Computational Geometry in Python

To illustrate a few advantages of the scipy stack in one of my upcoming talks, I have placed an ipython notebook with (a reduced version of) the current draft of Chapter 6 (Computational Geometry) of my upcoming book: Mastering SciPy.

The raw ipynb can be downloaded from my github repository [blancosilva/Mastering-Scipy/], or viewed directly from the nbviewer at [this other link]

I also made a selection with some fun examples for the talk. You can download the presentation by clicking in the image above.

Enjoy!

Robot stories

Every summer before school was over, I was assigned a list of books to read. Mostly nonfiction and historical fiction, but in fourth grade there that was that first science fiction book. I often remember how that book made me feel, and marvel at the impact that it had in my life. I had read some science fiction before—Well’s Time Traveller and War of the Worlds—but this was different. This was a book with witty and thought-provoking short stories by Isaac Asimov. Each of them delivered drama, comedy, mystery and a surprise ending in about ten pages. And they had robots. And those robots had personalities, in spite of their very simple programming: The Three Laws of Robotics.

1. A robot may not injure a human being or, through inaction, allow a human being to come to harm.
2. A robot must obey the orders given to it by human beings, except where such orders would conflict with the First Law.
3. A robot must protect its own existence as long as such protection does not conflict with the First or Second Law.

Back in the 1980s, robotics—understood as autonomous mechanical thinking—was no more than a dream. A wonderful dream that fueled many children’s imaginations and probably shaped the career choices of some. I know in my case it did.

Fast forward some thirty-odd years, when I met Astro: one of three research robots manufactured by the French company Aldebaran. This NAO robot found its way into the computer science classroom of Tom Simpson in Heathwood Hall Episcopal School, and quickly learned to navigate mazes, recognize some student’s faces and names, and even dance the Macarena! It did so with effortless coding: a basic command of the computer language python, and some idea of object oriented programming.

I could not let this opportunity pass. I created a small undergraduate team with Danielle Talley from USC (a brilliant sophomore in computer engineering, with a minor in music), and two math majors from Morris College: my Geometry expert Fabian Maple, and a McGyver-style problem solver, Wesley Alexander. Wesley and Fabian are supported by a Department of Energy-Environmental Management grant to Morris College, which funds their summer research experience at USC. Danielle is funded by the National Science Foundation through the Louis Stokes South Carolina-Alliance for Minority Participation (LS-SCAMP).

They spent the best of their first week on this project completing a basic programming course online. At the same time, the four of us reviewed some of the mathematical tools needed to teach Astro new tricks: basic algebra and trigonometry, basic geometry, and basic calculus and statistics. The emphasis—I need to point out in case you missed it—is in the word basic.

Talk the talk

The psychologist seated herself and watched Herbie narrowly as he took a chair at the other side of the table and went through the three books systematically.

At the end of half an hour, he put them down, “Of course, I know why you brought these.”

The corner of Dr. Calvin’s lip twitched, “I was afraid you would. It’s difficult to work with you, Herbie. You’re always a step ahead of me.”

“It’s the same with these books, you know, as with the others. They just don’t interest me. There’s nothing to your textbooks. Your science is just a mass of collected data plastered together by makeshift theory — and all so incredibly simple, that it’s scarcely worth bothering about.”

“It’s your fiction that interests me. Your studies of the interplay of human motives and emotions” – his mighty hand gestured vaguely as he sought the proper words.

Liar!
— Isaac Asimov

Astro can understand what I say, and maintain conversations with me. He has four microphones for voice recognition and sound localization, and two speakers for text-to-speech synthesis. But he sometimes gets confused: I say “acute”, and he understands “abide,” for example. This might sound frustrating by all standards. But mathematicians can’t be judged by any reasonable standard. Far from seeing this as a shortcoming, my team used as motivation the mechanics behind his speech recognition functionality, and they wrote some code to turn him into an expert Mastermind player.

Wait, what? How are these two things possibly connected?

Since Astro’s hearing is far from perfect, when he hears a word which is not clear, he gathers a short list of possible candidates. He then chooses the most likely guess based upon different considerations—using clues from context, for example. Based on similar strategies, he can be taught to be a perfect player of 5-letter-word Mastermind: Pick any five-letter word in the English language. Any word, provided it has no repeated letters. Astro chooses one random word as his first guess, and requests a score like in Mastermind.

To illustrate this process, I will show you how the game went when I chose Fabian’s favorite word: “acute.”

Astro: Is it firth?
Me   : NO
Astro: How many letters of my guess are in your word, in the right position?
       How many letters of my guess are in your word, in the wrong position?
Me   : 1, 0
Astro: Is it foams?
Me   : NO
Astro: How many letters of my guess are in your word, in the right position?
       How many letters of my guess are in your word, in the wrong position?
Me   : 0, 1
Astro: Is it junta?
Me   : NO
Astro: How many letters of my guess are in your word, in the right position?
       How many letters of my guess are in your word, in the wrong position?
Me   : 1, 2
Astro: Is it acute?
Me   : YES
Astro: Wonderful, I found it in 4 attempts!

I don’t want to get too technical here, but will mention some of the ideas. The main algorithm is based on techniques of numerical root finding and solving nonlinear equations — nothing complex: high-school level bracketing by bisection, or Newton’s method. To design better winning strategies, my team exploits the benefits of randomness. The analysis of this part is done with basic probability and statistics.

Walk the walk

Donovan’s pencil pointed nervously. “The red cross is the selenium pool. You marked it yourself.”

“Which one is it?” interrupted Powell. “There were three that MacDougal located for us before he left.”

“I sent Speedy to the nearest, naturally; seventeen miles away. But what difference does that make?” There was tension in his voice. “There are penciled dots that mark Speedy’s position.”

And for the first time Powell’s artificial aplomb was shaken and his hands shot forward for the man.

“Are you serious? This is impossible.”

“There it is,” growled Donovan.

The little dots that marked the position formed a rough circle about the red cross of the selenium pool. And Powell’s fingers went to his brown mustache, the unfailing signal of anxiety.

Donovan added: “In the two hours I checked on him, he circled that damned pool four times. It seems likely to me that he’ll keep that up forever. Do you realize the position we’re in?”

Runaround
— Isaac Asimov

Astro moves around too. It does so thanks to a sophisticated system combining one accelerometer, one gyrometer and four ultrasonic sensors that provide him with stability and positioning within space. He also enjoys eight force-sensing resistors and two bumpers. And that is only for his legs! He can move his arms, bend his elbows, open and close his hands, or move his torso and neck (up to 25 degrees of freedom for the combination of all possible joints). Out of the box, and without much effort, he can be coded to walk around, although in a mechanical way: He moves forward a few feet, stops, rotates in place or steps to a side, etc. A very naïve way to go from A to B retrieving an object at C, could be easily coded in this fashion as the diagram shows:

Fabian and Wesley devised a different way to code Astro taking full advantage of his inertial measurement unit. This will allow him to move around smoothly, almost like a human would. The key to their success? Polynomial interpolation and plane geometry. For advanced solutions, they need to learn about splines, curvature, and optimization. Nothing they can’t handle.

Sing me a song

He said he could manage three hours and Mortenson said that would be perfect when I gave him the news. We picked a night when she was going to be singing Bach or Handel or one of those old piano-bangers, and was going to have a long and impressive solo.

Mortenson went to the church that night and, of course, I went too. I felt responsible for what was going to happen and I thought I had better oversee the situation.

Mortenson said, gloomily, “I attended the rehearsals. She was just singing the same way she always did; you know, as though she had a tail and someone was stepping on it.”

One Night of Song
— Isaac Asimov

Astro has excellent eyesight and understanding of the world around him. He is equipped with two HD cameras, and a bunch of computer vision algorithms, including facial and shape recognition. Danielle’s dream is to have him read from a music sheet and sing or play the song in a toy piano. She is very close to completing this project: Astro is able now to identify partitures, and extract from them the location of the pentagrams. Danielle is currently working on identifying the notes and the clefs. This is one of her test images, and the result of one of her early experiments:

Most of the techniques Danielle is using are accessible to any student with a decent command of vector calculus, and enough scientific maturity. The extraction of pentagrams and the different notes on them, for example, is performed with the Hough transform. This is a fancy term for an algorithm that basically searches for straight lines and circles by solving an optimization problem in two or three variables.

The only thing left is an actual performance. Danielle will be leading Fabian and Wes, and with the assistance of Mr. Simpson’s awesome students Erica and Robert, Astro will hopefully learn to physically approach the piano, choose the right keys, and play them in the correct order and speed. Talent show, anyone?

Book presentation at the USC Python Users Group

Areas of Mathematics

For one of my upcoming talks I am trying to include an exhaustive mindmap showing the different areas of Mathematics, and somehow, how they relate to each other. Most of the information I am using has been processed from years of exposure in the field, and a bit of help from Wikipedia.

But I am not entirely happy with what I see: my lack of training in the area of Combinatorics results in a rather dry treatment of that part of the mindmap, for example. I am afraid that the same could be told about other parts of the diagram. Any help from the reader to clarify and polish this information will be very much appreciated.

And as a bonus, I included a script to generate the diagram with the aid of the tikz libraries.

\tikzstyle{level 2 concept}+=[sibling angle=40]
\begin{tikzpicture}[scale=0.49, transform shape]
  \path[mindmap,concept color=black,text=white]
    node[concept] {Pure Mathematics} [clockwise from=45]
      child[concept color=DeepSkyBlue4]{
        node[concept] {Analysis} [clockwise from=180]
          child { 
            node[concept] {Multivariate \& Vector Calculus}
              [clockwise from=120]
              child {node[concept] {ODEs}}}
              child { node[concept] {Functional Analysis}}
              child { node[concept] {Measure Theory}}
              child { node[concept] {Calculus of Variations}}
              child { node[concept] {Harmonic Analysis}}
              child { node[concept] {Complex Analysis}}
              child { node[concept] {Stochastic Analysis}}
              child { node[concept] {Geometric Analysis}
                [clockwise from=-40]
                child {node[concept] {PDEs}}}}
          child[concept color=black!50!green, grow=-40]{ 
            node[concept] {Combinatorics} [clockwise from=10]
              child {node[concept] {Enumerative}}
              child {node[concept] {Extremal}}
              child {node[concept] {Graph Theory}}}
          child[concept color=black!25!red, grow=-90]{ 
            node[concept] {Geometry} [clockwise from=-30]
              child {node[concept] {Convex Geometry}}
              child {node[concept] {Differential Geometry}}
              child {node[concept] {Manifolds}}
              child {node[concept,color=black!50!green!50!red,text=white] {Discrete Geometry}}
              child {
                node[concept] {Topology} [clockwise from=-150]
                  child {node [concept,color=black!25!red!50!brown,text=white]
                    {Algebraic Topology}}}}
          child[concept color=brown,grow=140]{ 
            node[concept] {Algebra} [counterclockwise from=70]
              child {node[concept] {Elementary}}
              child {node[concept] {Number Theory}}
              child {node[concept] {Abstract} [clockwise from=180]
                child {node[concept,color=red!25!brown,text=white] {Algebraic Geometry}}}
              child {node[concept] {Linear}}}
    node[extra concept,concept color=black] at (200:5) {Applied Mathematics} 
      child[grow=145,concept color=black!50!yellow] {
        node[concept] {Probability} [clockwise from=180]
          child {node[concept] {Stochastic Processes}}}
      child[grow=175,concept color=black!50!yellow] {node[concept] {Statistics}}
      child[grow=205,concept color=black!50!yellow] {node[concept] {Numerical Analysis}}
      child[grow=235,concept color=black!50!yellow] {node[concept] {Symbolic Computation}};
\end{tikzpicture}

More on Lindenmayer Systems

We briefly explored Lindenmayer systems (or L-systems) in an old post: Toying with Basic Fractals. We quickly reviewed this method for creation of an approximation to fractals, and displayed an example (the Koch snowflake) based on tikz libraries.

I would like to show a few more examples of beautiful curves generated with this technique, together with their generating axiom, rules and parameters. Feel free to click on each of the images below to download a larger version.

Note that any coding language with plotting capabilities should be able to tackle this project. I used once again tikz for , but this time with the tikzlibrary lindenmayersystems.

name : Dragon Curve axiom : X order : 11 step : 5pt angle : 90 rules : X -> X+YF+ Y -> -FX-Y	name : Gosper Space-filling Curve axiom : XF order : 5 step : 2pt angle : 60 rules : XF -> XF+YF++YF-XF--XFXF-YF+ YF -> -XF+YFYF++YF+XF--XF-YF
name : Quadric Koch Island axiom : F+F+F+F order : 4 step : 1pt angle : 90 rules : F -> F+F-F-FF+F+F-F	name : Sierpinski Arrowhead axiom : F order : 8 step : 3.5pt angle : 60 rules : G -> F+G+F F -> G-F-G
name : ? axiom : F+F+F+F order : 4 step : 2pt angle : 90 rules : F -> FF+F+F+F+F+F-F	name : ? axiom : F+F+F+F order : 4 step : 3pt angle : 90 rules : F -> FF+F+F+F+FF

Would you like to experiment a little with axioms, rules and parameters, and obtain some new pleasant curves with this method? If the mathematical properties of the fractal that they approximate are interesting enough, I bet you could attach your name to them. Like the astronomer that finds through her telescope a new object in the sky, or the zoologist that discover a new species of spider in the forest.

Some results related to the Feuerbach Point

Given a triangle the circle that goes through the midpoints of each side, is called the Feuerbach circle. It has very surprising properties:

• It also goes through the feet of the heights, points
• If denotes the orthocenter of the triangle, then the Feuerbach circle also goes through the midpoints of the segments For this reason, the Feuerbach circle is also called the nine-point circle.
• The center of the Feuerbach circle is the midpoint between the orthocenter and circumcenter of the triangle.
• The area of the circumcircle is precisely four times the area of the Feuerbach circle.

Most of these results are easily shown with sympy without the need to resort to Gröbner bases or Ritt-Wu techniques. As usual, we realize that the properties are independent of rotation, translation or dilation, and so we may assume that the vertices of the triangle are and for some positive parameters To prove the last statement, for instance we may issue the following:

>>> import sympy
>>> from sympy import *
>>> A=Point(0,0)
>>> B=Point(1,0)
>>> r,s=var('r,s')
>>> C=Point(r,s)
>>> D=Segment(A,B).midpoint
>>> E=Segment(B,C).midpoint
>>> F=Segment(A,C).midpoint
>>> simplify(Triangle(A,B,C).circumcircle.area/Triangle(D,E,F).circumcircle.area)
4

But probably the most amazing property of the nine-point circle, is the fact that it is tangent to the incircle of the triangle. With exception of the case of equilateral triangles, both circles intersect only at one point: the so-called Feuerbach point.

An Automatic Geometric Proof

We are familiar with that result that states that, on any given triangle, the circumcenter, centroid and orthocenter are always collinear. I would like to illustrate how to use Gröbner bases theory to prove that the incenter also belongs in the previous line, provided the triangle is isosceles.

We start, as usual, indicating that this property is independent of shifts, rotations or dilations, and therefore we may assume that the isosceles triangle has one vertex at , another vertex at and the third vertex at for some value In that case, we will need to work on the polynomial ring since we need the parameter free, the variables and are used to input the conditions for the circumcenter of the triangle, the variables and for centroid, and the variables and for the incenter (note that we do not need to use the orthocenter in this case).

We may obtain all six conditions by using sympy, as follows:

>>> import sympy
>>> from sympy import *
>>> A=Point(0,0)
>>> B=Point(1,0)
>>> s=symbols("s",real=True,positive=True)
>>> C=Point(1/2.,s)
>>> T=Triangle(A,B,C)
>>> T.circumcenter
Point(1/2, (4*s**2 - 1)/(8*s))
>>> T.centroid
Point(1/2, s/3)
>>> T.incenter
Point(1/2, s/(sqrt(4*s**2 + 1) + 1))

This translates into the following polynomials

(for circumcenter)
(for centroid)
(for incenter)

The hypothesis polynomial comes simply from asking whether the slope of the line through two of those centers is the same as the slope of the line through another choice of two centers; we could use then, for example, It only remains to compute the Gröbner basis of the ideal Let us use SageMath for this task:

sage: R.<s,x1,x2,x3,y1,y2,y3,z>=PolynomialRing(QQ,8,order='lex')
sage: h=[2*x1-1,8*r*y1-4*r**2+1,2*x2-1,3*y2-r,2*x3-1,(4*r*y3+1)**2-4*r**2-1]
sage: g=(x2-x1)*(y3-y1)-(x3-x1)*(y2-y1)
sage: I=R.ideal(1-z*g,*h)
sage: I.groebner_basis()
[1]

This proves the result.

Sympy should suffice

I have just received a copy of Instant SymPy Starter, by Ronan Lamy—a no-nonsense guide to the main properties of SymPy, the Python library for symbolic mathematics. This short monograph packs everything you should need, with neat examples included, in about 50 pages. Well-worth its money.

To celebrate, I would like to pose a few coding challenges on the use of this library, based on a fun geometric puzzle from cut-the-knot: Rhombus in Circles

Segments and are equal. Lines and intersect at Form four circumcircles: Prove that the circumcenters form a rhombus, with

Note that if this construction works, it must do so independently of translations, rotations and dilations. We may then assume that is the origin, that the segments have length one, and that for some parameters it is We let SymPy take care of the computation of circumcenters:

import sympy
from sympy import *

# Point definitions
M=Point(0,0)
A=Point(2,0)
B=Point(1,0)
a,theta=symbols('a,theta',real=True,positive=True)
C=Point((a+1)*cos(theta),(a+1)*sin(theta))
D=Point(a*cos(theta),a*sin(theta))

#Circumcenters
E=Triangle(A,C,M).circumcenter
F=Triangle(A,D,M).circumcenter
G=Triangle(B,D,M).circumcenter
H=Triangle(B,C,M).circumcenter

Finding that the alternate angles are equal in the quadrilateral is pretty straightforward:

In [11]: P=Polygon(E,F,G,H)

In [12]: P.angles[E]==P.angles[G]
Out[12]: True

In [13]: P.angles[F]==P.angles[H]
Out[13]: True

To prove it a rhombus, the two sides that coincide on each angle must be equal. This presents us with the first challenge: Note for example that if we naively ask SymPy whether the triangle is equilateral, we get a False statement:

In [14]: Triangle(E,F,G).is_equilateral()
Out[14]: False

In [15]: F.distance(E)
Out[15]: Abs((a/2 - cos(theta))/sin(theta) - (a - 2*cos(theta) + 1)/(2*sin(theta)))

In [16]: F.distance(G)
Out[16]: sqrt(((a/2 - cos(theta))/sin(theta) - (a - cos(theta))/(2*sin(theta)))**2 + 1/4)

Part of the reason is that we have not indicated anywhere that the parameter theta is to be strictly bounded above by (we did indicate that it must be strictly positive). The other reason is that SymPy does not handle identities well, unless the expressions to be evaluated are perfectly simplified. For example, if we trust the routines of simplification of trigonometric expressions alone, we will not be able to resolve this problem with this technique:

In [17]: trigsimp(F.distance(E)-F.distance(G),deep=True)==0
Out[17]: False

Finding that with SymPy is not that easy either. This is the second challenge.

How would the reader resolve this situation?

Instant SymPy Starter

Have a child, plant a tree, write a book

Or more importantly: rear your children to become nice people, water those trees, and make sure that your books make a good impact.

I recently enjoyed the rare pleasure of having a child (my first!) and publishing a book almost at the same time. Since this post belongs in my professional blog, I will exclusively comment on the latter: Learning SciPy for Numerical and Scientific Computing, published by Packt in a series of technical books focusing on Open Source software.

Keep in mind that the book is for a very specialized audience: not only do you need a basic knowledge of Python, but also a somewhat advanced command of mathematics/physics, and an interest in engineering or scientific applications. This is an excerpt of the detailed description of the monograph, as it reads in the publisher’s page:

It is essential to incorporate workflow data and code from various sources in order to create fast and effective algorithms to solve complex problems in science and engineering. Data is coming at us faster, dirtier, and at an ever increasing rate. There is no need to employ difficult-to-maintain code, or expensive mathematical engines to solve your numerical computations anymore. SciPy guarantees fast, accurate, and easy-to-code solutions to your numerical and scientific computing applications.

Learning SciPy for Numerical and Scientific Computing unveils secrets to some of the most critical mathematical and scientific computing problems and will play an instrumental role in supporting your research. The book will teach you how to quickly and efficiently use different modules and routines from the SciPy library to cover the vast scope of numerical mathematics with its simplistic practical approach that is easy to follow.

The book starts with a brief description of the SciPy libraries, showing practical demonstrations for acquiring and installing them on your system. This is followed by the second chapter which is a fun and fast-paced primer to array creation, manipulation, and problem-solving based on these techniques.

The rest of the chapters describe the use of all different modules and routines from the SciPy libraries, through the scope of different branches of numerical mathematics. Each big field is represented: numerical analysis, linear algebra, statistics, signal processing, and computational geometry. And for each of these fields all possibilities are illustrated with clear syntax, and plenty of examples. The book then presents combinations of all these techniques to the solution of research problems in real-life scenarios for different sciences or engineering — from image compression, biological classification of species, control theory, design of wings, to structural analysis of oxides.

The book is also being sold online in Amazon, where it has been received with pretty good reviews. I have found other random reviews elsewhere, with similar welcoming comments:

• Artificial Intelligence in Motion by Marcel Caraciolo
• The Endeavour, by John D. Cook

Buy my book!

Well, ok, it is not my book technically, but I am one of the authors of one of the chapters. And no, as far as I know, I don’t get a dime of the sales in concept of copyright or anything else.

As the title suggests (Modeling Nanoscale Imaging in Electron Microscopy), this book presents some recent advances that have been made using mathematical methods to resolve problems in electron microscopy. With improvements in hardware-based aberration software significantly expanding the nanoscale imaging capabilities of scanning transmission electron microscopes (STEM), these mathematical models can replace some labor intensive procedures used to operate and maintain STEMs. This book, the first in its field since 1998, covers relevant concepts such as super-resolution techniques (that’s my contribution!), special de-noising methods, application of mathematical/statistical learning theory, and compressed sensing.

We even got a nice review in Physics Today by Les Allen, no less!

Imaging with electrons, in particular scanning transmission electron microscopy (STEM), is now in widespread use in the physical and biological sciences. And its importance will only grow as nanotechnology and nano-Biology continue to flourish. Many applications of electron microscopy are testing the limits of current imaging capabilities and highlight the need for further technological improvements. For example, high throughput in the combinatorial chemical synthesis of catalysts demands automated imaging. The handling of noisy data also calls for new approaches, particularly because low electron doses are used for sensitive samples such as biological and organic specimens.

Modeling Nanoscale Imaging in Electron Microscopy addresses all those issues and more. Edited by Thomas Vogt and Peter Binev at the University of South Carolina (USC) and Wolfgang Dahmen at RWTH Aachen University in Germany, the book came out of a series of workshops organized by the Interdisciplinary Mathematics Institute and the NanoCenter at USC. Those sessions took the unusual but innovative approach of bringing together electron microscopists, engineers, physicists, mathematicians, and even a philosopher to discuss new strategies for image analysis in electron microscopy.

In six chapters, the editors tackle the ambitious challenge of bridging the gap between high-level applied mathematics and experimental electron microscopy. They have met the challenge admirably. I believe that high-resolution electron microscopy is at a point where it will benefit considerably from an influx of new mathematical approaches, daunting as they may seem; in that regard Modeling Nanoscale Imaging in Electron Microscopy is a major step forward. Some sections present a level of mathematical sophistication seldom encountered in the experimentally focused electron-microscopy literature.
The first chapter, by philosopher of science Michael Dickson, looks at the big picture by raising the question of how we perceive nano-structures and suggesting that a Kantian approach would be fruitful. The book then moves into a review of the application of STEM to nanoscale systems, by Nigel Browning, a leading experimentalist in the field, and other well-known experts. Using case studies, the authors show how beam-sensitive samples can be studied with high spatial resolution, provided one controls the beam dose and establishes the experimental parameters that allow for the optimum dose.

The third chapter, written by image-processing experts Sarah Haigh and Angus Kirkland, addresses the reconstruction, from atomic-resolution images, of the wave at the exit surface of a specimen. The exit surface wave is a fundamental quantity containing not only amplitude (image) information but also phase information that is often intimately related to the atomic-level structure of the specimen. The next two chapters, by Binev and other experts, are based on work carried out using the experimental and computational resources available at USC. Examples in chapter four address the mathematical foundations of compressed sensing as applied to electron microscopy, and in particular high-angle annular dark-field STEM. That emerging approach uses randomness to extract the essential content from low-information signals. Chapter five eloquently discusses the efficacy of analyzing several low-dose images with specially adapted digital-image-processing techniques that allow one to keep the cumulative electron dose low and still achieve acceptable resolution.

The book concludes with a wide-ranging discussion by mathematicians Amit Singer and Yoel Shkolnisky on the reconstruction of a three-dimensional object via projected data taken at random and initially unknown object orientations. The discussion is an extension of the authors’ globally consistent angular reconstitution approach for recovering the structure of a macromolecule using cryo-electron microscopy. That work is also applicable to the new generation of x-ray free-electron lasers, which have similar prospective applications, and illustrates nicely the importance of applied mathematics in the physical sciences.

Modeling Nanoscale Imaging in Electron Microscopy will be an important resource for graduate students and researchers in the area of high-resolution electron microscopy.

(Les J. Allen, Physics Today, Vol. 65 (5), May, 2012)

Table of contents	Preface	Sample chapter

Trigonometry

I have just realized that I haven’t posted a good puzzle here in a long time, so here it goes one on Trigonometry, that the average student of Calculus should be able to tackle: you can use anything you think it could help: derivatives, symmetry, periodicity, integration, summation, go to several variables, differential equations, etc

Prove that, for all real values it is

or, in a more compact notation,

Edge detection: The Convolution Approach

Today I would like to show a very basic technique of detection based on simple convolution of an image with small kernels (masks). The purpose of these kernels is to enhance certain properties of the image at each pixel. What properties? Those that define what means to be an edge, in a differential calculus way—exactly as it was defined in the description of the Canny edge detector. The big idea is to assign to each pixel a numerical value that expresses its strength as an edge: positive if we suspect that such structure is present at that location, negative if not, and zero if the image is locally flat around that point. Masks can be designed so that they mimic the effect of differential operators, but these can be terribly complicated and give rise to large matrices.

The first approaches were performed with simple kernels. For example, Faler came up with the following four simple masks that emulate differentiation:

Note that, adding all the values of each matrix, one obtains zero. This is consistent with the third property required for our kernels: in the event of a locally flat area around a given pixel, convolution with any of these will offer a value of zero.

Read more…

So you want to be an Applied Mathematician

The way of the Applied Mathematician is one full of challenging and interesting problems. We thrive by association with the Pure Mathematician, and at the same time with the no-nonsense, hands-in, hard-core Engineer. But not everything is happy in Applied Mathematician land: every now and then, we receive the disregard of other professionals that mistake either our background, or our efficiency at attacking real-life problems.

I heard from a colleague (an Algebrist) complains that Applied Mathematicians did nothing but code solutions of partial differential equations in Fortran—his skewed view came up after a naïve observation of a few graduate students working on a project. The truth could not be further from this claim: we do indeed occasionally solve PDEs in Fortran—I give you that—and we are not ashamed to admit it. But before that job has to be addressed, we have gone through a great deal of thinking on how to better code this simple problem. And you would not believe the huge amount of deep Mathematics that are involved in this journey: everything from high-level Linear Algebra, Calculus of Variations, Harmonic Analysis, Differential Geometry, Microlocal Analysis, Functional Analysis, Dynamical Systems, the Theory of Distributions, etc. Not only are we familiar with the basic background on all those fields, but also we are supposed to be able to perform serious research on any of them at a given time.

My soon-to-be-converted Algebrist friend challenged me—not without a hint of smugness in his voice—to illustrate what was my last project at that time. This was one revolving around the idea of frames (think of it as redundant bases if you please), and needed proving a couple of inequalities involving sequences of functions in —spaces, which we attacked using a beautiful technique: Bellman functions. About ninety minutes later he conceded defeat in front of the board where the math was displayed. He promptly admitted that this was no Fortran code, and showed a newfound respect and reverence for the trade.

It doesn’t hurt either that the kind of problems that we attack are more likely to attract funding. And collaboration. And to be noticed in the press.

Alright, so some of you are sold already. What is the next step? I am assuming that at his point you own your Calculus, Analysis, Probability and Statistics, Linear Programming, Topology, Geometry, Physics and you are able to solve most known ODEs. From here, as with any other field, my recommendation is to slowly build a Batman belt: acquire and devour a sequence of books and scientific articles, until you are very familiar with their contents. When facing a new problem, you should be able to recall from your Batman belt what technique could work best, in which book(s) you could get some references, and how it has been used in the past for related problems.

Following these lines, I have included below an interesting collection with the absolutely essential books that, in my opinion, every Applied Mathematician should start studying:

The ultimate metapuzzle

If you have been following this blog for a while, you already know of my fascination with metapuzzles. I have recently found one of the most beautiful examples of these riddles in this August’s edition of Ponder This, and following the advice of Travis, I will leave it completely open for the readers to give it a go. Enjoy!

A mathematician wanted to teach his children the value of cooperation, so he told them the following:

“I chose a secret triangle for which the lengths of its sides are all integers.

To you my dear son Charlie, I am giving the triangle’s perimeter. And to you, my beloved daughter Ariella, I am giving its area.

Since you are both such talented mathematicians, I’m sure that together you can find the lengths of the triangle’s sides.”

Instead of working together, Charlie and Ariella had the following conversation after their father gave each of them the information he promised.

Charlie: “Alas, I cannot deduce the lengths of the sides from my knowledge of the perimeter.”

Ariella: “I do not know the perimeter, but I cannot deduce the lengths of the sides from just knowing the area. Maybe our father is right and we should cooperate after all.”

Charlie: “Oh no, no need. Now I know the lengths of the sides.”

Ariella: “Well, now I know them as well.”

Find the lengths of the triangle’s sides.

Where are the powers of two?

The following construction gives an interesting pairing map between the positive integers and the lattice of integer-valued points in the plane:

• Place at the origin.
• For each level populate the points of the plane on the square with vertices starting from at the position and going counter-clockwise.

  

After pairing enough positive integers on the lattice, pay attention to the powers of two: they all seem to be located on the same two horizontal lines: and

Is this statement true?

Geolocation

Recall the First Spherical Law of Cosines:

Given a unit sphere, a spherical triangle on the surface of the sphere is defined by the great circles connecting three points , , and on the sphere. If the lengths of these three sides are (from to (from to and (from to and the angle of the corner opposite is then

In any decent device and for most computer languages, this formula should give well-conditioned results down to distances as small as around three feet, and thus can be used to compute an accurate geodetic distance between two given points in the surface of the Earth (well, ok, assuming the Earth is a perfect sphere). The geodetic form of the law of cosines is rearranged from the canonical one so that the latitude can be used directly, rather than the colatitude, and reads as follows: Given points and with positions and respectively, the distance between the two points is given by the following formula.

where is the radius of the Earth in miles (well, ok, the average radius of the Earth…)

A nice application of this formula can be used for geolocation purposes, and I recently had the pleasure to assist a software company (thumb-mobile.com) to write such functionality for one of their clients.

Go to www.lizardsthicket.com in your mobile device, and click on “Find a Location.” This fires up the location services of your browser. When you accept, your latitude and longitude are tracked. After a fast, reliable and resource-efficient algorithm, the page offers the location of the restaurant from the Lizard’s chain that is closest to you. Simple, right?

Apollonian gaskets and circle inversion fractals

Recall the definition of an iterated function system (IFS), and how on occasion, their attractors are fractal sets. What happens when we allow more general functions instead of mere affine maps?

The key to the design of this amazing fractal is in the notion of limit sets of circle inversions.

Toying with basic fractals

I will skip all the mathematical theory behind Fractals (dimensions, measures, etc) to focus directly into the description and implementation of some of the most basic examples. In this post, I will cover the ideas behind three classical techniques:

• Iterated function systems
• Membership tests
• Lindenmayer systems

Bertrand Paradox

Classically, we define the probability of an event as the ratio of the favorable cases, over the number of all possible cases. Of course, these possible cases need to be all equally likely. This works great for discrete settings, like dice rolls, card games, etc. But when facing non-discrete cases, this definition needs to be revised, as the following example shows:

Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle?

First example	Second example	Third example

Read more…

Voronoi mosaics

While looking for ideas to implement voronoi in sage, I stumbled upon a beautiful paper written by a group of japanese computer graphic professionals from the universities of Hokkaido and Tokyo: A Method for Creating Mosaic Images Using Voronoi Diagrams. The first step of their algorithm is simple yet brilliant: Start with any given image, and superimpose an hexagonal tiling of the plane. By a clever approximation scheme, modify the tiling to become a voronoi diagram that adaptively minimizes some approximation error. As a consequence, the resulting voronoi diagram is somehow adapted to the desired contours of the original image.

(Fig. 1)	(Fig. 2)	(Fig. 3)	(Fig. 4)

In a second step, they manually adjust the Voronoi image interactively by moving, adding, or deleting sites. They also take the liberty of adding visual effects by hand: emphasizing the outlines and color variations in each Voronoi region, so they look like actual pieces of stained glass (Fig. 4).

Triangulations

As part of a project developed by Professor Bradley J. Lucier, to code a PDE solver written in scheme, I worked in some algorithms to perform “good triangulations” of polygons with holes (“good triangulations” meaning here, those where all the triangles have their three angles as close to 60º as possible). I obtained the necessary theoretical background and coding strategies from the following references:

• Mark de Berg et al., “Computational Geometry by Example.“
• Francis Chin and Cao An Wang, “Finding the Constrained Delaunay Triangulation and Constrained Voronoi Diagram of a simple Polygon in Linear Time.“
• Joseph O’Rourke, “Computational Geometry in C.“
• Jim Ruppert, “A Delaunay Refinement Algorithm for Quality 2-dimensional Mesh Generation.“

Mechanical Geometry Theorem Proving

In 1977, Professor Wen-Tsun Wu succeeded in developing a method of mechanical geometry theorem proving. This method has been applied to prove or even discover hundreds of non-trivial difficult theorems in elementary and differential geometries on a computer in an almost trivial manner. Usign Ritt’s differential algebra, Wu established a method for solving algebraic and differential equations by transforming an equation system in the general form to equation systems in triangular form. This is the Ritt-Wu decomposition algorithm, that later on was shown to be equivalent to perform a series of operations on ideals, very easily carried out by means of Gröbner basis manipulation.

I wrote a script in MAPLE to perform evaluations of the validity of some simple theorems in Euclidean Geometry, and wrote a small paper (in Spanish) on one of my findings, that was published in Bol. Asoc. Prof. Puig Adams, in October’99: “Sobre demostración automática de un problema geométrico“.

The example I cover in that short article can be seen below.

Given: Circles , that intersect each other in points and , and given points , in circle , consider line through and , and line through and . The intersections of line with circle are and . The intersections of line with circle are and . Consider the segments (connecting with ) and (connecting with ).

To prove: Segments and are parallel.