Recognition of handwritten circles, diamonds and rectangles

Question 1

Given the possible variation in handwritten inputs I would suggest that a neural network approach is the way to go; you will find it difficult or impossible to accurately model these classes by hand. LastCoder's attempt works to a degree, but it does not cope with much variation or have promise for high accuracy if worked on further - this kind of hand-engineered approach was abandoned a very long time ago.

State-of-the-art results in handwritten character classification these days is typically achieved with convolutional neural networks (CNNs). Given that you have only 3 classes the problem should be easier than digit or character classification, although from experience with the MNIST handwritten digit dataset, I expect that your circles, squares and diamonds may occasionally end up being difficult for even humans to distinguish.

So, if it were up to me I would use a CNN. I would input binary images taken from the drawing area to the first layer of the network. These may require some preprocessing. If the drawn shapes cover a very small area of the input space you may benefit from bulking them up (i.e. increasing line thickness) so as to make the shapes more invariant to small differences. It may also be beneficial to centre the shape in the image, although the pooling step might alleviate the need for this.

I would also point out that the more training data the better. One is often faced with a trade-off between increasing the size of one's dataset and improving one's model. Synthesising more examples (e.g. by skewing, rotating, shifting, stretching, etc) or spending a few hours drawing shapes may provide more of a benefit than you could get in the same time attempting to improve your model.

Good luck with your app!

Question 2

Here's some working code for a shape classifier. http://jsfiddle.net/R3ns3/ I pulled the threshold numbers (*Threshold variables in the code) out of the ether, so of course they can be tweaked for better results.

I use the bounding box, average point in a sub-section, angle between points, polar angle from bounding box center, and corner recognition. It can classify hand drawn rectangles, diamonds, and circles. The code records points while the mouse button is down and tries to classify when you stop drawing.

HTML

<canvas id="draw" width="300" height="300" style="position:absolute; top:0px; left:0p; margin:0; padding:0; width:300px; height:300px; border:2px solid blue;"></canvas>

JS

var state = {
    width: 300,
    height: 300,
    pointRadius: 2,
    cornerThreshold: 125,
    circleThreshold: 145,
    rectangleThreshold: 45,
    diamondThreshold: 135,
    canvas: document.getElementById("draw"),
    ctx: document.getElementById("draw").getContext("2d"),
    drawing: false,
    points: [],
    getCorners: function(angles, pts) {
        var list = pts || this.points;
        var corners = [];
        for(var i=0; i<angles.length; i++) {
            if(angles[i] <= this.cornerThreshold) {
                corners.push(list[(i + 1) % list.length]);
            }
        }
        return corners;
    },
    draw: function(color, pts) {
        var list = pts||this.points;
        this.ctx.fillStyle = color;
        for(var i=0; i<list.length; i++) {
            this.ctx.beginPath();
            this.ctx.arc(list[i].x, list[i].y, this.pointRadius, 0, Math.PI * 2, false);
            this.ctx.fill();
        }
    },
    classify: function() {
        // get bounding box
        var left = this.width, right = 0, 
            top = this.height, bottom = 0;
        for(var i=0; i<this.points.length; i++) {
            var pt = this.points[i];
            if(left > pt.x) left = pt.x;
            if(right < pt.x) right = pt.x;
            if(top > pt.y) top = pt.y;
            if(bottom < pt.y) bottom = pt.y;
        }
        var center = {x: (left+right)/2, y: (top+bottom)/2};
        this.draw("#00f", [
            {x: left, y: top},
            {x: right, y: top},
            {x: left, y: bottom},
            {x: right, y: bottom},
            ]);
        // find average point in each sector (9 sectors)
        var sects = [
            {x:0,y:0,c:0},{x:0,y:0,c:0},{x:0,y:0,c:0},
            {x:0,y:0,c:0},{x:0,y:0,c:0},{x:0,y:0,c:0},
            {x:0,y:0,c:0},{x:0,y:0,c:0},{x:0,y:0,c:0}
            ];
        var x3 = (right + (1/(right-left)) - left) / 3;
        var y3 = (bottom + (1/(bottom-top)) - top) / 3;
        for(var i=0; i<this.points.length; i++) {
            var pt = this.points[i];
            var sx = Math.floor((pt.x - left) / x3);
            var sy = Math.floor((pt.y - top) / y3);
            var idx = sy * 3 + sx;
            sects[idx].x += pt.x;
            sects[idx].y += pt.y;
            sects[idx].c ++;
            if(sx == 1 && sy == 1) {
                return "UNKNOWN";
            }
        }
        // get the significant points (clockwise)
        var sigPts = [];
        var clk = [0, 1, 2, 5, 8, 7, 6, 3]
        for(var i=0; i<clk.length; i++) {
            var pt = sects[clk[i]];
            if(pt.c > 0) {
                sigPts.push({x: pt.x / pt.c, y: pt.y / pt.c});
            } else {
                return "UNKNOWN";
            }
        }
        this.draw("#0f0", sigPts);
        // find angle between consecutive 3 points
        var angles = [];
        for(var i=0; i<sigPts.length; i++) {
            var a = sigPts[i],
                b = sigPts[(i + 1) % sigPts.length],
                c = sigPts[(i + 2) % sigPts.length],
                ab = Math.sqrt(Math.pow(b.x-a.x,2)+Math.pow(b.y-a.y,2)),
                bc = Math.sqrt(Math.pow(b.x-c.x,2)+ Math.pow(b.y-c.y,2)),
                ac = Math.sqrt(Math.pow(c.x-a.x,2)+ Math.pow(c.y-a.y,2)),
                deg = Math.floor(Math.acos((bc*bc+ab*ab-ac*ac)/(2*bc*ab)) * 180 / Math.PI);
            angles.push(deg);                
        }
        console.log(angles);
        var corners = this.getCorners(angles, sigPts);
        // get polar angle of corners
        for(var i=0; i<corners.length; i++) {
            corners[i].t = Math.floor(Math.atan2(corners[i].y - center.y, corners[i].x - center.x) * 180 / Math.PI);
        }
        console.log(corners);
        // whats the shape ?
        if(corners.length <= 1) { // circle
            return "CIRCLE";
        } else if(corners.length == 2) { // circle || diamond
            // difference of polar angles
            var diff = Math.abs((corners[0].t - corners[1].t + 180) % 360 - 180);
            console.log(diff);
            if(diff <= this.circleThreshold) {
                return "CIRCLE";
            } else {
                return "DIAMOND";
            }
        } else if(corners.length == 4) { // rectangle || diamond
            // sum of polar angles of corners
            var sum = Math.abs(corners[0].t + corners[1].t + corners[2].t + corners[3].t); 
            console.log(sum);
            if(sum <= this.rectangleThreshold) {
                return "RECTANGLE";
            } else if(sum >= this.diamondThreshold) {
                return "DIAMOND";
            } else {
                return "UNKNOWN";
            }
        } else {
            alert("draw neater please");
            return "UNKNOWN";
        }
    }
};
state.canvas.addEventListener("mousedown", (function(e) {
    if(!this.drawing) {
        this.ctx.clearRect(0, 0, 300, 300);
        this.points = [];
        this.drawing = true;
        console.log("drawing start");
    }
}).bind(state), false);
state.canvas.addEventListener("mouseup", (function(e) {
    this.drawing = false;
    console.log("drawing stop");
    this.draw("#f00");
    alert(this.classify());
}).bind(state), false);
state.canvas.addEventListener("mousemove", (function(e) {
    if(this.drawing) {
        var x = e.pageX, y = e.pageY;
        this.points.push({"x": x, "y": y});
        this.ctx.fillStyle = "#000";
        this.ctx.fillRect(x-2, y-2, 4, 4);
    }
}).bind(state), false);

Question 3

A linear Hough transform of the square or the diamond ought to be easy to recognize. They will both produce four point masses. The square's will be in pairs at zero and 90 degrees with the same y-coordinates for both pairs; in other words, a rectangle. The diamond will be at two other angles corresponding to how skinny the diamond is, e.g. 45 and 135 or else 60 and 120.

For the circle you need a circular Hough transform, and it will produce a single bright point cluster in 3d (x,y,r) Hough space.

Both linear and circular Hough transforms are implemented in OpenCV, and it's possible to run OpenCV on Android. These implementations include thresholding to identify lines and circles. See pg. 329 and pg. 331 of the documentation here.

If you are not familiar with Hough transforms, the Wikipedia page is not bad.

Another algorithm you may find interesting and perhaps useful is given in this paper about polygon similarity. I implemented it many years ago, and it's still around here. If you can convert the figures to loops of vectors, this algorithm could compare them against patterns, and the similarity metric would show goodness of match. The algorithm ignores rotational orientation, so if your definition of square and diamond is with respect to the axes of the drawing surface, you will have to modify the algorithm a bit to differentiate these cases.

Question 4

What you have here is a fairly standard clasification task, in an arguably vision domain. You could do this several ways, but the best way isn't known, and can sometimes depend on fine details of the problem.

So, this isn't an answer, per se, but there is a website - Kaggle.com that runs competition for classifications. One of the sample/experiemental tasks they list is reading single hand written numeric digits. That is close enough to this problem, that the same methods are almost certainly going to apply fairly well.

I suggest you go to https://www.kaggle.com/c/digit-recognizer and look around.

But if that is too vague, I can tell you from my reading of it, and playing with that problem space, that Random Forests are a better basic starting place than Neural networks.

Question 5

In this case (your 3 simple objects) you could try RanSaC-fitting for ellipse (getting the circle) and lines (getting the sides of the rectangle or diamond) - on each connected object if there are several objects to classify at the same time. Based on the actual setting (expected size, etc.) the RanSaC-parameters (how close must a point be to count as voter, how many voters you need at minimun) must be tuned. When you have found a line with RanSaC-fitting, remove the points "close" to it and go for the next line. The angles of the lines should make a distinction between diamand and rectangle easy.

Question 6

A very simple approach optimized for classifying exactly these 3 objects could be the following:

compute the center of gravity of an object to classify
then compute the distances of the center to the object points as a function on the angle (from 0 to 2 pi).
classify the resulting graph based on the smoothness and/or variance and the position and height of the local maxima and minima (maybe after smoothing the graph).

Question 7

I propose a way to do it in following steps : -

Take convex hull of the image (consider the shapes being convex)
divide into segments using clustering algorithms
Try to fit a curves or straight line to it and measure & threshold using training set which can be used for classifications
For your application try to divide into 4 clusters .
once you classify clusters as line or curves you can use the info to derive whether curve is circle,rectangle or diamond

Question 8

I think the answers that are already in place are good, but perhaps a better way of thinking about it is that you should try to break the problem into meaningful pieces.

If possible avoid the problem entirely. For instance if you are recognizing gestures, just analyze the gestures in real time. With gestures you can provide feedback to the user as to how your program interpreted their gesture and the user will change what they are doing appropriately.
Clean up the image in question. Before you do anything come up with an algorithm to try to select what the correct thing is you are trying to analyze. Also use an appropriate filter (convolution perhaps) to remove image artifacts before you begin the process.
Once you have figured out what the thing is you are going to analyze then analyze it and return a score, one for circle, one for noise, one for line, and the last for pyramid.
Repeat this step with the next viable candidate until you come up with the best candidate that is not noise.

I suspect you will find that you don't need a complicated algorithm to find circle, line, pyramid but that it is more so about structuring your code appropriately.

Question 9

If I was you I'll use already available Image Processing libraries like "AForge".
Take A look at this sample article:
http://www.aforgenet.com/articles/shape_checker

Question 10

I have a jar on github that can help if you are willing to unpack it and obey the apache license. You can try to recreate it in any other language as well.

Its an edge detector. The best step from there could be to:

find the corners (median of 90 degrees)
find mean median and maximum radius
find skew/angle from horizontal
have a decision agent decide what the shape is

Play around with it and find what you want.

My jar is open to the public at this address. It is not yet production ready but can help.

Just thought I could help. If anyone wants to be a part of the project, please do.

Question 11

I did this recently with identifying circles (bone centers) in medical images.

Note: Steps 1-2 are if you are grabbing from an image.

Psuedo Code Steps

Step 1. Highlight the Edges
edges = edge_map(of the source image) (using edge detector(s))
(laymens: show the lines/edges--make them searchable)

Step 2. Trace each unique edge
I would (use a nearest neighbor search 9x9 or 25x25) to identify / follow / trace each edge, collecting each point into the list (they become neighbors), and taking note of the gradient at each point.
This step produces: a set of edges.
(where one edge/curve/line = list of [point_gradient_data_structure]s
(laymens: Collect a set of points along the edge in the image)

Step 3. Analyze Each Edge('s points and gradient data)
For each edge,
if the gradient similar for a given region/set of neighbors (a run of points along an edge), then we have a straight line.
If the gradient is changing gradually, we have a curve.
Each region/run of points that is a straight line or a curve, has a mean (center) and other gradient statistics.

Step 4. Detect Objects
We can use the summary information from Step 3 to build conclusions about diamonds, circles, or squares. (i.e. 4 straight lines, that have end points near each other with proper gradients is a diamond or square. One (or more) curves with sufficient points/gradients (with a common focal point) makes a complete circle).

Note: Using an image pyramid can improve algorithm performance, both in terms of results and speed.

This technique (Steps 1-4) would get the job done for well defined shapes, and also could detect shapes that are drawn less than perfectly, and could handle slightly disconnected lines (if needed).

Note: With some machine learning techniques (mentioned by other posters), it could be helpful/important to have good "classifiers" to basically break the problem down into smaller parts/components, so then a decider further down the chain could use to better understand/"see" the objects. I think machine learning might be a little heavy-handed for this question, but still could produce reasonable results. PCA(face detection) could potentially work too.