sampling section 4 3 l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Sampling (Section 4.3) PowerPoint Presentation
Download Presentation
Sampling (Section 4.3)

Loading in 2 Seconds...

play fullscreen
1 / 27

Sampling (Section 4.3) - PowerPoint PPT Presentation


  • 72 Views
  • Uploaded on

Sampling (Section 4.3). CS474/674 – Prof. Bebis. Sampling. How many samples should we get so that no information is lost during the sampling process? Hint: take enough samples so that the “continuous” image can be reconstructed from its samples. Example.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Sampling (Section 4.3)' - ivi


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
sampling section 4 3

Sampling (Section 4.3)

CS474/674 – Prof. Bebis

sampling
Sampling
  • How many samples should we get so that no information is lost during the sampling process?
  • Hint: take enough samples so that the “continuous” image can be reconstructed from its samples.
example
Example

Sampled signal looks like a sinusoidal of a lower frequency !

definition band limited functions
Definition: “band-limited” functions
  • A function whose spectrum is of finite duration
  • Are all functions band-limited?

max

frequency

NO!!

properties of band limited functions
Properties of band-limited functions
  • Band-limited functions have infinite duration in the time domain.
  • Functions with finite duration in the time domain have infinite duration in the frequency domain.
sampling a 1d function
Sampling a 1D function
  • Multiply f(x) with s(x)

sampled f(x)

x

Question: what is the DFT of f(x) s(x)?

Hint: use convolution theorem!

sampling a 1d function cont d
Sampling a 1D function (cont’d)
  • Suppose f(x) F(u)
  • What is the DFT of s(x)?
sampling a 2d function cont d
Sampling a 2D function (cont’d)
  • 2D train of impulses

s(x,y)

x

y

Δy

Δx

sampling a 2d function cont d10
Sampling a 2D function (cont’d)
  • DFT of 2D discrete function (i.e., image)

f(x,y)s(x,y) F(u,v)*S(u,v)

reconstructing f x from its samples

x

Reconstructing f(x) from its samples
  • Need to isolate a single period:
    • Multiply by a window G(u)
what is the effect of x
What is the effect of Δx?
  • Large Δx (i.e., few samples) results to overlapping periods!
effect of x cont d

x

Effect of Δx (cont’d)
  • But, if the periods overlap, we cannot anymore isolate
  • a single period  aliasing!
what is the effect of x cont d
What is the effect of Δx? (cont’d)
  • Smaller Δx (i.e., more samples) alleviates aliasing!
example17
Example
  • Suppose that we have an imaging system where the number of samples it can take is fixed at 96 x 96 pixels.
  • Suppose we use this system to digitize checkerboard patterns.
  • Such a system can resolve patterns that are up to 96 x 96 squares (i.e., 1 x 1 pixel squares).
  • What happens when squares are less than 1 x 1 pixels?
example18
Example

square size: 16 x 16 6 x 6

(same as

12 x 12

squares)

square size: 160.9174 0.4798

how to choose x
How to choose Δx?
  • The center of the overlapped region is at
how to choose x cont d
How to choose Δx? (cont’d)
  • Choose Δx as follows:

where W is the max frequency of f(x)

practical issues
Practical Issues
  • Band-limited functions have infinite duration in the time domain.
  • But, we can only sample a function over a finite interval!
practical issues cont d

x

=

Practical Issues (cont’d)
  • We would need to obtain a finite set of samples
  • by multiplying with a “box” function:
    • [s(x)f(x)]h(x)
practical issues cont d23
Practical Issues (cont’d)
  • This is equivalent to convolution in the frequency domain!
  • [s(x)f(x)]h(x)  [F(u)*S(u)] * H(u)
how does this affect things in practice
How does this affect things in practice?
  • Even if the Nyquist criterion is satisfied, recovering a function that has been sampled in a finite region is in general impossible!
  • Special case:periodic functions
    • If f(x) isperiodic, then a single period can be isolated assuming that the Nyquist theorem is satisfied!
    • e.g., sin/cos functions
anti aliasing
Anti-aliasing
  • In practice, aliasing in almost inevitable!
  • The effect of aliasing can be reduced by smoothing the input signal to attenuate its higher frequencies.
  • This has to be done before the function is sampled.
    • Many commercial cameras have true anti-aliasing filtering built in the lens of the surface of the sensor itself.
    • Most commercial software have a feature called “anti-aliasing” which is related to blurring the image to reduced aliasing artifacts (i.e., not trueanti-aliasing)
example27
Example

3 x 3 blurring and

50% less samples

50% less samples