We Keep Coding

Irregular Polygon Sub-division?

I have a problem in which I have an irregular polygon shaped container which needs to be sub-divided such that it contains certain smaller rectangles of specific dimensions. I have previously tried addressing this problem using a bin packing approach which seems to be too time consuming. As a result I wanted to try out sub-division based approaches where the algorithm ensures that the sub-division contains certain smaller rectangles of specific dimensions while the remaining space can be considered as empty space. Any suggestions on fast algorithms for the generation of these sub-divisions preferably using the shapely library?
The irregular polygon container would look some thing like the diagram below. The edges of the container will always be right angles and it is known that items to be packed can all fit in to the given container with additional remaining free space. The constraints are the same as regular bing packing where the items should not overlap and all items should be completely within the container. The items can be rotated (preferably at right angles).

Converting an AutoCAD model to a matrix of points/volumes with the mass density specified at each location

I am an experimental physicist (grad student) that is trying to take an AutoCAD model of the experiment I've built and find the gravitational potential from the whole instrument over a specified volume. Before I find the potential, I'm trying to make a map of the mass density at each point in the model.
What's important is that I already have a model and in the end I'll have a something that says "At (x,y,z) the value is d". If that's an crazy csv file, a numpy array, an excel sheet, or... whatever, I'll be happy.
Here's what I've come up with so far:
Step 1: I color code the AutoCAD file so that color associates with material.
Step 2: I send the new drawing/model to a slicer (made for 3D printing). This takes my 3D object and turns it into equally spaced (in z-direction) 2d objects... but then that's all output as g-code. But hey! G-code is a way of telling a motor how to move.
Step 3: This is the 'hard part' and the meat of this question. I'm thinking that I take that g-code, which is in essence just a set of instructions on how to move a nozzle and use it to populate a numpy array. Basically I have 3D array, each level corresponds to one position in z, and the grid left is my x-y plane. It reads what color is being put where, and follows the nozzle and puts that mass into those spots. It knows the mass because of the color. It follows the path by parsing the g-code.
When it is done with that level, it moves to the next grid and repeats.
Does this sound insane? Better yet, does it sound plausible? Or maybe someone has a smarter way of thinking about this.
Even if you just read all that, thank you. Seriously.

Does this sound insane? Better yet, does it sound plausible?
It's very reasonable and plausible. Using the g-code could do that, but it would require a g-code interpreter that could map the instructions to a 2D path. (Not 3D, since you mentioned that you're taking fixed z-slices.) That could be problematic, but, if you found one, it could work, but may require some parser manipulation. There are several of these in a variety of languages, that could be useful.
SUGGESTION
From what you describe, it's akin to doing a MRI scan of the object, and trying to determine its constituent mass profile along a given axis. In this case, and unlike MRI, you have multiple colors, so that can be used to your advantage in region selection / identification.
Even if you used a g-code interpreter, it would reproduce an image whose area you'll still have to calculate, so noting that and given that you seek to determine and classify material composition by path (in that the path defines the boundary of a particular material, which has a unique color), there may be a couple ways to approach this without resorting to g-code:
1) If the colors of your material are easily (or reasonably) distinguishable, you can create a color mask which will quantify the occupied area, from which you can then determine the mass.
That is, if you take a photograph of the slice, load the image into a numpy array, and then search for a specific value (say red), you can identify the area of the region. Then, you apply a mask on your array. Once done, you count the occupied elements within your array, and then you divide it by the array size (i.e. rows by columns), which would give you the relative area occupied. Since you know the mass of the material, and there is a constant z-thickness, this will give you the relative mass. An example of color masking using numpy alone is shown here: http://scikit-image.org/docs/dev/user_guide/numpy_images.html
As such, let's define an example that's analogous to your problem - let's say we have a picture of a red cabbage, and we want to know which how much of the picture contains red / purple-like pixels.
To simplify our life, we'll set any pixel above a certain threshold to white (RGB: 255,255,255), and then count how many non-white pixels there are:
from copy import deepcopy
import numpy as np
import matplotlib.pyplot as plt
def plot_image(fname, color=128, replacement=(255, 255, 255), plot=False):
# 128 is a reasonable guess since most of the pixels in the image that have the
# purplish hue, have RGB's above this value.
data = imread(fname)
image_data = deepcopy(data) # copy the original data (for later use if need be)
mask = image_data[:, :, 0] < color # apply the color mask over the image data
image_data[mask] = np.array(replacement) # replace the match
if plot:
plt.imshow(image_data)
plt.show()
return data, image_data
data, image_data = plot_image('cabbage.jpg') # load the image, and apply the mask
# Find the locations of all the pixels that are non-white (i.e. 255)
# This returns 3 arrays of the same size)
indices = np.where(image_data != 255)
# Now, calculate the area: in this case, ~ 62.04 %
effective_area = indices[0].size / float(data.size)
The selected region in question is shown here below:
Note that image_data contains the pixel information that has been masked, and would provide the coordinates (albeit in pixel space) of where each occupied (i.e. non-white) pixel occurs. The issue with this of course is that these are pixel coordinates and not a physical one. But, since you know the physical dimensions, extrapolating those quantities are easily done.
Furthermore, with the effective area known, and knowledge of the physical dimension, you have a good estimate of the real area occupied. To obtain better results, tweak the value of the color threshold (i.e. color). In your real-life example, since you know the color, search within a pixel range around that value (to offset noise and lighting issues).
The above method is a bit crude - but effective - and, it may be worth exploring using it in tandem with edge-detection, as that could help improve the region identification, and area selection. (Note that isn't always strictly true!) Also, color deconvolution may be useful: http://scikit-image.org/docs/dev/auto_examples/color_exposure/plot_ihc_color_separation.html#sphx-glr-auto-examples-color-exposure-plot-ihc-color-separation-py
The downside to this is that the analysis requires a high quality image, good lighting; and, most importantly, it's likely that you'll lose some of the more finer details of the edges, which would impact your masses.
2) Instead of resorting to camera work, and given that you have the AutoCAD model, you can use that and the software itself in addition to the above prescribed method.
Since you've colored each material in the model differently, you can use AutoCAD's slicing tool, and can do something similar to what the first method suggests doing physically: slicing the model, and taking pictures of the slice to expose the surface. Then, using a similar method described above of color masking / edge detection / region determination through color selection, you should obtain a much better and (arguably) very accurate result.
The downside to this, is that you're also limited by the image quality used. But, as it's software, that shouldn't be much of an issue, and you can get extremely high accuracy - close to its actual result.
The last suggestion to improve these results would be to script numerous random thin slicing of the AutoCAD model along a particular directional vector shared by every subsequent slice, exporting each exposed surface, analyzing each image in the manner described above, and then collecting those results to given you a Monte Carlo-like and statistically quantifiable determination of the mass (to correct for geometry effects due to slicing along one given axis).

Linear shift between 2 sets of coordinates

My Problem is the following:
For my work I need to compare images of scanned photographic plates with a catalogue of a sample of known stars within the general area of the sky the plates cover (I call it the master catalogue). To that end I extract information, like the brightness on the image and the position in the sky, of the objects in the images and save it in tables. I then use python to create a polynomial fit for the calibration of the magnitude of the stars in the image.
That works up to a certain accuracy pretty well, but unfortunately not well enough, since there is a small shift between the coordinates the object has in the photographic plates and in the master catalogue.
Here the green circles indicate the positions (center of the circle) of objects in the master catalogue. As you can see the actual stars are always situated to the upper left of the objects in the master catalogue.
I have looked a little bit in the comparison of images (i.e. How to detect a shift between images) but I'm a little at a loss now, because I'm not actually comparing images but arrays with the coordinates of the objects. An additional problem here is that (as you can see in the image) there are objects in the master catalogue that are not visible on the plates and not all plates have the same depth (meaning some show more stars than others do).
What I would like to know is a way to find and correct the linear shift between the 2 arrays of different size of coordinates in python. There shouldn't be any rotations, so it is just a shift in x and y directions. The arrays are normal numpy recarrays.

I would change #OphirYoktan's suggestion slightly. You have these circles. I assume you know the radius, and you have that radius value for a reason.
Instead of randomly choosing points, filter the master catalog for x,y within radius of your sample. Then compute however many vectors you need to compute for all possible master catalog entries within range of your sample. Do the same thing repeatedly, then collect a histogram of the vectors. Presumably a small number will occur repeatedly, those are the likely true translations. (Ideally, "small number" == 1.)

There are several possible solutions
Note - these are high level pointers, you'll need some work to convert it to working code
The original solution (cross correlation) can be adapted to the current data structure, and should work
A believe that RANSAC will be better in your case
basically it means:
create a model based on a small number of data points (the minimal number that are required to define a relevant model), and verify it's correctness using the full data set.
specifically, if you have only translation to consider (and not scale):
select one of your points
match it to a random point in the catalog [you may do "educated guesses", if you have some prior about what translation is more likely]
this matching gives you the translation
verify this translation matches the rest of your points
repeat until you find a good match

I'm assuming here the objects aren't necessarily in the same order in both the photo plate and master catalogue.
Consider the set of position vectors, A, of the objects in the photo plate, and the set of position vectors, B, of the objects in the master catalogue. You're looking for a vector, v, such that for each a in A, a + v is approximately some element in b.
The most obvious algorithm to me would be to say for each a, for each b, let v = b - a. Now, for each element in A, check that there is a corresponding element in B that is sufficiently close (within some distance e that you choose) to that element + v. Once you find the v that meets this condition, v is your shift.

Image registration using python and cross-correlation

I got two images showing exaktly the same content: 2D-gaussian-shaped spots. I call these two 16-bit png-files "left.png" and "right.png". But as they are obtained thru an slightly different optical setup, the corresponding spots (physically the same) appear at slightly different positions. Meaning the right is slightly stretched, distorted, or so, in a non-linear way. Therefore I would like to get the transformation from left to right.
So for every pixel on the left side with its x- and y-coordinate I want a function giving me the components of the displacement-vector that points to the corresponding pixel on the right side.
In a former approach I tried to get the positions of the corresponding spots to obtain the relative distances deltaX and deltaY. These distances then I fitted to the taylor-expansion up to second order of T(x,y) giving me the x- and y-component of the displacement vector for every pixel (x,y) on the left, pointing to corresponding pixel (x',y') on the right.
To get a more general result I would like to use normalized cross-correlation. For this I multiply every pixelvalue from left with a corresponding pixelvalue from right and sum over these products. The transformation I am looking for should connect the pixels that will maximize the sum. So when the sum is maximzied, I know that I multiplied the corresponding pixels.
I really tried a lot with this, but didn't manage. My question is if somebody of you has an idea or has ever done something similar.
import numpy as np
import Image
left = np.array(Image.open('left.png'))
right = np.array(Image.open('right.png'))
# for normalization (http://en.wikipedia.org/wiki/Cross-correlation#Normalized_cross-correlation)
left = (left - left.mean()) / left.std()
right = (right - right.mean()) / right.std()
Please let me know if I can make this question more clear. I still have to check out how to post questions using latex.
Thank you very much for input.
[left.png] http://i.stack.imgur.com/oSTER.png
[right.png] http://i.stack.imgur.com/Njahj.png
I'm afraid, in most cases 16-bit images appear just black (at least on systems I use) :( but of course there is data in there.
UPDATE 1
I try to clearify my question. I am looking for a vector-field with displacement-vectors that point from every pixel in left.png to the corresponding pixel in right.png. My problem is, that I am not sure about the constraints I have.
where vector r (components x and y) points to a pixel in left.png and vector r-prime (components x-prime and y-prime) points to the corresponding pixel in right.png. for every r there is a displacement-vector.
What I did earlier was, that I found manually components of vector-field d and fitted them to a polynom second degree:
So I fitted:
and
Does this make sense to you? Is it possible to get all the delta-x(x,y) and delta-y(x,y) with cross-correlation? The cross-correlation should be maximized if the corresponding pixels are linked together thru the displacement-vectors, right?
UPDATE 2
So the algorithm I was thinking of is as follows:
Deform right.png
Get the value of cross-correlation
Deform right.png further
Get the value of cross-correlation and compare to value before
If it's greater, good deformation, if not, redo deformation and do something else
After maximzied the cross-correlation value, know what deformation there is :)
About deformation: could one do first a shift along x- and y-direction to maximize cross-correlation, then in a second step stretch or compress x- and y-dependant and in a third step deform quadratic x- and y-dependent and repeat this procedure iterativ?? I really have a problem to do this with integer-coordinates. Do you think I would have to interpolate the picture to obtain a continuous distribution?? I have to think about this again :( Thanks to everybody for taking part :)

OpenCV (and with it the python Opencv binding) has a StarDetector class which implements this algorithm.
As an alternative you might have a look at the OpenCV SIFT class, which stands for Scale Invariant Feature Transform.
Update
Regarding your comment, I understand that the "right" transformation will maximize the cross-correlation between the images, but I don't understand how you choose the set of transformations over which to maximize. Maybe if you know the coordinates of three matching points (either by some heuristics or by choosing them by hand), and if you expect affinity, you could use something like cv2.getAffineTransform to have a good initial transformation for your maximization process. From there you could use small additional transformations to have a set over which to maximize. But this approach seems to me like re-inventing something which SIFT could take care of.
To actually transform your test image you can use cv2.warpAffine, which also can take care of border values (e.g. pad with 0). To calculate the cross-correlation you could use scipy.signal.correlate2d.
Update
Your latest update did indeed clarify some points for me. But I think that a vector field of displacements is not the most natural thing to look for, and this is also where the misunderstanding came from. I was thinking more along the lines of a global transformation T, which applied to any point (x,y) of the left image gives (x',y')=T(x,y) on the right side, but T has the same analytical form for every pixel. For example, this could be a combination of a displacement, rotation, scaling, maybe some perspective transformation. I cannot say whether it is realistic or not to hope to find such a transformation, this depends on your setup, but if the scene is physically the same on both sides I would say it is reasonable to expect some affine transformation. This is why I suggested cv2.getAffineTransform. It is of course trivial to calculate your displacement Vector field from such a T, as this is just T(x,y)-(x,y).
The big advantage would be that you have only very few degrees of freedom for your transformation, instead of, I would argue, 2N degrees of freedom in the displacement vector field, where N is the number of bright spots.
If it is indeed an affine transformation, I would suggest some algorithm like this:
identify three bright and well isolated spots on the left
for each of these three spots, define a bounding box so that you can hope to identify the corresponding spot within it in the right image
find the coordinates of the corresponding spots, e.g. with some correlation method as implemented in cv2.matchTemplate or by also just finding the brightest spot within the bounding box.
once you have three matching pairs of coordinates, calculate the affine transformation which transforms one set into the other with cv2.getAffineTransform.
apply this affine transformation to the left image, as a check if you found the right one you could calculate if the overall normalized cross-correlation is above some threshold or drops significantly if you displace one image with respect to the other.
if you wish and still need it, calculate the displacement vector field trivially from your transformation T.
Update
It seems cv2.getAffineTransform expects an awkward input data type 'float32'. Let's assume the source coordinates are (sxi,syi) and destination (dxi,dyi) with i=0,1,2, then what you need is
src = np.array( ((sx0,sy0),(sx1,sy1),(sx2,sy2)), dtype='float32' )
dst = np.array( ((dx0,dy0),(dx1,dy1),(dx2,dy2)), dtype='float32' )
result = cv2.getAffineTransform(src,dst)

I don't think a cross correlation is going to help here, as it only gives you a single best shift for the whole image. There are three alternatives I would consider:
Do a cross correlation on sub-clusters of dots. Take, for example, the three dots in the top right and find the optimal x-y shift through cross-correlation. This gives you the rough transform for the top left. Repeat for as many clusters as you can to obtain a reasonable map of your transformations. Fit this with your Taylor expansion and you might get reasonably close. However, to have your cross-correlation work in any way, the difference in displacement between spots must be less than the extend of the spot, else you can never get all spots in a cluster to overlap simultaneously with a single displacement. Under these conditions, option 2 might be more suitable.
If the displacements are relatively small (which I think is a condition for option 1), then we might assume that for a given spot in the left image, the closest spot in the right image is the corresponding spot. Thus, for every spot in the left image, we find the nearest spot in the right image and use that as the displacement in that location. From the 40-something well distributed displacement vectors we can obtain a reasonable approximation of the actual displacement by fitting your Taylor expansion.
This is probably the slowest method, but might be the most robust if you have large displacements (and option 2 thus doesn't work): use something like an evolutionary algorithm to find the displacement. Apply a random transformation, compute the remaining error (you might need to define this as sum of the smallest distance between spots in your original and transformed image), and improve your transformation with those results. If your displacements are rather large you might need a very broad search as you'll probably get lots of local minima in your landscape.
I would try option 2 as it seems your displacements might be small enough to easily associate a spot in the left image with a spot in the right image.
Update
I assume your optics induce non linear distortions and having two separate beampaths (different filters in each?) will make the relationship between the two images even more non-linear. The affine transformation PiQuer suggests might give a reasonable approach but can probably never completely cover the actual distortions.
I think your approach of fitting to a low order Taylor polynomial is fine. This works for all my applications with similar conditions. Highest orders probably should be something like xy^2 and x^2y; anything higher than that you won't notice.
Alternatively, you might be able to calibrate the distortions for each image first, and then do your experiments. This way you are not dependent on the distribution of you dots, but can use a high resolution reference image to get the best description of your transformation.
Option 2 above still stands as my suggestion for getting the two images to overlap. This can be fully automated and I'm not sure what you mean when you want a more general result.
Update 2
You comment that you have trouble matching dots in the two images. If this is the case, I think your iterative cross-correlation approach may not be very robust either. You have very small dots, so overlap between them will only occur if the difference between the two images is small.
In principle there is nothing wrong with your proposed solution, but whether it works or not strongly depends on the size of your deformations and the robustness of your optimization algorithm. If you start off with very little overlap, then it may be hard to find a good starting point for your optimization. Yet if you have sufficient overlap to begin with, then you should have been able to find the deformation per dot first, but in a comment you indicate that this doesn't work.
Perhaps you can go for a mixed solution: find the cross correlation of clusters of dots to get a starting point for your optimization, and then tweak the deformation using something like the procedure you describe in your update. Thus:
For a NxN pixel segment find the shift between the left and right images
Repeat for, say, 16 of those segments
Compute an approximation of the deformation using those 16 points
Use this as the starting point of your optimization approach

You might want to have a look at bunwarpj which already does what you're trying to do. It's not python but I use it in exactly this context. You can export a plain text spline transformation and use it if you wish to do so.

Detect the location of an image within a larger image

How do you detect the location of an image within a larger image? I have an unmodified copy of the image. This image is then changed to an arbitrary resolution and placed randomly within a much larger image which is of an arbitrary size. No other transformations are conducted on the resulting image. Python code would be ideal, and it would probably require libgd. If you know of a good approach to this problem you'll get a +1.

There is a quick and dirty solution, and that's simply sliding a window over the target image and computing some measure of similarity at each location, then picking the location with the highest similarity. Then you compare the similarity to a threshold, if the score is above the threshold, you conclude the image is there and that's the location; if the score is below the threshold, then the image isn't there.
As a similarity measure, you can use normalized correlation or sum of squared differences (aka L2 norm). As people mentioned, this will not deal with scale changes. So you also rescale your original image multiple times and repeat the process above with each scaled version. Depending on the size of your input image and the range of possible scales, this may be good enough, and it's easy to implement.
A proper solution is to use affine invariants. Try looking up "wide-baseline stereo matching", people looked at that problem in that context. The methods that are used are generally something like this:
Preprocessing of the original image
Run an "interest point detector". This will find a few points in the image which are easily localizable, e.g. corners. There are many detectors, a detector called "harris-affine" works well and is pretty popular (so implementations probably exist). Another option is to use the Difference-of-Gaussians (DoG) detector, it was developed for SIFT and works well too.
At each interest point, extract a small sub-image (e.g. 30x30 pixels)
For each sub-image, compute a "descriptor", some representation of the image content in that window. Again, many descriptors exist. Things to look at are how well the descriptor describes the image content (you want two descriptors to match only if they are similar) and how invariant it is (you want it to be the same even after scaling). In your case, I'd recommend using SIFT. It is not as invariant as some other descriptors, but can cope with scale well, and in your case scale is the only thing that changes.
At the end of this stage, you will have a set of descriptors.
Testing (with the new test image).
First, you run the same interest point detector as in step 1 and get a set of interest points. You compute the same descriptor for each point, as above. Now you have a set of descriptors for the target image as well.
Next, you look for matches. Ideally, to each descriptor from your original image, there will be some pretty similar descriptor in the target image. (Since the target image is larger, there will also be "leftover" descriptors, i.e. points that don't correspond to anything in the original image.) So if enough of the original descriptors match with enough similarity, then you know the target is there. Moreover, since the descriptors are location-specific, you will also know where in the target image the original image is.

You probably want cross-correlation. (Autocorrelation is correlating a signal with itself; cross correlating is correlating two different signals.)
What correlation does for you, over simply checking for exact matches, is that it will tell you where the best matches are, and how good they are. Flip side is that, for a 2-D picture, it's something like O(N^3), and it's not that simple an algorithm. But it's magic once you get it to work.
EDIT: Aargh, you specified an arbitrary resize. That's going to break any correlation-based algorithm. Sorry, you're outside my experience now and SO won't let me delete this answer.

http://en.wikipedia.org/wiki/Autocorrelation is my first instinct.

Take a look at Scale-Invariant Feature Transforms; there are many different flavors that may be more or less tailored to the type of images you happen to be working with.