Enjoy(nt) the Adjoints

Most of the modeling we do requires linear operations that predict data from models. However, in Imaging, the usual task is to find inverse of these operators to find the models from the data. This is where the Adjoint operator comes to rescue as a part of the inversion process. Adjoint of a matrix is … Continue reading Enjoy(nt) the Adjoints


Singular Value Decomposition

An intuitive approach: https://www.quora.com/What-is-an-intuitive-explanation-of-singular-value-decomposition-SVD/answer/Jason-Liu-21?srid=uSFD Two part in-depth description: https://jeremykun.com/2016/04/18/singular-value-decomposition-part-1-perspectives-on-linear-algebra/ https://jeremykun.com/2016/05/16/singular-value-decomposition-part-2-theorem-proof-algorithm/ Use of SVD in finding pseudo inverse of a matrix: https://www.cse.unr.edu/~bebis/CS791E/Notes/SVD.pdf https://math.stackexchange.com/questions/1541986/what-is-the-best-way-to-compute-the-pseudoinverse-of-a-matrix https://math.stackexchange.com/questions/1939962/singular-value-decomposition-and-inverse-of-square-matrix

Angular wavenumber

A very interesting thing I re-discovered today is the angular wavenumber. Although it seems like I had understood that pretty well long back, I had forgotten quite some things about it. The rumination came from my continuous learning process of Fourier Transform. While I seem to have gotten a hang of temporal Fourier transform, spacial … Continue reading Angular wavenumber


My notes on Laplace transform

The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), which is a unilateral transform defined by: $latex F(s)=\int_{0}^{\infty} f(t)e^{-st} dt$ where s is a complex number frequency parameter $latex s=\sigma +i\omega $, with real numbers $latex \sigma$ and $latex \omega$. In the case of the Laplace transform, … Continue reading My notes on Laplace transform


Hessian and Jacobian Matrix

Hessian Matrix "We can use Hessian to find out how big a step we need to make from our current point to the point of minima or maxima, depending on the sign of the Hessian (i.e. the determinant of the Hessian matrix). By repeatedly performing an update that takes into account the inverse of the … Continue reading Hessian and Jacobian Matrix


Hilbert transform and the curious case of enveloping [My notes]

Hilbert transform in frequency domain -It is a peculiar sort of filter that changes the phase of the spectral components depending on the sign of their frequency. -It only effects the phase of the signal. It has no effect on the amplitude at all.For any signal g(t), its Hilbert Transform has the following property: $latex … Continue reading Hilbert transform and the curious case of enveloping [My notes]


Conjugate gradient method

Disclaimer: The notes are made from information available here and wikipedia. Conjugate gradient is effective for solving systems of the form: $latex \mathbf{A}x=b$ $latex x$ is an unknown vector $latex \mathbf{A}$ is a known positive-definite, square, symmetric matrix $latex b$ is a known vector The CG method is most effective in solving systems where $latex \mathbf{A}$ is … Continue reading Conjugate gradient method