Who says math is not beautiful? Anyone who doubts the beauty in math must check out algebraic surfaces.
An implicit functions has the form:
where, is an arbitrary constant.
For example, the implicit equation of the unit circle is and the equation describes a sphere of radius 1.
An algebraic surface is described by an implicit function . It may be rendered using Mayavi by evaluating the expression on a regular (specified) grid, and then generate isosurface of contour value equal to 0.
Here are some examples of algebraic surfaces plotted using Mayavi:
I wrote a quick function called implicit_plot for plotting algebraic surfaces using Mayavi. The most important argument to the function is of course the expression as a string. It is probably not the best function, but the idea is to show how to plot implicit surfaces. The code snippet is included at the end of this post. Suggestions for improvement are always welcome.
Summary: Most cellphone cameras today come with “large aperture” lens such as F/2.0 or F/2.2. Since the optical resolution of a “perfect” lens (a lens devoid of any aberration) is inversely proportional to the F/# (the relation is shown in equation 1), one may assume that a F/2.0 cellphone camera lens should be able to resolve fine details on an object as well as a F/2.0 DSLR lens.
The above image shows two lenses of same F/#, and also of equal normalized focal length. The one on the left is a lens whose 35 mm focal length is about 28 mm (with complete camera module, including the image sensor) from a cellphone, and the lens on the right is a 28 mm DSLR lens.
The optical resolution of a lens determines how close two line-objects or point-objects be placed before the objects cannot be distinguished from each other when viewing through the lens. The minimum resolvable separation, , is the inverse of . The larger the value of (usually measured in line-pairs-per-mm), the better is the resolving ability of the lens. So if the f-number, , of a cellphone lens matches the of a DSLR lens, then the equation (1) seems to suggest that they have the same optical resolution! However, as it is shown in the following paragraphs, the ability to resolve fine details of a F/2.0 DSLR lens is in fact much better than a F/2.0 cellphone camera lens. Concretely, if the focal length and the aperture diameter of the cellphone lens is 1/k (k>1) times the respective parameters of the DSLR lens, then the of the two systems are equal but the resolving ability of the cellphone lens is times that of the DSLR lens. For example, a 50 mm, F/2.0 lens (D = 25 mm), which is a first order approximation of a 50 mm DSLR lens, can resolve details as fine as 54 microns focused at a distance of 2 meters. Whereas a 5 mm, F/2.0 lens (D = 2.5 mm), a close approximation of a typical cellphone camera lens, can resolve details only up to about 540 microns focused at the same distance. This is essentially a manifestation of the difference in magnifications (or angular resolution) of the lenses.
ZEMAX (also known as OpticStudio), a leading tool for design and analysis of optical systems, provides an in-built DDE server that enables access to ZEMAX’s features externally using any Windows application. The ability to access and control ZEMAX externally allows it to be used in many interesting and powerful ways, some of which has been documented in the “ZEMAX Extensions” chapter of the ZEMAX manual. An excellent toolbox called MZDDE, developed by Derek Griffith at CSIR, already exists for users of MATLAB. When I tried to look for an equivalent toolbox in Python, I couldn’t find one, so I decided to write one myself.
PyZDDE is a Python-based toolbox for communicating with ZEMAX using the DDE messaging protocol. It is similar to MZDDE and very much inspired by it. Currently PyZDDE is not as extensive as MZDDE as it is a work in progress. However, it has some distinguishing features — it can be used with regular Python scripts and also in an interactive environment such as an IPython shell, QtConsole or IPython notebook. The ability to interact with ZEMAX from an IPython notebook using PyZDDE can be a powerful tool for both teaching and documentation. Please visit the github repository for downloading and using the code. The github repository also contains an accompanying wiki page which describes briefly on how to use the PyZDDE toolbox.
Here I would like to briefly document the communication process between ZEMAX and Python using PyZDDE. But before that I would like to show you how simple it is to communicate with Zemax using PyZDDE using a “Hello world” example (it gets and prints the running Zemax version):
import pyzdde.zdde as pyz
link = pyz.createLink()
print(&amp;amp;quot;Hello Zemax version: &amp;amp;quot;, link.zGetVersion())
In this post I will describe the process of installing OpenCV (both versions 2.4.2 and 2.4.3) on Debian Linux (especially Debian-6). After installing, we will do some tests to verify the installation and also see some examples. As I was trying to install OpenCV on Debian I found that although there is lot of information on similar topics (see the references at the end), I still had to dig around a bit to understand the process completely. Some of the excellent guides are now outdated. Also, I found a few that have all the “what to do” but not “why to do”. Then as I started jotting down my own notes into my OneNote notebook, I realized that I should share these notes with others who are interested. I hope to update this document as I find more useful information in future.
1. Before installing the Prerequisites:
It is recommended [Reference 2] that you update and upgrade your current system before installing OpenCV. You can do this from the terminal using the following two commands:
Humans have covered the face of the earth with deep concrete forests in the name of modernization and advancement. Once there used to be green trees everywhere I looked below while floating miles and miles across the great plains, mountains, and rivers. Nature used mostly organic forms on structures and engravings on the surface of the earth — very similar to my own form (when I have one!), and I felt happy as I could relate their shape to my own. Now I see regular geometric patterns — circles, triangles, squares, rectangles, trapezoids everywhere I look. I feel sad; I feel lost; I feel lonely. I don’t seem to belong here. I feel blue!