| Line 43: | Line 43: | ||
let's consider the diagrams below: | let's consider the diagrams below: | ||
[[Image:Slide1.jpg]] | [[Image:Slide1.jpg]] | ||
| − | *This shows the basic schematic of an interpolator. D in the circle is how many zeros we fill between samples and the Low Pass Filter removes the extraneous copies of the signal shown in the output below. | + | *This shows the basic schematic of an interpolator. D in the circle is how many zeros we fill between samples and the Low Pass Filter removes the extraneous copies of the signal beyond W/D shown in the output below. |
| + | *Let's pretend we waved the math wand and got the frequency domain as below. | ||
---- | ---- | ||
[[Image:Slide2.jpg]] | [[Image:Slide2.jpg]] | ||
| + | ---- | ||
| + | == Application to Actual Signal == | ||
| + | * Now, let's see if this can be applied to an actual example. | ||
| + | * From the introduction, our main aim was to take a badly sampled signal(our friendly sine wave shown below) and make it "good" or sampled at a better resolution | ||
| + | *According to the math, this whole process should give the same result if we were to take a "good" sampling of the original ct signal | ||
[[Image:sin_bad.jpg]] | [[Image:sin_bad.jpg]] | ||
Revision as of 00:11, 23 September 2009
Contents
Discrete Time Interpolation
Mathematically, sure. Realistically? Let's find out.
Introduction
My fascination with the concept of discrete time interpolation began, when I asked Prof. Boutin if a discrete time interpolator, can ideally make a low-resolution image, a high resolution one.
The answer was, yes. Ideally, it can.
The equations that led to the concept seemed impeccable and mathematically, it seemed to make perfect sense.
But essentially, all we are doing is:
- adding zeros in between samples (the result of which looks horrible by the way)
- Low pass filtering,
and Voila! hi-res image. Impossible right?
After 6 hours of coding, and processing the image of my dog "Milo"(shown below), countless times, I am proud to say, that it is
"Almost Possible in the Real World"
Page Map
This page contains
- Application to a simple 1-D signal
- Application to an actual real-world image
Mathematical Basis
To put the math in a nutshell, so that we can get started with the cool stuff,
let's consider the diagrams below:
- This shows the basic schematic of an interpolator. D in the circle is how many zeros we fill between samples and the Low Pass Filter removes the extraneous copies of the signal beyond W/D shown in the output below.
- Let's pretend we waved the math wand and got the frequency domain as below.
Application to Actual Signal
- Now, let's see if this can be applied to an actual example.
- From the introduction, our main aim was to take a badly sampled signal(our friendly sine wave shown below) and make it "good" or sampled at a better resolution
- According to the math, this whole process should give the same result if we were to take a "good" sampling of the original ct signal
