Good day.
I have the task of finding the set of points in 2D space for which the sum of the distances to the rectangles is minimal. For example, for two rectangles, the result will be the next area (picture). Any point in this area has the minimum sum of lengths to A and B rectangles.
Which algorithm is suitable for finding a region, all points of which have the minimum sum of lengths? The number of rectangles can be different, they are randomly located. They can even overlap each other. The sides of the rectangles are parallel to the coordinate axes and cannot be rotated. The region must be either a rectangle or a line or a point.
Hint:
The distance map of a rectangle (function that maps any point (x,y) to the closest distance to the rectangle) is made of four slanted planes (slope 45°), four quarter of cones and the rectangle itself, which is at ground level, forming a continuous surface.
To obtain the global distance map, it "suffices" to sum the distance maps of the individual rectangles. A pretty complex surface will result. Depending on the geometries, the minimum might be achieved on a single vertex, a whole edge or a whole face.
The construction of the global map seems more difficult than that of a line arrangement, due to the conic patches. A very difficult problem in the general case, though the axis-aligned constraint might ease it.
Add on Yves's answer.
As Yves described, each rectangle 'divide' plane into 9 parts and adds different distance method in to the sum. Middle part (rectangle) add distance 0, side parts add coordinate distance to that side, corner parts add point distance to that corner. With that approach plan has to be divided into 9^n parts, and distance sum is calculated by adding appropriate rectangle distance functions. That is feasible if number of rectangles is not too large.
Probably it is not needed to calculate all parts since it is easy to calculate some bound on part min value and check is it needed to calculate part at all.
I am not sure, but it seems to me that global distance map is convex function. If that is the case than it can be solved iteratively by similar idea as in linear programming.
what are best features to detect a contour by comparison to a saved one?
for now i have the following on the table and will use ratios between some of them (preliminary results are good):
-enclosing area circle
-bounding rectangle area
-bounding rectangle height over width
-perimeter (arc length)
-convex hull polygon area (using the polygon area formula)
which ones will be most distinct between different shapes and will be most accurate when compared (ratioA - ratioB < threshold) so i dont waste computing power on all together?
ps: should also work with scale and rotation
There is a module shape in OpenCV that contains two shape comparators: ShapeContextDistanceExtractor and HausdorffDistanceExtractor. Both require only two contours as arguments and return single number (a distance between these contours in some space) that you can treat as a measure of dissimilarity.
Easier way is to try cv::matchShapes(): this function calculates two vectors of invariant Hu moments for each contour and then calculates their dissimilarity based on these vectors as well.
I want to generalize this question: How to crop away convexity defects.
Situation described there is clear to me but I have more complex situation: I have a contour which can have not only convex, but a concave defects:
Blue is the contour I have and red is a rectangle I want - it can be rotated rectangle in general, so I need 4-point rectangular contour with area is as close as possible to the given contour. I'm looking for algorithm on how this transformation function could be implemented and it would be nice if it will be illustrated with code.
My idea would be to perform a Hough transform on the contour and choose the top 4 most prominent lines as the edges of the rectangle.
You should draw the contour as a b&w image and pass that to HoughLines. It will return a vector of (r, theta) pairs, which encode the detected lines in polar coordinates. See this example for how to convert them back to cartesian coordinates.
The lines will be ordered by decreasing number of "votes" (i.e. the sum of pixel intensities along the line).
Note that HoughLinesP is less suitable for the problem at hand, because it returns line segments, not lines.
The approach from the linked answer can only handle convexity defects based on deviation from the convex hull, I don't think you can use the opposite approach to remove 'concavity defects' (which grow the convex hull and make other features the defects).
You could try the following:
Run the solution from the linked answer to remove convex defects
On the resulting feature, iteratively call cv::ApproxPoly with varying accuracy parameter (e.g. 0.0, 0.5, 1.0...) until it approximates to a 4 sided shape.
I have an image of a circle but my circle is not perfect
Firstly I found transition coordinates
Detecting Circles without using Hough Circles
and than ı use this formula https://math.stackexchange.com/questions/675203/calculating-centre-of-rotation-given-point-coordinates-at-different-positions/1414344#1414344
Finally ı have fourn radiuses which the longest and shortest
Now I have this image:
BUt they are radius I need to find diameter How to find diameter from image??
Or How to I can find mutual/symetric/ point in a circle
For this image, the approaches mentioned are overkill. Just find the bounding box of the non-black pixels. Because of sampling artifacts, the horizontal and vertical side lengths may differ by one or two pixels.
If I am right, the outer circle is 277 x 273 pixels. If you consider the difference to be significant, then this is an ellipse, not a circle.
3 ways to do this:
I think you need to measure it from image:
so use edge detection (blue line width from left to right = width of bounding box of the blue pixels) then count pixels.
If you need then convert to any unit you want like inch (using pixel per inch).
If your circle is not perfect circle (stretched) measure in many direction so you can find its deviation too.
There is another way named Monte-Carlo method:
first generate random x and y (inside square) and then evaluate that point (x, y) is inside the circle or not and count the number of inside occurrences, then you can calculate the Area of circle using ratio on inside count/total (and therefore diameter).
without using random numbers:
fill(color) circle inside then simply count black pixels this is = to Area outside circle => Total Area(Square Area)- black pixel Area = Circle Area => calculate diameter.
I can successfully threshold images and find edges in an image. What I am struggling with is trying to extract the angle of the black edges accurately.
I am currently taking the extreme points of the black edge and calculating the angle with the atan2 function, but because of aliasing, depending on the point you choose the angle can come out with some degree of variation. Is there a reliable programmable way of choosing the points to calculate the angle from?
Example image:
For example, the Gimp Measure tool angle at 3.12°,
If you're writing your own library, then creating a robust solution for this problem will allow you to develop several independent chunks of code that you can string together to solve other problems, too. I'll assume that you want to find the corners of the checkerboard under arbitrary rotation, under varying lighting conditions, in the presence of image noise, with a little nonlinear pincushion/barrel distortion, and so on.
Although there are simple kernel-based techniques to find whole pixels as edge pixels, when working with filled polygons you'll want to favor algorithms that can find edges with sub-pixel accuracy so that you can perform accurate line fits. Even though the gradient from dark square to white square crosses several pixels, the "true" edge will be found at some sub-pixel point, and very likely not the point you'd guess by manually clicking.
I tried to provide a simple summary of edge finding in this older SO post:
what is the relationship between image edges and gradient?
For problems like yours, a robust solution is to find edge points along the dark-to-light transitions with sub-pixel accuracy, then fit lines to the edge points, and use the line angles. If you are processing a true camera image, and if there is an uncorrected radial distortion in the image, then there are some potential problems with measurement accuracy, but we'll ignore those.
If you want to find an accurate fit for an edge, then it'd be great to scan for sub-pixel edges in a direction perpendicular to that edge. That presupposes that we have some reasonable estimate of the edge direction to begin with. We can first find a rough estimate of the edge orientation, then perform an accurate line fit.
The algorithm below may appear to have too many steps, but my purpose is point out how to provide a robust solution.
Perform a few iterations of erosion on black pixels to separate the black boxes from one another.
Run a connected components algorithm (blob-finding algorithm) to find the eroded black squares.
Identify the center (x,y) point of each eroded square as well as the (x,y) end points defining the major and minor axes.
Maintain the data for each square in a structure that has the total area in pixels, the center (x,y) point, the (x,y) points of the major and minor axes, etc.
As needed, eliminate all components (blobs) that are too small. For example, you would want to exclude all "salt and pepper" noise blobs. You might also temporarily ignore checkboard squares that are cut off by the image edges--we can return to those later.
Then you'll loop through your list of blobs and do the following for each blob:
Determine the direction roughly perpendicular to the edges of the checkerboard square. How you accomplish this depends in part on what data you calculate when you run your connected components algorithm. In a general-purpose image processing library, a standard connected components algorithm will determine dozens of properties and measurements for each individual blob: area, roundness, major axis direction, minor axis direction, end points of the major and minor axis, etc. For rectangular figures, it can be sufficient to calculate the topmost, leftmost, rightmost, and bottommost points, as these will define the four corners.
Generate edge scans in the direction roughly perpendicular to the edges. These must be performed on the original, unmodified image. This generally assumes you have bilinear interpolation implemented to find the grayscale values of sub-pixel (x,y) points such as (100.35, 25.72) since your scan lines won't fall exactly on whole pixels.
Use a sub-pixel edge point finding technique. In general, you'll perform a curve fit to the edge points in the direction of the scan, then find the real-valued (x,y) point at maximum gradient. That's the edge point.
Store all sub-pixel edge points in a list/array/collection.
Generate line fits for the edge points. These can use Hough, RANSAC, least squares, or other techniques.
From the line equations for each of your four line fits, calculate the line angle.
That algorithm finds the angles independently for each black checkerboard square. It may be overkill for this one application, but if you're developing a library maybe it'll give you some ideas about what sub-algorithms to implement and how to structure them. For example, the algorithm would rely on implementations of these techniques:
Image morphology (e.g. erode, dilate, close, open, ...)
Kernel operations to implement morphology
Thresholding to binarize an image -- the Otsu method is worth checking out
Connected components algorithm (a.k.a blob finding, or the OpenCV contours function)
Data structure for blob
Moment calculations for blob data
Bilinear interpolation to find sub-pixel (x,y) values
A linear ray-scanning technique to find (x,y) gray values along a specific direction (which will also rely on bilinear interpolation)
A curve fitting technique and means to determine steepest tangent to find edge points
Robust line fit technique: Hough, RANSAC, and/or least squares
Data structure for line equation, related functions
All that said, if you're willing to settle for a slight loss of accuracy, and if you know that the image does not suffer from radial distortion, etc., and if you just need to find the angle of the parallel lines defined by all checkboard edges, then you might try..
Simple kernel-based edge point finding technique (Laplacian on Gaussian-smoothed image)
Hough line fit to edge points
Choose the two line fits with the greatest number of votes, which should be one set of horizontal-ish lines and the other set of vertical-ish lines
There are also other techniques that are less accurate but easier to implement:
Use a kernel-based corner-finding operator
Find the angles between corner points.
And so on and so on. As you're developing your library and creating robust implementations of standalone functions that you can string together to create application-specific solutions, you're likely to find that robust solutions rely on more steps than you would have guessed, but it'll also be more clear what the failure mode will be at each incremental step, and how to address that failure mode.
Can I ask, what C++ library are you using to code this?
Jerry is right, if you actually apply a threshold to the image it would be in 2bit, black OR white. What you may have applied is a kind of limiter instead.
You can make a threshold function (if you're coding the image processing yourself) by applying the limiter you may have been using and then turning all non-white pixels black. If you have the right settings, the squares should be isolated and you will be able to calculate the angle.
Once this is done you can use a path finding algorithm to find some edge, any edge will do. If you find a more or less straight path, you can use the extreme points as you are doing now to determine the angle. Since the checker-board rotation is only relevant within 90 degrees, your angle should be modulo 90 degrees or pi over 2 radians.
I'm not sure it's (anywhere close to) the right answer, but my immediate reaction would be to threshold twice: once where anything but black is treated as white, and once where anything but white is treated as black.
Find the angle for each, then interpolate between the two angles.
Your problem have few solutions but all have one very important issue which you seem to neglect. Note: When you are trying to make geometrical calculation in the image, the points you use must be as far as possible one from the other. You are taking 2 points inside a single square. Those points are very close one to another, so a slight error in pixel location of of the points leads to a large error in the angle. Why do you use only a single square, when you have many squares in the image?
Here are few solutions:
Find the line angle of every square. You have at least 9 squares in the image, 4 lines in each square which give you total of 36 angles (18 will be roughly at 3[deg] and 18 will be ~93[deg]). Remove the 90[degrees] and you get 36 different measurements of the angle. Sort them and take the average of the middle 30 (disregarding the lower 3 and higher 3 measurements). This will give you an accurate result
Second solution, find the left extreme point of the leftmost square and the right extreme point of the rightmost square. Now calculate the angle between them. The result will be much more accurate because the points are far away.
A third algorithm which will give you accurate results because it doesn't involve finding any points and no need for thresholding. Just smooth the image, calculate gradients in X and Y directions (gx,gy), calculate the angle of the gradient in each pixel atan(gy,gx) and make histogram of the angles. You will have 2 significant peaks near the 3[deg] and 93[deg]. Just find the peaks by searching the maximum in the histogram. This will work even if you have a lot of noise in the image, even with antialising and jpg artifacts, and even if you have other drawings on the image. But remember, you must smooth the image a lot before calculating the derivatives.