2010-09-10 12:12:25 8 Comments

I'm helping a veterinary clinic measuring pressure under a dogs paw. I use Python for my data analysis and now I'm stuck trying to divide the paws into (anatomical) subregions.

I made a 2D array of each paw, that consists of the maximal values for each sensor that has been loaded by the paw over time. Here's an example of one paw, where I used Excel to draw the areas I want to 'detect'. These are 2 by 2 boxes around the sensor with local maxima's, that together have the largest sum.

So I tried some experimenting and decide to simply look for the maximums of each column and row (can't look in one direction due to the shape of the paw). This seems to 'detect' the location of the separate toes fairly well, but it also marks neighboring sensors.

So what would be the best way to tell Python which of these maximums are the ones I want?

**Note: The 2x2 squares can't overlap, since they have to be separate toes!**

Also I took 2x2 as a convenience, any more advanced solution is welcome, but I'm simply a human movement scientist, so I'm neither a real programmer or a mathematician, so please keep it 'simple'.

Here's a version that can be loaded with `np.loadtxt`

## Results

So I tried @jextee's solution (see the results below). As you can see, it works very on the front paws, but it works less well for the hind legs.

More specifically, it can't recognize the small peak that's the fourth toe. This is obviously inherent to the fact that the loop looks top down towards the lowest value, without taking into account where this is.

Would anyone know how to tweak @jextee's algorithm, so that it might be able to find the 4th toe too?

Since I haven't processed any other trials yet, I can't supply any other samples. But the data I gave before were the averages of each paw. This file is an array with the maximal data of 9 paws in the order they made contact with the plate.

This image shows how they were spatially spread out over the plate.

## Update:

**I have set up a blog for anyone interested** and I have setup a SkyDrive with all the raw measurements. So to anyone requesting more data: more power to you!

## New update:

So after the help I got with my questions regarding paw detection and paw sorting, I was finally able to check the toe detection for every paw! Turns out, it doesn't work so well in anything but paws sized like the one in my own example. Off course in hindsight, it's my own fault for choosing the 2x2 so arbitrarily.

Here's a nice example of where it goes wrong: a nail is being recognized as a toe and the 'heel' is so wide, it gets recognized twice!

The paw is too large, so taking a 2x2 size with no overlap, causes some toes to be detected twice. The other way around, in small dogs it often fails to find a 5th toe, which I suspect is being caused by the 2x2 area being too large.

After trying the current solution on all my measurements I came to the staggering conclusion that for nearly all my small dogs it didn't find a 5th toe and that in over 50% of the impacts for the large dogs it would find more!

So clearly I need to change it. My own guess was changing the size of the `neighborhood`

to something smaller for small dogs and larger for large dogs. But `generate_binary_structure`

wouldn't let me change the size of the array.

Therefore, I'm hoping that anyone else has a better suggestion for locating the toes, perhaps having the toe area scale with the paw size?

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## 22 comments

## @jtlz2 2019-02-08 09:27:13

There are several and extensive pieces of software available from the astronomy and cosmology community - this is a significant area of research both historically and currently.

Do not be alarmed if you are not an astronomer - some are easy to use outside the field. For example, you could use astropy/photutils:

https://photutils.readthedocs.io/en/stable/detection.html#local-peak-detection

[It seems a bit rude to repeat their short sample code here.]

An incomplete and slightly biased list of techniques/packages/links that might be of interest is given below - do add more in the comments and I will update this answer as necessary. Of course there is a trade-off of accuracy vs compute resources. [Honestly, there are too many to give code examples in a single answer such as this so I am not sure whether this answer will fly or not.]

Source Extractor https://www.astromatic.net/software/sextractor

MultiNest https://github.com/farhanferoz/MultiNest [+ pyMultiNest]

ASKAP/EMU source-finding challenge: https://arxiv.org/abs/1509.03931

You could also search for Planck and/or WMAP source-extraction challenges.

...

## @Ivan 2010-09-11 03:38:07

I detected the peaks using a

local maximum filter. Here is the result on your first dataset of 4 paws:I also ran it on the second dataset of 9 paws and it worked as well.

Here is how you do it:

All you need to do after is use

`scipy.ndimage.measurements.label`

on the mask to label all distinct objects. Then you'll be able to play with them individually.Notethat the method works well because the background is not noisy. If it were, you would detect a bunch of other unwanted peaks in the background. Another important factor is the size of theneighborhood. You will need to adjust it if the peak size changes (the should remain roughly proportional).## @Ryan Soklaski 2017-06-22 02:12:55

There is a simpler solution than (eroded_background ^ local_peaks). Just do (foreground & local peaks)

## @frank 2018-09-26 20:50:34

Here is paws.txt.

## @S. Huber 2017-11-08 21:41:43

Using persistent homology to analyze your data set I get the following result (click to enlarge):

This is the 2D-version of the peak detection method described in this SO answer. The above figure simply shows 0-dimensional persistent homology classes sorted by persistence.

I did upscale the original dataset by a factor of 2 using scipy.misc.imresize(). However, note that I did consider the four paws as one dataset; splitting it into four would make the problem easier.

Methodology.The idea behind this quite simple: Consider the function graph of the function that assigns each pixel its level. It looks like this:Now consider a water level at height 255 that continuously descents to lower levels. At local maxima islands pop up (birth). At saddle points two islands merge; we consider the lower island to be merged to the higher island (death). The so-called persistence diagram (of the 0-th dimensional homology classes, our islands) depicts death- over birth-values of all islands:

The

persistenceof an island is then the difference between the birth- and death-level; the vertical distance of a dot to the grey main diagonal. The figure labels the islands by decreasing persistence.The very first picture shows the locations of births of the islands. This method not only gives the local maxima but also quantifies their "significance" by the above mentioned persistence. One would then filter out all islands with a too low persistence. However, in your example every island (i.e., every local maximum) is a peak you look for.

Python code can be found here.

## @Thomio 2018-04-04 21:52:53

just wanna tell you guys there is a nice option to find local maxima in images with python.

or for skimage 0.8.0

http://scikit-image.org/docs/0.8.0/api/skimage.feature.peak.html

## @Rick Hull 2010-09-10 17:56:08

I am not sure this answers the question, but it seems like you can just look for the n highest peaks that don't have neighbors.

Here is the gist. Note that it's in Ruby, but the idea should be clear.

## @Ivo Flipse 2010-09-10 18:58:45

I'm going to try and have a look and see if I can convert it to Python code :-)

## @agf 2012-04-14 00:41:19

Please include the code in the post itself, rather than linking to a gist, if it's a reasonable length.

## @astromax 2013-08-13 21:02:03

I'm sure you have enough to go on by now, but I can't help but suggest using the k-means clustering method. k-means is an unsupervised clustering algorithm which will take you data (in any number of dimensions - I happen to do this in 3D) and arrange it into k clusters with distinct boundaries. It's nice here because you know exactly how many toes these canines (should) have.

Additionally, it's implemented in Scipy which is really nice (http://docs.scipy.org/doc/scipy/reference/cluster.vq.html).

Here's an example of what it can do to spatially resolve 3D clusters:

What you want to do is a bit different (2D and includes pressure values), but I still think you could give it a shot.

## @Eric O Lebigot 2010-09-10 13:05:10

Here is an idea: you calculate the (discrete) Laplacian of the image. I would expect it to be (negative and) large at maxima, in a way that is more dramatic than in the original images. Thus, maxima could be easier to find.

Here is another idea: if you know the typical size of the high-pressure spots, you can first smooth your image by convoluting it with a Gaussian of the same size. This may give you simpler images to process.

## @Ron 2010-09-11 01:07:08

thanks for the raw data. I'm on the train and this is as far as I've gotten (my stop is coming up). I massaged your txt file with regexps and have plopped it into a html page with some javascript for visualization. I'm sharing it here because some, like myself, might find it more readily hackable than python.

I think a good approach will be scale and rotation invariant, and my next step will be to investigate mixtures of gaussians. (each paw pad being the center of a gaussian).

## @Ivo Flipse 2010-09-11 09:27:38

I reckon this is a proof of concept that the recommended Gaussian techniques could work, now if only someone could prove it with Python ;-)

## @CakeMaster 2010-09-10 14:54:44

This is an image registration problem. The general strategy is:

prioron the data.roughlyaligned in the first place.Here's a rough and ready approach, "the dumbest thing that could possibly work":To counteract the orientation problem, you could have 8 or so initial settings for the basic directions (North, North East, etc). Run each one individually and throw away any results where two or more toes end up at the same pixel. I'll think about this some more, but this kind of thing is still being researched in image processing - there are no right answers!

Slightly more complex idea: (weighted) K-means clustering.It's not that bad.Then iterate until convergence:

This method will almost certainly give much better results, and you get the mass of each cluster which may help in identifying the toes.

(Again, you've specified the number of clusters up front. With clustering you have to specify the density one way or another: Either choose the number of clusters, appropriate in this case, or choose a cluster radius and see how many you end up with. An example of the latter is mean-shift.)

Sorry about the lack of implementation details or other specifics. I would code this up but I've got a deadline. If nothing else has worked by next week let me know and I'll give it a shot.

## @Ivo Flipse 2010-09-10 15:12:56

The problem is, that the paws change their orientation and I don't have any calibration/baseline of a correct paw to start with. Plus I fear that a lot of the image recognition algorithms are a bit out of my league.

## @CakeMaster 2010-09-10 16:14:27

The "rough and ready" approach is pretty simple - maybe I didn't the idea well. I'll put in some pseudocode to illustrate.

## @Ivo Flipse 2010-09-10 17:31:25

I have a feeling your suggestion will help fix the recognition of the hind paws, I just don't know 'how'

## @CakeMaster 2010-09-10 19:19:47

I've added another idea. By the way if you have a load of good data it would be cool to put it online somewhere. It could be useful for people studying image processing / machine learning and you might get some more code out of it...

## @Ivo Flipse 2010-09-10 19:22:55

I was just thinking about writing down my data processing on a simple Wordpress blog, simply to be of use for other and I have to write it down anyway. I like all your suggestions, but I fear I'll have to wait for someone without a deadline ;-)

## @sastanin 2010-09-10 14:09:34

## Solution

Data file: paw.txt. Source code:

Output without overlapping squares. It seems that the same areas are selected as in your example.

## Some comments

The tricky part is to calculate sums of all 2x2 squares. I assumed you need all of them, so there might be some overlapping. I used slices to cut the first/last columns and rows from the original 2D array, and then overlapping them all together and calculating sums.

To understand it better, imaging a 3x3 array:

Then you can take its slices:

Now imagine you stack them one above the other and sum elements at the same positions. These sums will be exactly the same sums over the 2x2 squares with the top-left corner in the same position:

When you have the sums over 2x2 squares, you can use

`max`

to find the maximum, or`sort`

, or`sorted`

to find the peaks.To remember positions of the peaks I couple every value (the sum) with its ordinal position in a flattened array (see

`zip`

). Then I calculate row/column position again when I print the results.## Notes

I allowed for the 2x2 squares to overlap. Edited version filters out some of them such that only non-overlapping squares appear in the results.

## Choosing fingers (an idea)

Another problem is how to choose what is likely to be fingers out of all the peaks. I have an idea which may or may not work. I don't have time to implement it right now, so just pseudo-code.

I noticed that if the front fingers stay on almost a perfect circle, the rear finger should be inside of that circle. Also, the front fingers are more or less equally spaced. We may try to use these heuristic properties to detect the fingers.

Pseudo code:

This is a brute-force approach. If N is relatively small, then I think it is doable. For N=12, there are C_12^5 = 792 combinations, times 5 ways to select a rear finger, so 3960 cases to evaluate for every paw.

## @Johannes Charra 2010-09-10 14:22:50

He'll have to filter out the paws manually, given your result list ... picking the four topmost results will give him the four possibilities to construct a 2x2 square containing the maximum value 6.8

## @Ivo Flipse 2010-09-10 14:26:53

The 2x2 boxes can't overlap, since if I want to do statistics, I don't want to be using the same region, I want to compare regions :-)

## @sastanin 2010-09-10 15:58:27

I edited the answer. Now there are no overlapping squares in the results.

## @Ivo Flipse 2010-09-10 17:22:43

I tried it out and it seems to work for the front paws, but less so for the hind ones. Guess we'll have to try something that knows where to look

## @sastanin 2010-09-10 22:16:22

I see what's the problem. I'll think how to recognize the best "constellation" of peaks to select. What do you think about an approach of "four in a row and one aside" or "four on a circle and one inside"?

## @Ivo Flipse 2010-09-10 22:53:30

As my second image indicates (here's a link for all paws), all peaks get marked if you check for the maxes per row and column, so perhaps rather than just going through the list sorted top down, we could check which of these maximums are the highest, while having no neighbor (ignoring everything close to the maximum). Perhaps even looking which sum of 2x2 is the largest for each row and column.

## @sastanin 2010-09-10 23:23:35

I explained my idea how fingers can be detected in pseudo code. If you like it, I may try to implement it tomorrow evening.

## @Ivo Flipse 2010-09-11 00:07:13

If we use some heuristics to determine the 'most likely' candidates for the two highest toes and perhaps based on the shape, the rear toe, it should be doable to reduce the amount of combinations. Also from looking at other suggestions using Gaussian filters, perhaps this would increase the effectiveness of your suggestion

## @dmckee 2010-09-10 22:49:13

This problem has been studied in some depth by physicists. There is a good implementation in ROOT. Look at the TSpectrum classes (especially TSpectrum2 for your case) and the documentation for them.

References:

...and for those who don't have access to a subscription to NIM:

## @Ivo Flipse 2010-09-10 23:59:44

For glancing over the article it does seem to describe the same data processing as what I'm trying here, however I fear it greatly surpassed my programming skills :(

## @dmckee 2010-09-11 00:13:04

@Ivo: I've never tried to implement it myself. I just use ROOT. There are python bindings, none-the-less, but be aware that ROOT is a pretty heavy package.

## @physicsmichael 2010-09-11 22:55:24

@Ivo Flipse: I agree with dmckee. You have a lot of promising leads in other answers. If they all fail and you feel like investing some time, you can delve into ROOT and it will (probably) do what you need it to. I've never known anyone who's tried to learn ROOT via the python bindings (rather than it's natural C++), so I wish you luck.

## @Bob Mottram 2010-09-10 22:39:09

Interesting problem. The solution I would try is the following.

Apply a low pass filter, such as convolution with a 2D gaussian mask. This will give you a bunch of (probably, but not necessarily floating point) values.

Perform a 2D non-maximal suppression using the known approximate radius of each paw pad (or toe).

This should give you the maximal positions without having multiple candidates which are close together. Just to clarify, the radius of the mask in step 1 should also be similar to the radius used in step 2. This radius could be selectable, or the vet could explicitly measure it beforehand (it will vary with age/breed/etc).

Some of the solutions suggested (mean shift, neural nets, and so on) probably will work to some degree, but are overly complicated and probably not ideal.

## @Ivo Flipse 2010-09-10 22:58:39

I have 0 experience with convolution matrices and Gaussian filters, so would you like to show how it would work on my example?

## @mbq 2010-09-10 19:24:39

Physicist's solution:

Define 5 paw-markers identified by their positions

`X_i`

and init them with random positions. Define some energy function combining some award for location of markers in paws' positions with some punishment for overlap of markers; let's say:(

`S(X_i)`

is the mean force in 2x2 square around`X_i`

,`alfa`

is a parameter to be peaked experimentally)Now time to do some Metropolis-Hastings magic:

1. Select random marker and move it by one pixel in random direction.

2. Calculate dE, the difference of energy this move caused.

3. Get an uniform random number from 0-1 and call it r.

4. If

`dE<0`

or`exp(-beta*dE)>r`

, accept the move and go to 1; if not, undo the move and go to 1.This should be repeated until the markers will converge to paws. Beta controls the scanning to optimizing tradeoff, so it should be also optimized experimentally; it can be also constantly increased with the time of simulation (simulated annealing).

## @Ivo Flipse 2010-09-10 19:43:52

Care to show how this would work on my example? As I'm really not into high level math, so I already have a hard time unraveling the formula you proposed :(

## @mbq 2010-09-10 20:08:28

This is high school math, probably my notation is just obfuscated. I have a plan to check it, so stay tuned.

## @dmckee 2010-09-11 00:43:26

I'm a particle physicist. For a long time the go-to software tool in our discipline was called PAW, and it had a entity related to graphs called a "marker". You can imagine how confusing I found this answer on the first couple of times around...

## @Aidan Brumsickle 2010-09-10 19:19:18

Perhaps you can use something like Gaussian Mixture Models. Here's a Python package for doing GMMs (just did a Google search) http://www.ar.media.kyoto-u.ac.jp/members/david/softwares/em/

## @joshua 2010-09-10 19:05:25

a rough outline...

you'd probably want to use a connected components algorithm to isolate each paw region. wiki has a decent description of this (with some code) here: http://en.wikipedia.org/wiki/Connected_Component_Labeling

you'll have to make a decision about whether to use 4 or 8 connectedness. personally, for most problems i prefer 6-connectedness. anyway, once you've separated out each "paw print" as a connected region, it should be easy enough to iterate through the region and find the maxima. once you've found the maxima, you could iteratively enlarge the region until you reach a predetermined threshold in order to identify it as a given "toe".

one subtle problem here is that as soon as you start using computer vision techniques to identify something as a right/left/front/rear paw and you start looking at individual toes, you have to start taking rotations, skews, and translations into account. this is accomplished through the analysis of so-called "moments". there are a few different moments to consider in vision applications:

central moments: translation invariant normalized moments: scaling and translation invariant hu moments: translation, scale, and rotation invariant

more information about moments can be found by searching "image moments" on wiki.

## @antirez 2010-09-10 18:33:32

It's probably worth to try with neural networks if you are able to create some training data... but this needs many samples annotated by hand.

## @Ivo Flipse 2010-09-10 18:52:14

If it's worth the trouble, I wouldn't mind annotating a large sample by hand. My problem would be: how do I implement this, since I know nothing about programming neural networks

## @Paulus 2010-09-10 18:21:33

Heres another approach that I used when doing something similar for a large telescope:

1) Search for the highest pixel. Once you have that, search around that for the best fit for 2x2 (maybe maximizing the 2x2 sum), or do a 2d gaussian fit inside the sub region of say 4x4 centered on the highest pixel.

Then set those 2x2 pixels you have found to zero (or maybe 3x3) around the peak center

go back to 1) and repeat till the highest peak falls below a noise threshold, or you have all the toes you need

## @Ivo Flipse 2010-09-10 19:16:28

Care to share a code example that does this? I can follow what you're trying to do, but have no idea how to code it myself

## @Paulus 2010-09-10 20:38:06

I have some matlab code, will that help?

## @Ivo Flipse 2010-09-10 20:55:05

I actually come from working with Matlab, so yes that would already help. But if you use really foreign functions, it might be hard for me to replicate it with Python

## @geoff 2010-09-10 17:45:53

It seems you can cheat a bit using jetxee's algorithm. He is finding the first three toes fine, and you should be able to guess where the fourth is based off that.

## @Justin Peel 2010-09-10 14:46:33

Well, here's some simple and not terribly efficient code, but for this size of a data set it is fine.

I basically just make an array with the position of the upper-left and the sum of each 2x2 square and sort it by the sum. I then take the 2x2 square with the highest sum out of contention, put it in the

`best`

array, and remove all other 2x2 squares that used any part of this just removed 2x2 square.It seems to work fine except with the last paw (the one with the smallest sum on the far right in your first picture), it turns out that there are two other eligible 2x2 squares with a larger sum (and they have an equal sum to each other). One of them is still selects one square from your 2x2 square, but the other is off to the left. Fortunately, by luck we see to be choosing more of the one that you would want, but this may require some other ideas to be used to get what you actually want all of the time.

## @Ivo Flipse 2010-09-10 17:44:29

I reckon your results are the same as the ones in @Jextee's answer. Or at least so it seems from me testing it.

## @Cedric H. 2010-09-10 13:05:50

What if you proceed step by step: you first locate the global maximum, process if needed the surrounding points given their value, then set the found region to zero, and repeat for the next one.

## @Ivo Flipse 2010-09-10 13:10:43

Hmmm that setting to zero would at least remove it from any further calculations, that would be useful.

## @Daniyar 2010-09-10 17:48:49

Instead of setting to zero, you may calculate a gaussian function with hand picked parameters and subtract the found values from the original pressure readings. So if the toe is pressing your sensors, then by finding the highest pressing point, you use it to decrease the effect of that toe on the sensors, thus, eliminating the neighbouring cells with high pressure values. en.wikipedia.org/wiki/File:Gaussian_2d.png

## @Ivo Flipse 2010-09-10 18:39:09

Care to show an example based on my sample data @Daniyar? As I'm really not familiar with such kind of data processing

## @Johannes Charra 2010-09-10 13:00:29

Maybe a naive approach is sufficient here: Build a list of all 2x2 squares on your plane, order them by their sum (in descending order).

First, select the highest-valued square into your "paw list". Then, iteratively pick 4 of the next-best squares that don't intersect with any of the previously found squares.

## @Ivo Flipse 2010-09-10 13:08:29

I actually made a list with all the 2x2 sums, but when I had them ordered I had no idea how to iteratively compare them. My problem was that when I sorted it, I lost track of the coordinates. Perhaps I could stick them in a dictionary, with the coordinates as the key.

## @Johannes Charra 2010-09-10 13:31:09

Yes, some kind of dictionary would be necessary. I would have assumed that your representation of the grid is some sort of dictionary already.

## @Ivo Flipse 2010-09-10 13:53:09

Well the image you see above is a numpy array. The rest is currently stored in multidimensional lists. It would probably be better to stop doing that, though I'm not as familiar with iterating over dictionaries

## @ChrisC 2010-09-10 12:38:45

Just a couple of ideas off the top of my head:

You might also want to take a look at OpenCV, it's got a fairly decent Python API and might have some functions you'd find useful.

## @Ivo Flipse 2010-09-10 13:37:52

With gradient, you mean I should calculate the steepness of the slopes, once this get's above a certain value I know there's 'a peak'? I tried this, but some of the toes only have very low peaks (1.2 N/cm) compared to some of the others (8 N/cm). So how should I handle peaks with a very low gradient?

## @ChrisC 2010-09-10 14:27:20

What's worked for me in the past if I couldn't use the gradient directly was to look at gradient and the maxima, e.g. if the gradient is a local extrema and I'm at a local maxima, then I'm at a point of interest.