This module extends the ideas of the Discrete Fourier Transform (DFT) into two-dimensions, which is necessary for any image processing.
2d dft
To perform image restoration (and many other useful image
processing algorithms) in a computer, we need a FourierTransform (FT) that is discrete and two-dimensional.
where the above equation (
)
has finite support for an
$N$ x
$N$ image.
Inverse 2d dft
As with our regular fourier transforms, the 2D DFT also has
an inverse transform that allows us to reconstruct an imageas a weighted combination of complex sinusoidal basis
functions.
Below we go through the steps of convolving two
two-dimensional arrays. You can think of
$f$ as representing an image and
$h$ represents a PSF, where
$h(m, n)=0$ for
$m\land n> 1$ and
$m\land n< 0$ .
$$h=\begin{pmatrix}h(0, 0) & h(0, 1)\\ h(1, 0) & h(1, 1)\\ \end{pmatrix}$$$$f=\begin{pmatrix}f(0, 0) & & f(0, N-1)\\ & & \\ f(N-1, 0) & & f(N-1, N-1)\\ \end{pmatrix}$$ Step 1 (Flip
$h$ ):
We use the standard 2D convolution equation (
) to find the first element of
our convolved image. In order to better understand what ishappening, we can think of this visually. The basic idea is
to take
$h(-m, -n)$ and place it "on top" of
$f(k, l)$ , so that just the bottom-right element,
$h(0, 0)$ of
$h(-m, -n)$ overlaps with the top-left element,
$f(0, 0)$ , of
$f(k, l)$ . Then, to get the next element of our convolved
image, we slide the flipped matrix,
$h(-m, -n)$ , over one element to the right and get the
following result:
$$g(0, 1)=h(0, 0)f(0, 1)+h(0, 1)f(0, 0)$$ We continue in this fashion to find all of the elements ofour convolved image,
$g(m, n)$ . Using the above method we define the general
formula to find a particular element of
$g(m, n)$ as:
Using this equation, we can calculate the value for each
position on our final image,
$\stackrel{~}{g}(m, n)$ . For example, due to the periodic extension of
the images, when circular convolution is applied we willobserve a
wrap-around effect.
where in the above equation,
$\sum_{n=0}^{N-1} f(m, n)e^{-i\frac{2\pi ln}{N}}$ is simply a 1D DFT over
$n$ .
This means that we will take the 1D FFT of each row; if wehave
$N$ rows, then it will
require
$N\lg N$ operations per row. We can rewrite this as
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?