More precisely, these are the areas where the change of the gray – for light to dark or dark to light – is the fastest. Then one needs a threshold so that all pixels where this change is higher that this number are considered “edges”:

Mathematically, we deal with

 the rate of change of the gray level

             = the gradient of the gray scale function.

(In fact, one only needs the norm of the gradient.) Computation of the derivative however in the digital (discrete) context is a challenge as it is severely affected by noise. Consider the image of coins and its version with noise added.

If now edge detection is run, the results are unsatisfactory – too many irrelevant contours. 

Of course it may be possible to filter out the smaller contours. In this particular case it’s impossible because they are parts of large ones. In fact they form large fractal-like structures. This is the reason why edge detection may have to be preceded by smoothing of the image.

Read more…

Under Review summary (in Output tab) Pixcavator shows the data about the objects found in the image. Pixcavator displays the total area of dark and the total area of light objects – as percentages of the total size of the image (second row).

Under certain circumstances though, the contours of the same kind may be “nested” and, as a results, these percentages may be wrong or even above 100%.

Example below (measuring grass coverage): the dark shows the 151% coverage.

The number is certainly meaningless (there will be a warning about that in the next release).

Why is it above 100%? Because the area is covered several times by these objects. If you click “Color objects”, you’ll see one large object with red contour and many others inside of it.

What happens is easier to see in this simpler image:

The results of image analysis may considered “good” here, but only in the sense that we have captured some 3D information. In general, we restrict our attention to image with mostly 2d information (see Images appropriate for analysis).

What exactly happens here? The way Pixcavator’s sliders operate is this: the contour is allowed to grow until its size (or contrast) is over the bound set by the corresponding slider. Practically, this means that each potential contour C is compared to a contour C’ corresponding to the previous gray level. Then, if C passes but C’ does not then C is plotted.

For more, see Nested boundaries.

Now, the area covered should be just a step towards what we are really interested in – the height of the vegetation (or volume, even better).

Let’s consider how one can compute the height of vegetation from a digital image. The idea is very simple:

 the average height = the area / the width.

Consider now what we see in the image.

Views from a side (vegetation in green) and from above:

Assumptions:

1. The board is a square and its dimensions are known.
2. The board is vertical (otherwise it’s impossible to know where the bottom is).
3. The bottom of the board is horizontal on the horizontal (along the board) ground.
4. The field of view of the camera includes the edge of the vegetation and the top of the board.

Then, the average height computed as below is independent from:

• the deviation of the angle of the camera from the horizontal,
• the distance from the camera to the board,
• the height of the position of the camera above the ground.

The measurements (the image in black, the bottom of the board in red):

These come from image analysis:

 A = the area of the board visible above the vegetation (sq pixel),
 W = the width of the board (pixel).

This is known:

 S = the length of the side of the board (in).

Then average height of the vegetation above the ground (in) is:

  H = S * (1 - A / W2).

Computations here.

I will be supervising 2 students in 1-2 of these areas:

• image analysis, and/or
• image-to-image search, and/or
• topological data analysis.

Temporary page for the projects: Computational science training: 2010 projects.

The power of the software comes from much newer algorithms. Some of them are described in the papers:

• M. Mrozek, P. Pilarczyk, N. Zelazna, Homology algorithm based on acyclic subspace, Computers and Mathematics with Applications, 55 (2008), 2395 –2412.
• M. Mrozek, B. Batko, Coreduction homology algorithm, Discrete and Computational Geometry, 41 (2009), 96-118.
• M. Mrozek, Cech Type Approach to Computing Homology of Maps
  Discrete and Computational Geometry, accepted
• and a few more which are just in preparation.

We just finish[ed] writing a new, much stronger version of the software which will accept not only cubical complexes but also simplicial complexes and general CW complexes and will produce broader output, in particular homology generators, homology maps and persistence intervals for filtered complexes.

The new version of our software at first will be available from the webpage
of our CAPD group at Jagiellonian University, Krakow, Poland:
http://capd.ii.uj.edu.pl/.

Take a look also at our Homology Software page.

What is the goal? I would like the site to cover a big chunk of the math curriculum, interlinked within and with the computer vision / image analysis topics (see The Mathematics of Computer Vision). Even though the format is identical to Wikipedia the presentation is very different. This is a textbook: more details, more examples, exercises, etc. It can still be used for reference.

The content comes directly from my lectures. I use Tablet PC with Windows Journal. I started doing this last fall and I really love the results: bright, colorful slides, but with the spontaneity and flexibility of a chalkboard. Later I transcribe the lectures into text, put it on the site, and simply copy the illustrations. (Plus, I don’t have to deal with chalk on my shoes, pants, and lungs!) I think this approach has huge advantages over the common practice of simply posting video lectures online: searchability, cross-linking, speed of download, the person can read and work at his own pace, etc.

Q: We “need to verify the internal diameter, external diameter and the wall thickness between the ID, OD and the reinforcement yarn. One issue we have is that the wall is not always concentric. We have a minimum wall thickness specification so we would like to measure the wall thickness at the thinnest point to determine if it meets our spec or not.”  

I analyzed one of the images. I found fairly good contours that capture the inner (red) and outer (green) borders of the hose with the settings that you can see in the screenshot. The measurements for this contours can be seen in the Pixcavator’s output table.

The area inside the red contour is 130,966. Assuming this is a circle, the area is equal to π*R², so the radius is

 R = √(130,966/3.14) = 204 pixels.

Then the external diameter is 408 pixels (one would have to do calibration at this point to convert to inches).

The area inside the green contour is 96,595. Assuming this is a circle, the radius is

 R = √(96,595/3.14) = 175 pixels.

Then the internal diameter is 350 pixels.

This suggests that the thickness of the wall should be 204-175=29 pixels. This is the average thickness of a ring with these measurements. To verify this number one can drop the assumption that these are circles and use the perimeters of the contours taken from the output table. Then

 average thickness
   = (area of the wall)/(average perimeter)
   = (130,966-96,595)/((1,547+1,283)/2)
   = 24 pixels.

A similar computation is presented here: Wall of a blood vessel.

Other examples of image analysis

Below is the abstract of a paper I am working on.

Suppose we have conducted 1000 experiments with a set of 100 various measurements in each. Then each experiment is a string of 100 numbers or simply a vector of dimension 100. The result is a collection of disconnected 1000 points (aka point cloud) in the 100-dimensional Euclidean space.

It is impossible to visualize this data as any representation that one can see is lim

ited to dimension 3 (by using colors one gets 6, time – 7). Yet we still need to answer the same questions about the object behind the point cloud: is it one piece or more? Is there a tunnel or a void? And what about possible 100-dimensional topological features?

This is a common approach to the problem.

For a point cloud in a euclidean space, suppose we are given a threshold r so that any two points within r from each other are to be considered “close”. Then each pair of such points is connected by an edge. If three points are “close”, we add a face, etc. The result is a cell complex (more precisely, simplicial complex) that approximates the manifold M behind the point cloud.

We want to count the number of topological features in M by means of the Betti numbers: the number of connected components in M, the number tunnels, the number of voids, etc. This information is contained in the homology of the complex.

Further, to deal with noise and other uncertainty one needs to evaluate the significance of these topological features. For each value of the threshold r we build a separate cell complex, then combine the homology groups of these complexes in a single structure, and count the features with a high measure of robustness. This measure, called persistence, is the length of the interval of values of r for which each of the topological features is present.

Even more important than these “global” properties may be the local topology of the data. For example, in both of the images above the datasets are 3-dimensional but what’s behind is 2-dimensional (surfaces). This is called dimensionality reduction.

Most of the links here are dead but the article will be fixed by the time I am done with the topology course.

A more detailed outline is here: Homological methods in manifold learning (warning: heavy math).

Outline
This is an introductory, two semester course on algebraic topology and its applications. It is intended for advanced undergraduate and beginning graduate students.

Part 1. Introduction to algebraic topology
Starts with topological issues in digital image analysis, informal introduction of homology

Part 2. Homology theory
Cubical complexes, their homology, and maps

Part 3. Overview of point-set topology
Minimized to the extreme (still could have cut even more)

Part 4. Homology groups
A more formal, group theory based, exposition

Part 5. Homology and uncertainty
Applications in computer vision, image analysis and data analysis

Part 6. Beyond homology
The fundamental group and cohomology

Also, I ran across this white paper from Hewlett-Packard: Algebraic topology for computer vision. Good review and an honest attempt to convince the practitioners to that this is something that they might need to know (good luck with that!).

From simplicity comes complexity. And beauty!