## Solving mazes using image analysis

I found an interesting submission on the File Exchange. It uses a very simple set of mathematical morphology filters to find the solution to simple mazes. I have several other solutions, so I decided to write an entry about the subject.

### Original solution

This section is paraphrased from File Exchange “Image Analyst”‘s solution. His code uses MATLAB’s image processing toolbox, here we want to use DIPimage, of course.

We start by reading in the image with the maze:

`inputmaze = readim('maze_solution_01.png')` Next we convert it to a grey-value image and threshold it to obtain a binary image. The maze walls are true and shown in red:

```maze = colorspace(inputmaze,'grey');
maze = ~threshold(maze)``` When we label the binary image, we can clearly see two discrete, separate walls:

```maze = label(maze);
dipshow(maze*2,'labels')``` It is fairly easy to see the path from start to finish: it’s the path in between the two colors. To draw this path over the image we can follow this simple sequence of commands, it’s all mathematical morphology. We ignore everything except the first of the two walls:

`path = maze==1` Then we dilate the walls by a few pixels, and fill all the holes:

```pathwidth = 5;
path = bdilation(path,pathwidth);
path = fillholes(path)``` Then we erode by the same amount of pixels and take the difference:

`path = path - berosion(path,pathwidth);`

The function `overlay` makes a nice display:

`overlay(inputmaze,path)` However, when the maze is more complex, things start to go wrong. In this maze I’ve added a photograph, and cut two holes to create more paths:

`inputmaze = readim('maze_solution_07.png')` The result after labeling now shows more than two colors:

```maze = colorspace(inputmaze,'grey');
maze = ~threshold(maze);
maze = label(maze);
dipshow(maze,'labels')``` This is not a big deal, but when we follow the first wall, we’ll find only one of the possible paths, and it’s not necessarily the shortest path. Also, we are lucky that object with ID=1 is a good object to follow. If you pick the wrong object ID here, you’ll end up with a path that doesn’t connect the input and output!

```path = maze==1;
pathwidth = 5;
path = bdilation(path,pathwidth);
path = fillholes(path);
path = path - berosion(path,pathwidth);
overlay(inputmaze,path)``` ### Alternate solution

A more robust solution involves using the skeleton. This is a morphological operation that, in rough terms, erodes and object under certain constraints, until the whole object is a single pixel thick. The constraints are that a pixel is never removed if removing it would change the geometry of the object. That is, it will never split an object in two. Let’s go back to the binarised maze image, but now we want the path to be the object, rather than the walls:

```inputmaze = readim('maze_solution_07.png');
maze = colorspace(inputmaze,'grey');
maze = threshold(maze);```

Because of the way that the skeleton is implemented in DIPimage, we need to add a 2-pixel border around the maze. The skeleton ignores the pixels in this border, so if we don’t add it, part of our maze wouldn’t be taken into account:

`maze = extend(maze,imsize(maze)+4);`

The skeleton now produces a 1-pixel-thick version of the maze:

`path = bskeleton(maze)` All the dead ends can be easily found, by looking at pixels that have only one neighbor:

```deadends = getendpixel(path); One of the options of the `bskeleton` function is to remove these dead-end pixels iteratively. This leaves the skeleton as a set of dots and closed loops. By setting the area outside the image to the value 1, we make sure that the bits of the skeleton connected to the image border are kept:

`path = bskeleton(maze,1,'looseendsaway')` Notice that there are several loops and dots not connected to the image border. These are caused by the image I added to the maze. We can remove those by keeping only those objects that are connected to the image border:

`path = path-brmedgeobjs(path,2)` Finally we’ll remove the temporary image border we added to compute the skeleton, and make the path a little thicker for display:

```path = cut(path,imsize(inputmaze));
path = bdilation(path,1);
dipshow(overlay(inputmaze,path,[255,0,0]))``` This returned all possible paths from start to finish. But how do we select only the optimal path?

### A better solution

The optimal path selection in graph theory is found using Dijkstra’s algorithm. We could easily implement this algorithm by extracting the paths found by the skeleton as a graph. But we want to do this here using image analysis algorithms.

Let’s again start by reading in the maze image and obtaining a binary representation:

```inputmaze = readim('maze_solution_07.png');
maze = colorspace(inputmaze,'grey');
maze = ~threshold(maze);```

Our first task is to find where the start and stop points are for our path. We’re simply looking for black pixels connected to the image edge:

```markers = bpropagation(newim(maze,'bin'),~maze,3,1,1);
markers = label(markers);```

Let’s make sure we have exactly two exits, and let’s get their coordinates:

```if max(markers)~=2
error('The maze doesn''t have two exits!');
end
m = measure(markers,[],'center');
start = round(m(1).center)
stop = round(m(2).center)```
```start =
401
37

stop =
365
401
```

The most important step to find the shortest path is the distance transform. This function simply computes the distance from every object pixel in the image to the background. But instead of using Euclidean distances, which take the straight-line distance between the pixel and the background, we use a constrained distance. This constraint forces the distances computed to follow the possible paths through the maze. One of the simplest ways of constraining the distance transform is to weigh the distance with the image’s grey value. That is, the distance between two points is computed as the integral of the grey values along the path between the two points. This is called the grey-weighted distance transform. Incidentally, the grey-weighted distance transform is computed with something very similar to Dijkstra’s algorithm.

By giving the maze walls very high grey values, we can make sure that going around the maze through some long-winded path produces a shorter (weighted) distance than taking a “short-cut” through a wall. We give the maze floors a grey value of 1, so that the distance travelled through the maze is equal to the path length, and the maze walls a value of 106, larger than any path you could possibly draw in such a small image. Furthermore, as with the skeleton, we need to add a little border around the image:

```region = extend(markers~=2,imsize(maze)+4); % We compute the distance to the stop point
weights = extend(1+(maze*1e6),imsize(maze)+4);
distance = gdt(region,weights,3);
distance = cut(distance,imsize(maze));
dipshow(distance,[0,2000])``` As you can see, the further from the stop point, the higher the grey value of the output is. There is a region in the top-left that cannot be reached at all from the stop point. The distance from start to finish is:

`D = double(distance(start(1),start(2)))`
```D =
1.0716e+03
```

To find the path all we now need to do is follow a steepest descent path from the start all the way to the finish. We start at the start point, and look at all its neighbours for the one that has the lowest value. That then becomes the current point and we repeat until we are at the bottom (where `distance==0`).

```current = start;
path = current;
while distance(current(1),current(2))>0
n = distance(current(1)+[-1,0,1],current(2)+[-1,0,1]);
[n,step] = min(n);
step = step-1;
current = current + step';
path = [path,current];
end```

We can now plot the path on the input image:

```dipshow(inputmaze)
hold on
plot(path(1,:),path(2,:),'r','linewidth',3)``` ### 16 Responses to “Solving mazes using image analysis”

1. On April 17th, 2010, at 21:22, vahid said:

Thanks .

2. On April 26th, 2010, at 22:09, Thomas said:

A”maz”ing use of image processing, thank you for taking the time to share it!

3. On April 28th, 2010, at 15:22, Matteo said:

This is fantastic – puzzle + programming – thank you!!

4. On May 19th, 2010, at 16:47, Tweets that mention Cris’s Image Analysis Blog » Blog Archive » Solving mazes using image analysis -- Topsy.com said:

[…] This post was mentioned on Twitter by Matthew McDonnell, chibaf. chibaf said: RT @mattmcd: 'Solving mazes using image analysis' http://is.gd/cfDnU (using the #MATLAB Image Processing) See also http://is.gd/cfDyT #maze […]

5. On May 19th, 2010, at 17:10, Cris’s Image Analysis Blog » Blog Archive » Solving mazes using image analysis « MATLAB Stuff on the Web said:

[…] File Exchange submission by Image Analyst inspires a blog post by Loren Shure and another blog post by Chris […]

6. On February 17th, 2011, at 3:08, vidya said:

hello !sorry if this sounds dumb …..when there are multiple paths the node at which the paths branch have number of 1s…how do u select one from them and make sure u do not follow that path the next time….

7. On February 17th, 2011, at 14:49, Cris Luengo said:

Hi Vidya,

I’m not sure which of the methods you are referring to. Using the grey-weighted distance transform, the distance to every point within the maze is computed, then the steepest path is taken from the exit back to the start point. In this case, at every intersection, there is always one direction in which the grey values decrease faster. That indicates the direction of the shortest path. You don’t need to go back, because you never go into dead ends.

In you are talking about simply walking along a maze from pixel to pixel until you find the exit, at each intersection you would push the coordinates on a stack, together with the possible directions to try. When you then hit a dead end, you take the last set of coordinates on the stack and continue from there.

8. On May 8th, 2011, at 15:19, sardar said:

9. On May 9th, 2011, at 9:58, Cris Luengo said:

Sardar: Sorry, but I don’t have time to write specific code for people.

10. On November 6th, 2011, at 9:44, xiao said:

it’s interesting^_^

11. On July 6th, 2012, at 18:48, 用迷宫困住死理性派？没门！ : 临溪笔记 said:

[…] 另一类迷宫的特点是你时刻可以看到它的全貌，比如人们经常玩的“纸上迷宫”。它的解法很容易，只要用铅笔把所有看到的死胡同涂黑，正确的路径就显现出来了。另一种略显粗暴而又不失技术含量的解法是 图像分析法 。这里举个简单的例子： […]

12. On March 18th, 2013, at 12:01, Resolviendo laberintos utilizando el programa Paint | Francis (th)E mule Science's News said:

[…] Trata de resolver este laberinto (la entrada está en el lateral derecho, en la parte de arriba, y la salida en el lateral inferior, a la derecha). Imagina que copias esta figura en un programa de dibujo (como Paint de Windows). ¿Cómo podrías resolver este laberinto utilizando dicho programa? Piensa un poco, porque es muy fácil. Si al final no logras resolverlo, sigue este enlace para ver la solución. Por supuesto, este método gráfico es muy viejo, fue publicado por E. W. Dijkstra, “A note on two problems in connexion with graphs,” Numerische Mathematik 1: 269-271, 1959 [copias gratis en pdf]. Quien prefiera una explicación for dummies (de donde he sacado este ejemplo) y una implementación en Matlab preferirá leer a Cris Luengo, “Solving mazes using image analysis,” Cris’s Image Analysis Blog, April 17, 2010. […]

13. On April 14th, 2015, at 19:11, Matteo said:

Hi Cris

I often come back to read this article again, it has it all, the fun and the techniques.

A couple of questions:

If you were to superimpose in one single image all the path results to compare them, see where they are the same, and where they differ, what would be the best way, in your opinion?

A bit of an aside: if you had two skeleton results, one from an input image, one from an enhanced image (sharpened or otherwise filtered) and you wanted to compare them numerically (e.g. to see if the filtering really has enhanced the image), what would you do? One tough I had was perhaps to count the number of segments required to an edge linking operation of some kind to get the image to have the same number of labeled component. A similar process could involve opening instead of edge linking. What do you think?

Thanks

Matteo

14. On April 15th, 2015, at 15:34, Cris Luengo said:

Hi Matteo,

As the paths are sets of pixels, I would superimpose them by adding their binary masks. Pixels used by three paths will have a value of three. Some color mapping will then help in seeing how paths merge or split, and which portions are the most “popular”.

Regarding the skeleton, your suggestion could work. You could also split the skeleton at the branch points, and count the number and length of each of the remaining segments. A less noisy image would have fewer and longer segments, I presume?

15. On April 16th, 2015, at 17:51, Matteo said:

Hi Cris

Thanks for your comments. For the latter, I will try your suggestion. For the former, I’d thought about summation, perhaps preceded by a very small amount of dilation to get a smoother “popularity”.

Matteo

16. On December 25th, 2015, at 10:47, Maze | Gaurish4Math said:

[…] Also, there is an interesting article on Maze solving using image analysis: http://www.crisluengo.net/index.php/archives/277 […]

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