played around with these kinds of things I have always found that experimentation is the best way to get cool new things. Of course that could be just my lack of imagination. :)
I know I am not very math heavy in this answer but that is up to you and experimentation. Just remember that COS and SIN are your friends and there are 360 degrees in a cricle (or 2pi radians depending on your math package).
EDIT: Adding some math for the "Flower"
Starting with a center of (Xo, Yo) and a flower radius of r...
The tips of the petals (P0, P1, etc) are determined by...
X = Xo + (sin((n * pi)/3 + (pi / 6)) * r)
Y = Yo - (cos((n * pi)/3 + (pi / 6)) * r)
where n is the petal number (0..5)
Once you compute a petal tip, just draw the petal and then start a new flower at the tip. You would also set a bounding circle so that any point outside that circle would not be drawn.
I would try to create a PDF with iText in Java. PDF supports vector graphics, so it should scale without problems. I don't know how well iText scales w.r.t. performance when you have a really big number of graphic elements.
A1. You might want to look at turtle-graphics, l-systems, iterated function systems, space filling curves, and probably a lot of other approaches I'm not familiar with or haven't thought of yet.
A2. You can program any of these with any of the languages you suggest. I like Mathematica, but I know that not everyone has a copy of it and I have a copy 'cos I work in number-crunching and get to play with it for making pretty pictures. But Processing, which is free, was designed to be artist-friendly and might be a better starting point for you. Both Mathematica and Processing do the graphics right there and then, no calls to external libraries (or worrying which ones to use).
And, while I agree with everyone who says that vectors are the way to go, don't forget that the final productions step, onto paper or screen, is rendering so give some thought to how that will be done. This might, for example, lead you to Postscript or PDF for an output format.
Have fun
Mark
Well, I used to draw flowers of life with a compass, back then in junior school ... very simple actually ... but I don't think that's the answer you're looking for.
Basically it consists of drawing a circle of the same radius, from every point, until you encounter the big circle (limit).