lecture 22: sampling and quantization - mit opencourseware · pdf filetime (second) 0 0.5 1...
TRANSCRIPT
![Page 1: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/1.jpg)
6.003: Signals and Systems
Sampling and Quantization
November 29, 2011 1
![Page 2: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/2.jpg)
Last Time: Sampling
Sampling allows the use of modern digital electronics to process,
record, transmit, store, and retrieve CT signals.
• audio: MP3, CD, cell phone
• pictures: digital camera, printer
• video: DVD
• everything on the web
2
![Page 3: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/3.jpg)
x[n] Impulse
Reconstruction
xp(t) =∑x[n]δ(t− nT )
xr(t)
LPF
−ωs2
ωs2
ωT
x[n] xr(t)Impulse
Reconstruction
xp(t) =∑x[n]δ(t− nT )
LPF
Last Time: Sampling Theory
Sampling
x(t) → x[n] = x(nT )
Impulse Reconstruction
2π Bandlimited Reconstruction ωs =
T
ωsSampling Theorem: If X(jω) = 0 ∀ |ω| > then xr(t) = x(t).2
3
![Page 4: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/4.jpg)
Aliasing
−ωs ωsFrequencies outside the range < ω < alias.2 2
ω
Xp(jω) = 12π (X(j · ) ∗ P (j · ))(ω)
−ωs2
ωs2
1T
ω
P (jω)
−ωs ωs
2πT
ω
X(jω)1
4
![Page 5: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/5.jpg)
Anti-Aliasing Filter
To avoid aliasing, remove frequency components that alias before
sampling.
−ωs2
ωs2
ω1
×−ωs
2ωs2
ωT
x(t)
p(t)
xr(t)xp(t)
ReconstructionFilter
Anti-aliasingFilter
5
![Page 6: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/6.jpg)
Anti-Aliasing
−ωs ωsRemove frequencies outside the range < ω < before sampling 2 2 to avoid aliasing.
ω
Xp(jω) = 12π (X(j · ) ∗ P (j · ))(ω)
−ωs2
ωs2
1T
ω
P (jω)
−ωs ωs
2πT
ω
Anti-aliased X(jω)
6
![Page 7: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/7.jpg)
Today
Digital recording, transmission, storage, and retrieval requires dis
crete representations of both time (e.g., sampling) and amplitude.
• audio: MP3, CD, cell phone
• pictures: digital camera, printer
• video: DVD
• everything on the web
Quantization: discrete representations for amplitudes
7
![Page 8: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/8.jpg)
Quantization
We measure discrete amplitudes in bits.
11
0
0
1
1Input voltage
Outputvoltage 2 bits 3 bits 4 bits
00
01
10
0 0.5 11
0
1
Time (second)0 0.5 1
Time (second)0 0.5 1
Time (second)
1 0 1Input voltage
1 0 1Input voltage
Bit rate = (# bits/sample)×(# samples/sec)
8
![Page 9: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/9.jpg)
Check Yourself
We hear sounds that range in amplitude from 1,000,000 to 1.
How many bits are needed to represent this range?
1. 5 bits
2. 10 bits
3. 20 bits
4. 30 bits
5. 40 bits
9
![Page 10: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/10.jpg)
Check Yourself
How many bits are needed to represent 1,000,000:1?
bits
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20
range
2 4 8
16 32 64
128 256 512
1, 024 2, 048 4, 096 8, 192
16, 384 32, 768 65, 536
131, 072 262, 144 524, 288
1, 048, 576 10
![Page 11: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/11.jpg)
Check Yourself
We hear sounds that range in amplitude from 1,000,000 to 1.
How many bits are needed to represent this range? 3
1. 5 bits
2. 10 bits
3. 20 bits
4. 30 bits
5. 40 bits
11
![Page 12: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/12.jpg)
Quantization Demonstration
Quantizing Music
• 16 bits/sample
• 8 bits/sample
• 6 bits/sample
• 4 bits/sample
• 3 bits/sample
• 2 bit/sample
J.S. Bach, Sonata No. 1 in G minor Mvmt. IV. Presto
Nathan Milstein, violin
12
![Page 13: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/13.jpg)
Quantization
We measure discrete amplitudes in bits.
11
0
0
1
1Input voltage
Outputvoltage 2 bits 3 bits 4 bits
00
01
10
0 0.5 11
0
1
Time (second)0 0.5 1
Time (second)0 0.5 1
Time (second)
1 0 1Input voltage
1 0 1Input voltage
2 channels× 16 bits
sample× 44, 100 samples
sec× 60 sec
min× 74 min ≈ 6.3 G bits
≈ 0.78 G bytes
Example: audio CD
13
![Page 14: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/14.jpg)
Quantizing Images
Converting an image from a continuous representation to a discrete
representation involves the same sort of issues.
This image has 280 × 280 pixels, with brightness quantized to 8 bits.
14
![Page 15: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/15.jpg)
Quantizing Images
8 bit image 7 bit image
15
![Page 16: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/16.jpg)
Quantizing Images
8 bit image 6 bit image
16
![Page 17: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/17.jpg)
Quantizing Images
8 bit image 5 bit image
17
![Page 18: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/18.jpg)
Quantizing Images
8 bit image 4 bit image
18
![Page 19: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/19.jpg)
Quantizing Images
8 bit image 3 bit image
19
![Page 20: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/20.jpg)
Quantizing Images
8 bit image 2 bit image
20
![Page 21: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/21.jpg)
Quantizing Images
8 bit image 1 bit image
21
![Page 22: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/22.jpg)
Check Yourself
What is the most objectionable artifact of coarse quantization?
8 bit image 4 bit image
22
![Page 23: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/23.jpg)
Dithering
One very annoying artifact is banding caused by clustering of pixels
that quantize to the same level.
Banding can be reduced by dithering.
Dithering: adding a small amount (±12 quantum) of random noise to
the image before quantizing.
Since the noise is different for each pixel in the band, the noise
causes some of the pixels to quantize to a higher value and some to
a lower. But the average value of the brightness is preserved.
23
![Page 24: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/24.jpg)
Quantizing Images with Dither
7 bit image 7 bits with dither
24
![Page 25: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/25.jpg)
Quantizing Images with Dither
6 bit image 6 bits with dither
25
![Page 26: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/26.jpg)
Quantizing Images with Dither
5 bit image 5 bits with dither
26
![Page 27: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/27.jpg)
Quantizing Images with Dither
4 bit image 4 bits with dither
27
![Page 28: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/28.jpg)
Quantizing Images with Dither
3 bit image 3 bits with dither
28
![Page 29: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/29.jpg)
Quantizing Images with Dither
2 bit image 2 bits with dither
29
![Page 30: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/30.jpg)
Quantizing Images with Dither
1 bit image 1 bit with dither
30
![Page 31: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/31.jpg)
Check Yourself
What is the most objectionable artifact of dithering?
3 bit image 3 bit dithered image
31
![Page 32: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/32.jpg)
Check Yourself
What is the most objectionable artifact of dithering?
One annoying feature of dithering is that it adds noise.
32
![Page 33: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/33.jpg)
Quantization Schemes
Example: slowly changing backgrounds.
Quantization: y = Q(x)
Quantization with dither: y = Q(x + n)
33
![Page 34: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/34.jpg)
Check Yourself
What is the most objectionable artifact of dithering?
One annoying feature of dithering is that it adds noise.
Robert’s technique: add a small amount (±12 quantum) of random
noise before quantizing, then subtract that same amount of random
noise.
34
![Page 35: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/35.jpg)
Quantization Schemes
Example: slowly changing backgrounds.
Quantization: y = Q(x)
Quantization with dither: y = Q(x + n)
Quantization with Robert’s technique: y = Q(x + n) − n
35
![Page 36: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/36.jpg)
Quantizing Images with Robert’s Method
7 bits with dither 7 bits with Robert’s method
36
![Page 37: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/37.jpg)
Quantizing Images with Robert’s Method
6 bits with dither 6 bits with Robert’s method
37
![Page 38: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/38.jpg)
Quantizing Images with Robert’s Method
5 bits with dither 5 bits with Robert’s method
38
![Page 39: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/39.jpg)
Quantizing Images with Robert’s Method
4 bits with dither 4 bits with Robert’s method
39
![Page 40: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/40.jpg)
Quantizing Images with Robert’s Method
3 bits with dither 3 bits with Robert’s method
40
![Page 41: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/41.jpg)
Quantizing Images with Robert’s Method
2 bits with dither 2 bits with Robert’s method
41
![Page 42: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/42.jpg)
Quantizing Images with Robert’s Method
1 bits with dither 1 bit with Robert’s method
42
![Page 43: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/43.jpg)
Quantizing Images: 3 bits
8 bits 3 bits
dither Robert’s
43
![Page 44: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/44.jpg)
Quantizing Images: 2 bits
8 bits 2 bits
dither Robert’s
44
![Page 45: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/45.jpg)
Quantizing Images: 1 bit
8 bits 1 bit
dither Robert’s
45
![Page 46: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/46.jpg)
Progressive Refinement
Trading precision for speed.
Start by sending a crude representation, then progressively update
with increasing higher fidelity versions.
46
![Page 47: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/47.jpg)
Discrete-Time Sampling (Resampling)
DT sampling is much like CT sampling.
×x[n] xp[n]
p[n] =∑k δ[n− kN ]
n
xp[n]
0
n
x[n]
0
n
p[n]
0
47
![Page 48: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/48.jpg)
Discrete-Time Sampling
jΩ).As in CT, sampling introduces additional copies of X(e
×x[n] xp[n]
p[n] =∑k δ[n− kN ]
0 2π3
4π3
2π−2π3
−4π3
−2π
13
Ω
Xp(ejΩ)
0 2π−2π
1
Ω
X(ejΩ)
0 2π3
4π3
2π−2π3
−4π3
−2π
2π3
Ω
P (ejΩ)
48
![Page 49: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/49.jpg)
Discrete-Time Sampling
Sampling a finite sequence gives rise to a shorter sequence.
n
xb[n]
0
n
xp[n]
0
n
x[n]
0
−jΩn −jΩn −jΩk/3 jΩ/3)Xb(ejΩ) = xb[n]e = xp[3n]e = xp[k]e = Xp(en n k
49
![Page 50: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/50.jpg)
Discrete-Time Sampling
But the shorter sequence has a wider frequency representation.
0 2π3
4π3
2π−2π3
−4π3
−2π
13
Ω
Xb(ejΩ) = Xp(ejΩ/3)
0
0 2π−2π
13
Ω
Xp(ejΩ)
2π−2π
1
Ω
X(ejΩ)
0
50
![Page 51: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/51.jpg)
Discrete-Time Sampling
51
![Page 52: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/52.jpg)
Discrete-Time Sampling
52
![Page 53: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/53.jpg)
Discrete-Time Sampling
53
![Page 54: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/54.jpg)
Discrete-Time Sampling
54
![Page 55: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/55.jpg)
Discrete-Time Sampling
55
![Page 56: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/56.jpg)
Discrete-Time Sampling
56
![Page 57: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/57.jpg)
Discrete-Time Sampling
57
![Page 58: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/58.jpg)
Discrete-Time Sampling
Insert zeros between samples to upsample the images.
n
xb[n]
0
n
xp[n]
0
n
x[n]
0
−jΩn −jΩn −jΩk/3 jΩ/3)Xb(ejΩ) = xb[n]e = xp[3n]e = xp[k]e = Xp(en n k
58
∑ ∑ ∑
![Page 59: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/59.jpg)
Discrete-Time Sampling
Then filter out the additional copies in frequency.
0 2π3
4π3
2π−2π3
−4π3
−2π
13
Ω
Xb(ejΩ) = Xp(ejΩ/3)
0
0 2π−2π
13
Ω
Xp(ejΩ)
2π−2π
1
Ω
X(ejΩ)
0
59
![Page 60: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/60.jpg)
Discrete-Time Sampling: Progressive Refinement
60
![Page 61: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/61.jpg)
Discrete-Time Sampling: Progressive Refinement
61
![Page 62: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/62.jpg)
Discrete-Time Sampling: Progressive Refinement
62
![Page 63: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/63.jpg)
Discrete-Time Sampling: Progressive Refinement
63
![Page 64: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/64.jpg)
Discrete-Time Sampling: Progressive Refinement
64
![Page 65: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/65.jpg)
Discrete-Time Sampling
65
![Page 66: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/66.jpg)
Perceptual Coding
Quantizing in the Fourier domain: JPEG.
66
![Page 67: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/67.jpg)
JPEG
Example: JPEG (“Joint Photographic Experts Group”) encodes im
ages by a sequence of transformations:
• color encoding
• DCT (discrete cosine transform): a kind of Fourier series
• quantization to achieve perceptual compression (lossy)
• Huffman encoding: lossless information theoretic coding
We will focus on the DCT and quantization of its components.
• the image is broken into 8 × 8 pixel blocks
• each block is represented by its 8 × 8 DCT coefficients
• each DCT coefficient is quantized, using higher resolutions for
coefficients with greater perceptual importance
67
![Page 68: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/68.jpg)
JPEG
Discrete cosine transform (DCT) is similar to a Fourier series, but
high-frequency artifacts are typically smaller.
Example: imagine coding the following 8 × 8 block.
For a two-dimensional transform, take the transforms of all of the
rows, assemble those results into an image and then take the trans
forms of all of the columns of that image.
68
![Page 69: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/69.jpg)
JPEG
Periodically extend a row and represent it with a Fourier series.
x[n] = x[n+ 8]
n0 8
There are 8 distinct Fourier series coefficients. 1 2π−jkΩ0n ak = x[n]e ; Ω0 = 8 8
n=<8>
69
∑
![Page 70: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/70.jpg)
JPEG
DCT is based on a different periodic representation, shown below.
y[n] = y[n+ 16]
n0 16
70
![Page 71: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/71.jpg)
Check Yourself
Which signal has greater high frequency content?
x[n] = x[n+ 8]
n0 8
y[n] = y[n+ 16]
n0 16
71
![Page 72: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/72.jpg)
Check Yourself
The first signal, x[n], has discontinuous amplitude. The second sig
nal, y[n] is not discontinuous, but has discontinuous slope.
x[n] = x[n+ 8]
n0 8
y[n] = y[n+ 16]
n0 16
The magnitude of its Fourier series coefficients decreases faster with
k for the second than for the first.
72
![Page 73: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/73.jpg)
Check Yourself
Which signal has greater high frequency content? x[n]
x[n] = x[n+ 8]
n0 8
y[n] = y[n+ 16]
n0 16
73
![Page 74: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/74.jpg)
JPEG
Periodic extension of an 8 × 8 pixel block can lead to a discontinuous
function even when the “block” was taken from a smooth image.
original row
n0
8 pixel ”block”
n0
x[n] = x[n+ 8]
n0 8
74
![Page 75: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/75.jpg)
JPEG
Periodic extension of the type done for JPEG generates a continuous
function from a smoothly varying image.
original row
n0
8 pixel ”block”
n0
y[n] = y[n+ 16]
n0 16
75
![Page 76: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/76.jpg)
JPEG
Although periodic in N = 16, y[n] can be represented by just 8 distinct
DCT coefficients.
y[n] = y[n+ 16]
n0 16
bk = 7
n=0
y[n] cos
πk N
n + 1 2
This results because y[n] is symmetric about n = −12 , and this sym
metry introduces redundancy in the Fourier series representation.
Notice also that the DCT of a real-valued signal is real-valued.
76
∑
![Page 77: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/77.jpg)
JPEG
The magnitudes of the higher order DCT coefficients are smaller
than those of the Fourier series.
−3−2−1 0 1 2 3 4 m−3−2−1
01234n Fourier series
m
|ak|
0 1 2 3 4 5 6 7 m01234567n DCT
m
bk
77
![Page 78: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/78.jpg)
JPEG
Humans are less sensitive to small deviations in high frequency com
ponents of an image than they are to small deviations at low frequen
cies. Therefore, the DCT coefficients are quantized more coarsely
at high frequencies.
Divide coefficient b[m, n] by q[m, n] and round to nearest integer.
q[m, n] m →
16 11 10 16 24 40 51 61
12 12 14 19 26 58 60 55
14 13 16 24 40 57 69 56
n 14 17 22 29 51 87 80 62
↓ 18 22 37 56 68 109 103 77
24 35 55 64 81 104 113 92
49 64 78 87 103 121 120 101
72 92 95 98 112 100 103 99 78
![Page 79: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/79.jpg)
Check Yourself
Which of the following tables of q[m, n] (top or bottom)
will result in higher “quality” images?
q[m, n] m → 16 11 10 16 24 40 51 61 12 12 14 19 26 58 60 55 14 13 16 24 40 57 69 56
n 14 17 22 29 51 87 80 62 ↓ 18 22 37 56 68 109 103 77
24 35 55 64 81 104 113 92 49 64 78 87 103 121 120 101 72 92 95 98 112 100 103 99
q[m, n] m → 32 22 20 32 48 80 102 122 24 24 28 38 52 116 120 110 28 26 32 48 80 114 139 112
n 28 34 44 58 102 174 160 124 ↓ 36 44 74 112 136 218 206 154
48 70 110 128 162 208 226 194 98 128 156 174 206 256 240 202 144 184 190 196 224 200 206 198
79
![Page 80: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/80.jpg)
Check Yourself
Which of the following tables of q[m, n] (top or bottom)
will result in higher “quality” images? top
q[m, n] m → 16 11 10 16 24 40 51 61 12 12 14 19 26 58 60 55 14 13 16 24 40 57 69 56
n 14 17 22 29 51 87 80 62 ↓ 18 22 37 56 68 109 103 77
24 35 55 64 81 104 113 92 49 64 78 87 103 121 120 101 72 92 95 98 112 100 103 99
q[m, n] m → 32 22 20 32 48 80 102 122 24 24 28 38 52 116 120 110 28 26 32 48 80 114 139 112
n 28 34 44 58 102 174 160 124 ↓ 36 44 74 112 136 218 206 154
48 70 110 128 162 208 226 194 98 128 156 174 206 256 240 202 144 184 190 196 224 200 206 198
80
![Page 81: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/81.jpg)
JPEG
Finally, encode the DCT coefficients for each block using “run
length” encoding followed by an information theoretic (lossless)
“Huffman” scheme, in which frequently occuring patterns are
represented by short codes.
The “quality” of the image can be adjusted by changing the values
of q[m, n]. Large values of q[m, n] result in large “runs” of zeros,
which compress well.
81
![Page 82: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/82.jpg)
JPEG: Results
1%: 1666 bytes 10%: 2550 bytes 20%: 3595 bytes
40%: 5318 bytes 80%: 100%: 47k bytes
10994 bytes 82
![Page 83: Lecture 22: Sampling and quantization - MIT OpenCourseWare · PDF fileTime (second) 0 0.5 1 Time (second) 0 0.5 1 Time (second) 1 0 1 Input voltage ... We will focus on the DCT and](https://reader031.vdocuments.pub/reader031/viewer/2022022002/5a8d970f7f8b9a4a268cc296/html5/thumbnails/83.jpg)
MIT OpenCourseWarehttp://ocw.mit.edu
6.003 Signals and SystemsFall 2011
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.