I have a 3D shape in a 3D binary image. Therefore, I have a list of all of the x,y,z points.
If I am to analyze a shape for various identification, such as "sphericity", "spiky"-ness, volume, surface area, etc., what are some of the choices do I have here?
Could you post a sample shape? Do you have a complete set of points on the surface and interior of the shape? Are the points evenly spaced? Is this synthetic data, or perhaps a point cloud from a 3D scan?
A few ideas:
Calculate the 3D convex hull of the points. This will give you the outer "envelope" of the points and is useful for comparison with other measurements. For example, you can compare the surface area of the convex hull to the surface area of the outer surface points.
Find the difference between "on" voxels in the convex hull and "on" voxels in the raw point set. You can then determine how many points are different, whether there is one big clump, etc. If the original shape is a doughnut, the convex hull will be a disk, and the difference will be the shape of the hole.
To calculate spikiness, you can think of comparing the Euclidean distance between two points (the "straight line" distance) and the shortest distance on the outer surface between those two points.
Compare the surface area of the raw data to the surface area after a 3D morphological "close" operation or some other smoothing operation.
To suggest a type of volume calculation, we'd need to know more about the point set.
Consider the Art Gallery Problem to 3D. Are there points on the surface not visible to certain points in the interior? Is the shape convex or star convex?
http://en.wikipedia.org/wiki/Art_gallery_problem
http://en.wikipedia.org/wiki/Star-convex_set
A good reference for geometric algorithms is Geometric Tools for Computer Graphics by Schneider and Eberly. It's pricey new, but you can probably find a cheap used copy in good condition at addall.com. I suspect you'll find all the answers you want and more in that book.
http://www.amazon.com/Geometric-Computer-Graphics-Morgan-Kaufmann/dp/1558605940
One of the authors maintains a site on the same subject:
http://www.geometrictools.com/
Another good textbook is Computational Geometry in C by Joseph O'Rourke.
http://www.amazon.com/Computational-Geometry-Cambridge-Theoretical-Computer/dp/0521649765/ref=sr_1_1?s=books&ie=UTF8&qid=1328939654&sr=1-1
Related
I have a non-planar object with 9 points with known dimensions in 3D i.e. length of all sides is known. Now given a 2D projection of this shape, I want to reconstruct the 3D model of it. I basically want to retrieve the shape of this object in the real world i.e. angles between different sides in 3D. For eg: given all the dimensions of every part of the table and a 2D image, I'm trying to reconstruct its 3D model.
I've read about homography, perspective transform, procrustes and fundamental/essential matrix so far but haven't found a solution that'll apply here. I'm new to this, so might have missed out something. Any direction on this will be really helpful.
In your question, you mention that you want to achieve this using only a single view of the object. In that case, homographies or Essential/Fundamental matrices wont help you, because these require at least two views of the scene to make sense. If you don't have any priors on the shape of the objects that you want to reconstruct, the key information that you'll be missing is (relative) depth, and in that case I think those are the two possible solutions:
Leverage a learning algorithm. There is a rich literature on 6dof object pose estimation with deep networks, see this paper for example. You wont have to deal with depth directly if you use those since those networks are trained end to end to estimate a pose in SO(3).
Add many more images and use a dense photometric SLAM/SFM pipeline, such as elastic fusion. However, in that case you will need to segment the resulting models since the estimation they produce is of the entire environment, which can be difficult depending on the scene.
However, as you mentioned in your comment, it is possible to reconstruct the model up to scale if you have very strong priors on its geometry. In the case of a planar object (a cuboid will just be an extension of that), you can use this simple algorithm (that is more or less what they do here, there are other methods but I find them a bit messy, equation-wise):
//let's note A,B,C,D the rectangle in 3d that we are after, such that
//AB is parellel with CD. Let's also note a,b,c,d their respective
//reprojections in the image, i.e. a=KA where K is the calibration matrix, and so on.
1) Compute the common vanishing point of AB and CD. This is just the intersection
of ab and cd in the image plane. Let's call it v_1.
2) Do the same for the two other edges, i.e bc and da. Let's call this
vanishing point v_2.
3) Now, you can compute the vanishing line, which will just be
crossproduct(v_1, v_2), i.e. the line going through both v_1 and v_2. This gives
you the orientation of your plane. Let's write its normal N.
5) All you need to find now is the boundaries of the rectangle. To do
that, just consider any plane with normal N that doesn't go through
the camera center. Now find the intersections of K^{-1}a, K^{-1}b,
K^{-1}c, K^{-1}d with that plane.
If you need a refresher on vanishing points and lines, I suggest you take a look at pages 213 and 216 of Hartley-Zisserman's book.
Using Emgu CV I have extracted a set of closed polygons from the contours in an image of a road network. The polygons represent road outlines. The result is shown below, plotted over an OpenStreetMaps map (the polygons in 'pixel' form from Emgu CV have been converted to latitude/longitude form to be plotted).
Set of polygons representing road outlines:
I would now like to compute the Voronoi diagram of this set of polygons, which will help me find the centerline of the road. But in Emgu CV I can only find a way to get the Voronoi diagram of a set of points. This is done by finding the Delaunay triangulation of the set of points (using the Subdiv2D class) and then computing the voronoi facets with GetVoronoiFacets.
I have tried computing the Voronoi diagram of the points defined by all the polygons in the set (each polygon is a list of points), but this gives me an extremely complicated Voronoi diagram, as one might expect:
Voronoi diagram of set of points:
This image shows a smaller portion of the first picture (for clarity, since it is so convoluted). Indeed some of the lines in the diagram seem to represent the road centerline, but there are so many other lines, it will be tough to find a criterion to extract the "good" lines.
Another potential problem that I am facing is that, as you should be able to tell from the first picture, some polygons are in the interior of others, so we are not in the standard situation of a set of disjoint closed polygons. That is, sometimes the road is between the outer boundary of one polygon and the inner boundary of another.
I'm looking for suggestions as to how to compute the Voronoi graph of the set of polygons using Emgu CV (or Open CV), hopefully overcoming this second problem I've outlined as well. I'm also open to other suggestions for how to acheive this without using Emgu CV.
If you already have polygons, you can try computing the Straight Skeleton.
I haven't tried it, but CGAL has an implementation. Note that this particular function license is GPL.
A possible issue may be:
The current version of this CGAL package can only construct the
straight skeleton in the interior of a simple polygon with holes, that
is it doesn't handle general polygonal figures in the plane.
Probably there are workarounds for that. For example you can include all polygons in a bigger rectangle (this way the original polygons will be holes of the new rectangle). This may not work well if the original polygons have holes. To solve that, you could execute the algorithm for each polygon with holes and then put all polygons in a rectangle, removing all holes and execute the algorithm again.
When using OpenCV for example, algorithms like SIFT or SURF are often used to detect keypoints. My question is what actually are these keypoints?
I understand that they are some kind of "points of interest" in an image. I also know that they are scale invariant and are circular.
Also, I found out that they have orientation but I couldn't understand what this actually is. Is it an angle but between the radius and something? Can you give some explanation? I think I need what I need first is something simpler and after that it will be easier to understand the papers.
Let's tackle each point one by one:
My question is what actually are these keypoints?
Keypoints are the same thing as interest points. They are spatial locations, or points in the image that define what is interesting or what stand out in the image. Interest point detection is actually a subset of blob detection, which aims to find interesting regions or spatial areas in an image. The reason why keypoints are special is because no matter how the image changes... whether the image rotates, shrinks/expands, is translated (all of these would be an affine transformation by the way...) or is subject to distortion (i.e. a projective transformation or homography), you should be able to find the same keypoints in this modified image when comparing with the original image. Here's an example from a post I wrote a while ago:
Source: module' object has no attribute 'drawMatches' opencv python
The image on the right is a rotated version of the left image. I've also only displayed the top 10 matches between the two images. If you take a look at the top 10 matches, these are points that we probably would want to focus on that would allow us to remember what the image was about. We would want to focus on the face of the cameraman as well as the camera, the tripod and some of the interesting textures on the buildings in the background. You see that these same points were found between the two images and these were successfully matched.
Therefore, what you should take away from this is that these are points in the image that are interesting and that they should be found no matter how the image is distorted.
I understand that they are some kind of "points of interest" of an image. I also know that they are scale invariant and I know they are circular.
You are correct. Scale invariant means that no matter how you scale the image, you should still be able to find those points.
Now we are going to venture into the descriptor part. What makes keypoints different between frameworks is the way you describe these keypoints. These are what are known as descriptors. Each keypoint that you detect has an associated descriptor that accompanies it. Some frameworks only do a keypoint detection, while other frameworks are simply a description framework and they don't detect the points. There are also some that do both - they detect and describe the keypoints. SIFT and SURF are examples of frameworks that both detect and describe the keypoints.
Descriptors are primarily concerned with both the scale and the orientation of the keypoint. The keypoints we've nailed that concept down, but we need the descriptor part if it is our purpose to try and match between keypoints in different images. Now, what you mean by "circular"... that correlates with the scale that the point was detected at. Take for example this image that is taken from the VLFeat Toolbox tutorial:
You see that any points that are yellow are interest points, but some of these points have a different circle radius. These deal with scale. How interest points work in a general sense is that we decompose the image into multiple scales. We check for interest points at each scale, and we combine all of these interest points together to create the final output. The larger the "circle", the larger the scale was that the point was detected at. Also, there is a line that radiates from the centre of the circle to the edge. This is the orientation of the keypoint, which we will cover next.
Also I found out that they have orientation but I couldn't understand what actually it is. It is an angle but between the radius and something?
Basically if you want to detect keypoints regardless of scale and orientation, when they talk about orientation of keypoints, what they really mean is that they search a pixel neighbourhood that surrounds the keypoint and figure out how this pixel neighbourhood is oriented or what direction this patch is oriented in. It depends on what descriptor framework you look at, but the general jist is to detect the most dominant orientation of the gradient angles in the patch. This is important for matching so that you can match keypoints together. Take a look at the first figure I have with the two cameramen - one rotated while the other isn't. If you take a look at some of those points, how do we figure out how one point matches with another? We can easily identify that the top of the cameraman as an interest point matches with the rotated version because we take a look at points that surround the keypoint and see what orientation all of these points are in... and from there, that's how the orientation is computed.
Usually when we want to detect keypoints, we just take a look at the locations. However, if you want to match keypoints between images, then you definitely need the scale and the orientation to facilitate this.
I'm not as familiar with SURF, but I can tell you about SIFT, which SURF is based on. I provided a few notes about SURF at the end, but I don't know all the details.
SIFT aims to find highly-distinctive locations (or keypoints) in an image. The locations are not merely 2D locations on the image, but locations in the image's scale space, meaning they have three coordinates: x, y, and scale. The process for finding SIFT keypoints is:
blur and resample the image with different blur widths and sampling rates to create a scale-space
use the difference of gaussians method to detect blobs at different scales; the blob centers become our keypoints at a given x, y, and scale
assign every keypoint an orientation by calculating a histogram of gradient orientations for every pixel in its neighborhood and picking the orientation bin with the highest number of counts
assign every keypoint a 128-dimensional feature vector based on the gradient orientations of pixels in 16 local neighborhoods
Step 2 gives us scale invariance, step 3 gives us rotation invariance, and step 4 gives us a "fingerprint" of sorts that can be used to identify the keypoint. Together they can be used to match occurrences of the same feature at any orientation and scale in multiple images.
SURF aims to accomplish the same goals as SIFT but uses some clever tricks in order to increase speed.
For blob detection, it uses the determinant of Hessian method. The dominant orientation is found by examining the horizontal and vertical responses to Haar wavelets. The feature descriptor is similar to SIFT, looking at orientations of pixels in 16 local neighborhoods, but results in a 64-dimensional vector.
SURF features can be calculated up to 3 times faster than SIFT features, yet are just as robust in most situations.
For reference:
A good SIFT tutorial
An introduction to SURF
I'm looking for an efficient way of selecting a relatively large portion of points (2D Euclidian graph) that are the furthest away from the center. This resembles the convex hull, but would include (many) more points. Further criteria:
The number of points in the selection / set ("K") must be within a specified range. Most likely it won't be very narrow, but it most work for different ranges (eg. 0.01*N < K < 0.05*N as well as 0.1*N < K < 0.2*N).
The algorithm must be able to balance distance from the center and "local density". If there are dense areas near the upper part of the graph range, but sparse areas near the lower part, then the algorithm must make sure to select some points from the lower part even if they are closer to the center than the points in the upper region. (See example below)
Bonus: rather than simple distance from center, taking into account distance to a specific point (or both a point and the center) would be perfect.
My attempts so far have focused on using "pigeon holing" (divide graph into CxR boxes, assign points to boxes based on coordinates) and selecting "outer" boxes until we have sufficient points in the set. However, I haven't been successful at balancing the selection (dense regions over-selected because of fixed box size) nor at using a selected point as reference instead of (only) the center.
I've (poorly) drawn an Example: The red dots are the points, the green shape is an example of what I want (outside the green = selected). For sparse regions, the bounding shape comes closer to the center to find suitable points (but doesn't necessarily find any, if they're too close to the center). The yellow box is an example of what my Pigeon Holing based algorithms does. Even when trying to adjust for sparser regions, it doesn't manage well.
Any and all ideas are welcome!
I don't think there are any standard algorithms that will give you what you want. You're going to have to get creative. Assuming your points are embedded in 2D Euclidean space here are some ideas:
Iteratively compute several convex hulls. For example, compute the convex hull, keep the points that are part of the convex hull, then compute another convex hull ignoring the points from the original convex hull. Continue to do this until you have a sufficient number of points, essentially plucking off points on the perimeter for each iteration. The only problem with this approach is that it will not work well for concavities in your data set (e.g., the one on the bottom of your sample you posted).
Fit a Gaussian to your data and keep everything > N standard
deviations away from the mean (where N is a value that you'd have to
choose). This should work pretty well if your data is Gaussian. If
it isn't, you could always model it with several Gaussians (instead
of one), and keep points with a joint probability less than some threshold. Using multiple Gaussians will probably handle concavities decently. References:
http://en.wikipedia.org/wiki/Gaussian_function
How to fit a gaussian to data in matlab/octave?\
Use Kernel Density Estimation - If you create a kernel density
surface, you could slice the surface at some height (e.g., turning
it into a plateau), giving you a perimeter shape (the shape of the
plateau) around the points. The trick would be to slice it at the
right location though, because you could end up getting no points
outside of the shape, but with the right selection you could easily
get the green shape you drew. This approach will work well and give you the green shape in your example if you choose the slice point wisely (which may be difficult to do). The big drawback of this approach is that it is very computationally expensive. More information:
http://en.wikipedia.org/wiki/Multivariate_kernel_density_estimation
Use alpha shapes to get a general shape the wraps tightly around
the outside perimeter of the point set. Then erode the shape a
little to force some points outside of the shape. I don't have a lot of experience with alpha shapes, but this approach will also be quite computationally expensive. More info:
http://doc.cgal.org/latest/Alpha_shapes_2/index.html
I'm trying to implement user-assisted edge detection using OpenCV.
Assume you have an image in which we need to find a polygonal shape. For the sake of discussion, let's say we need to find the top of a rectangular table in a picture. The user will click on the four corners of the table to help us narrow things down. Connecting those four points gives us a polygon, or four vectors.
But the user is not very accurate when clicking on those corners. So I'd like to use edge information from the image to increase the accuracy.
I'm using a Canny edge detector with a fairly high treshold to determine important edges in my image. (more precisely, I'm scaling down, blurring, converting to grayscale, then run Canny). How can I compute whether a vector aligns with an edge in my image? If I have a way to compute "alignment", my overal algorithm comes down to perturbating the location of the four edge points, computing the total "alignment" of my polygon with the edges in the image, until I find an optimum.
What is a good way to define and compute this "alignment" metric?
You may want to try to use FindContours to detect your table or any other contour. Then build a contour also from the user input points. After this you can read about Contour Moments by which you can compare contours. You can compare all the contours from the image with the one built from the user points and then select the closest match.