Total Magnification

If a microscope has objective magnification (Mo) of 10x and eyepiece magnification (Me) of 10x, then total magnification (Mt) is given as:

Mt = Mo * Me = 100x

What is a Virtual Object in Graphical Ray Tracing?

• A virtual object is simply the object on the wrong side (right side) of the lens.

Radius v/s Clear Semi-Diameter in Zemax OpticStudio

- In Zemax, Radius defines how curved is the lens surface. This value is 'infinity' for a flat surface.

- Whereas the Clear Semi-Diameter defines how big is the optic/lens. Its is the vertical distance from the optic axis to the periphery. Because for radius values very large, we cannot have a camera infinitely tall!

Snell's Law and Fermat's Principle

Snell's Law:

n1 * sin(theta_1) = n2 * sin(theta_2)

Fermat's Principle:

Light rays take the path that requires the least travel time.

Fraser's Spiral

 

(Image Source: Wikipedia)

It turns out that there is actually no spiral in the above image. The above image only has concentric circles. The spiral is just an incorrect human perception. This is called the Fraser's Spiral!

"While there is no computer vision system that is as versatile as the human vision system as of today; there are many computer vision systems that are more precise and reliable than the human vision system. In short, for many tasks that require vision the human vision system might be the wrong system to emulate. Furthermore, human visual system is more fallible than we would like to believe." - Dr. Shree K. Nayar

Importance of Luminance and Chrominance in Image Processing

    The human eye is more sensitive to the luminance(brightness) in an image as compared to the chrominance(color) information in an image. Thus, image recovery algorithms define a cost function in such a way that the regularization term applied to luminance component tries to preserve edges(high frequency information), while the regularization term applied to the chrominance component tries to ensure smoothness. The chrominance regularization thus removes color artifacts which are more sensitive and objectionable to the human eye.

(Source: S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, “Advances and challenges in super-resolution,” International Journal of Imaging Systems and Technology 14, 47–57 (2004).)

Field Number of a Microscope

    For typical objective lenses in a microscope, the magnification range is from 10X to 100X. These objectives form an intermediate image which is a circle, typically 50-60mm in diameter. Out of this circular image, only 19-23mm from the center is well corrected for aberrations. The ocular/eyepiece clips the image to this inner diameter which is called as the Field Number of a Microscope.

(Source: M. Levoy, R. Ng, A. Adams, M. Footer, and M. Horowitz, “Light field microscopy,” in Proceedings of ACM SIGGRAPH. (2006) 924–934.)

Mean and Expected Value

The mean and expected values are closely related but there is a difference.

For example, we have 5 coins with 2 coins of five cents and 3 coins of 10 cents respectively. In this case, we calculate the mean by averaging the coin's values as Cmean = (1/5)*(5+5+10+10+10) = 8. Here, 8 is the Mean and not the expected value as the system state (coin's value) is not hidden.

Now, we have five different weight measures of the same person as: 79.8kg, 80kg, 80.1kg, 79.8kg, and 80.2kg. Here, the person is a system and the person's weight is the system's state. The measurements are different due to random errors in the weight scales and therefore the persons weight is a hidden state. We do not know the exact value of the person's weight but can make an estimate by averaging the scale's measurements as Wavg = (1/5)*(79.8 + 80 + 80.1 + 79.8 + 80.2) = 79.98kg. The outcome of this estimate is called the Expected Value.

(Source: https://www.kalmanfilter.net/background.html)

Human Eye has a Blind Spot

    The human eye has a blind spot on the retina, meaning there are no image sensors(rods and/or cones) in this region. This is the place where image gets transferred to the brain using optic nerves. To realize this blind spot, draw a cross and a circle at two distinct ends on a piece of paper respectively. Now, close your one eye and move back and forth while continuously focusing on the cross. At a certain point, the circle disappears from your vision and this is because the circle’s image was formed at the location of your blind spot. If you further continue moving in the direction you were already moving, the circle appears again.

Dual nature of light and matter

    Isaac Newton was able to assert the particle theory of light by describing phenomena like refraction, reflection and dispersion. However, this particle theory of light could not explain everything that people saw light do. Christiaan Huygens thought that light was actually a wave and came up with the concept of wavefronts and his idea was able to model diffraction. Later on, Thomas Young's famous two-slit experiment (demonstrated interference and diffraction) and James Clerk Maxwell's idea showing it's the electric and magnetic fields that wave in light, supported wave theory of light.

    The wave theory seemed an appropriate description of light until scientists began looking at the photoelectric effect in the early 1900s. To observe this effect, two electrodes connected to a battery were placed in a glass ball (with vacuum). As light waves incident on these electrodes, the electrons were supposed to absorb energy. As the electrons absorbed enough energy, they were expected to eject from the atoms and travel as current between the electrodes. The results of this experiment showed four problems with the wave theory of light:

1. No electrons were ejected if the frequency of light was below a certain cutoff frequency, no matter how bright the light was. According to wave theory, electrons would just wait until they absorbed enough energy to leave the metal.
2. The maximum kinetic energy of electrons was independent of the incident light intensity with frequency above cutoff frequency. As per wave theory, greater intensity would mean more energy thereby increasing kinetic energy of electrons.
3. The maximum kinetic energy increased with the frequency of incident light. As per wave theory kinetic energy should depend on incident intensity and not incident frequency.
4. The electrons were ejected from the electrodes instantaneously even when the incident intensity was very low. Wave theory suggested electrons should have waited before being ejected until they absorb enough energy from incident light intensity.

    Wave theory could not explain the above results from the photoelectric effect. In 1905, Albert Einstein came up with the idea of light having a dual nature, both particle and wave. He proposed light being a wave but with finite extent unlike the wave theory. He called this localized wave as a quanta and also later suggested that it's energy depended on the frequency of light. In 1923, Louis de Broglie postulated that matter, similar to photons, has both wave and particle properties. Using Einstein's idea about momentum and energy for photons, he suggested similar properties for electrons, protons, neutrons, atoms, etc. With light, it's the electric and magnetic fields that are waving. With particles, what is waving is not clear. Wave function for a particle is a probability distribution function (likelihood of finding a particle in a particular location) in quantum mechanics.

    As time progressed, things got more interesting as single electrons were sent through Young's double-slit setup. More number of electrons were collected at some locations (in places like a bright fringe in case of light) and almost no electrons were collected at other locations (in places like a dark fringe in case of light). This result was equivalent to the interference pattern produced by light. Electrons also exhibited diffraction when passed through crystals and analyzing the angle and spacing of the atoms that did the scattering led to the first experimental confirmation of the wavelength of the electrons. Also, these results exactly matched de Broglie's hypothesis. Concentrated electron counts in some areas and almost zero elsewhere could only be explained by the wave theory.

    In a similar fashion, single photons were passed through Young's double-slit. It was observed that a single photon exists at both slits at the same time which showed there might be more to the wave theory. With that, perhaps photons can communicate with each other even though they are separated by vacuum. This property is referred to as entanglement. Entangled photons are two or more photons with a special link. This phenomenon is so mysterious that Albert Einstein called it a spooky action. Recently, many experiments have shown that entangled photons can exchange information instantly even if the distance between them is in miles. Although bizarre, photon entanglement opens possibilities for applications in the realm of super-secure communication and computing.

(These are notes taken from the book "Optics For Dummies" by Galen Duree, Jr., PhD.)

Depth of Field

    It is the range of object distances over which the image is "sufficiently well" focused, i.e. the range over which the blur(b) is less than the pixel size of a camera sensor.

(Taken from a YouTube video of Shree K. Nayar from Columbia University)

Unsupervised Learning in Bioinformatics

    There are a few methods in machine learning that can be applied in the field of bioinformatics like studying a genome or classification of different proteins/sub-genes in a gene. Such methods known to me are classified under unsupervised learning approaches and are listed as follows:

1) Hierarchical Clustering (Dendrograms).

2) Matrix Factorization.

Eigenfaces: An application of Principal Component Analysis (precisely Dimensionality Reduction)

(The above image represents 16 eigenfaces for about a 1000 different images of human faces. Image Source: same link as for "towardsdatascience" page mentioned below.)

    I just came across the concept of having human faces as eigen vectors and this is simply awesome! This concept can be used to store compressed version of individual human faces(this saves storage space) and those individual faces can still be reconstructed to have a good approximation of their original versions. All this can be done using the basic concept of machine learning called Principal Component Analysis! Please checkout this crisp explanation from "towardsdatascience" page to know more about this topic.

Link: https://towardsdatascience.com/eigenfaces-recovering-humans-from-ghosts-17606c328184

Interpretability v/s Flexibility for different statistical learning methods

 

(Source: An Intoduction to Statistical Learning with Applications in R written by James G., Witten D., Hastie T. and Tibshirani R. and published by Springer Text in Statistics. This is a part of my machine learning specialization course from CU Boulder.)

    I learned from this topic that we can choose a method from multiple statistical learning topics based on our goal. This book shares an example saying that if we are interested in predicting the accurate values in the stock market and not much concerned about inference (how each predictor variable is associated with the output), we can use high-flexibility approaches like deep learning. However, if we are concerned about the inference, we can use high-interpretability models like Lasso or Least-squares. The book also states that sometimes the high-interpretability models give more accurate results.

Birefringence

    Aside from dispersion, materials usually have the same index of refraction for all orientations of the electric field in the light. However, some materials, usually nearly pure crystals, have two indices of refraction for the same wavelength of light. The difference is due only to the orientation of the light's electric field. Therefore, light with its electric field oscillating in the vertical direction experiences a different index of refraction from the index of refraction for an electric field oscillating in the horizontal direction. This is called birefringence. (Source: Optics for Dummies by Duree)

Variables in a Linear Regression model (machine learning)

There are two types of variables in a dataset in general:
1) Real valued (ex.: 1,2,3,4)
2) Categorical (ex.: Blood Group)
    Categorical variables have two sub-types:
    a) Ordinal (ex.: Age group 25-55, or Grades A-D)
    b) Non-ordinal (ex.: Male/Female, Plant/Animal)

    It is comparatively easy to build linear regression model around real-values and ordinal-categorical variables that can relate to regression. However, nonordinal-categorical variables pose a problem for a "regression" model. This is  because the regression equations,

y = a0 + a1X1 + a2X2 +....

cannot rely on their arbitrary nature to predict output values. Nonetheless, these nonordinal categorical variables can always be transformed into real-valued or binary forms to make use of them in a regression problem.

Mangin Mirror to Resolve Spherical Aberration

Most imaging systems face the problem of spherical aberration when trying to focus light on an image plane. A Mangin mirror, which is a mirror with a positive focal length (concave shape for mirrors) when placed alongside a lens with negative meniscus, is known to resolve for spherical aberration.

Figure 1. Mangin mirror with a negative meniscus lens in blue color and the mirror in red color. (Source: Wikipedia)

Important Difference: Method Overloading v/s Method Overriding

Here is a very important difference which leads to confusion for many new programmers:

1) Method Overloading: In the same class, we have two methods with the same name but with different parameters.

2) Method Overriding: In the sub-class, we have a method with the same name as in parent class and with the same parameters. This is usually done to have a different functionality in the sub-class.

Need of Inheritance in Object-oriented Programming

These days, I'm working on my Java programming skills especially object-oriented programming. I have taken a Coursera course for this purpose from UC San Diego. Today, I happened to come across an interesting example from Leo Porter demonstrating the need of inheritance in object-oriented programming. His explanation went as follows:

Inheritance:

Suppose you are working as a Software Engineer in a university. You have written a Person class that provides a blue-print for information of all individuals and it's working as expected. After a few days, the university administration comes to you saying we cannot have the same class for all individuals. We need to have separate classes for students and faculty respectively. One of the options to tackle this problem is to use a boolean operator and set if-else conditions for every method in your class.

Okay, so that works fine for a few days as well. Again, the university administration comes to you saying we cannot have the same class for all the students. We need separate classes for undergrad and grad students. Now, maybe you go for a nested if-else condition. But again you are asked to sort ful-time and part-time students. Now a nested if-else starts getting complicated.

Further, you are being told that you cannot have the same methods for students and faculty. Now you create two classes and name them Students and Faculty respectively. What do you think can go wrong with this?

The problem comes when you have to modify the basic structure of these classes. For example, earlier you had a single variable for storing name of a person. And now you have to store first name and last name in two separate variables. So now you have to make similar changes in both classes. The required changes might be necessary at multiple lines inside a class and the same have to be done line-by-line in another class as well. This is where the job gets tedious.

The solution to this is inheritance! Inheritance allows you to keep some common functionality in the parent class (say Person) and have distinct functionalities in child classes (Students and Faculty in this case). 

A brief study on 'Dynamic Imaging in STED'

The innovation of STED technique was a breakthrough in far-field fluorescence light microscopy. We can now image information beyond Abbe's diffraction limit and enter a new world of super-resolution imaging using light microscopy. Let's discuss advancements in STED as an effective method for dynamic imaging.

Please follow the links below to watch a video presentation on this topic:

https://www.youtube.com/watch?v=Rz0KMdvwVz0&t=15s

OR

https://www.linkedin.com/feed/update/urn:li:activity:6346789826188099584