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.)