EENG 851: Advanced Signal Processing

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EENG 851 Advanced Signal Processing

• Class 15 Outline
• Introduction to Image Processing
• Introduction to Image Compression
• Introduction to the Discrete Wavelet Transform
• References
• Digital Image Processing by Gonzalez and Woods,
  Addison-Wesely 1993
• A Simplified Approach to Image Processing, by
  Randy Crane, Prentice Hall, 1997
• -Ripples in Mathematics The Discrete Wavelet
  Transform by Jensen, A. and la Cour-Harbo, A.,
  Springer 2001

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Introduction to Wavelets

• Image Examples
• Pixel Definition
• Precision

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Image Examples
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Image Examples
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Image Examples
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Image Examples
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Image Examples
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Image Examples
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Image Examples
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Introduction to Image Compression
Consider a 512-by-512 pixel image updated at 30
frames/sec with 3 bytes/pixel (color). If such
a process must store the data, then 23.6
MBytes/sec are required. With MPEG-1 compression,
full motion video can be compressed down to 187
KBytes/sec. Suppose you can rent a video
elctronically and then download it over the
phone line. Since one secon of
uncompressed video requires 23.6 MBytes of
storage, your two hour video would need 169
GBytes. If your high bandwidth phone line can
transfer 180 MBytes/sec, it would take 15.7
minutes to transfer your movie (about 1/8 the
time to watch the movie). If the movie
was compressed with MPEG-1. The storage
requirement would be 1.3GBytes and the transfer
time would be 7.48 sec.
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Introduction to Image Compression (Cont)
Clearly image compression is required. There
are a tremendous number of hardware and software
products decicated solely to compress images. As
computer graphics attain higher resolution and
applications require higher intensity
resolution (more bits/pixel), the need for image
compression will increase. Medical imagery is a
prime example of images increasing in
both spatial resolution and intensity resolution.
Although humans dont need more than 8
bits/pixel to view gray scale images,
computer vision can analyze dat of much higher
intensity resolution.
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Introduction to Image Compression (Cont)

• There are two basic types of Image Compression.
• 1. Lossless- Encodes and decodes the data
  exactly and the
• resulting image matches the original perfectly.
• 2. Lossy
• Allow redundant and nonessential information to
  be lost.
• There is a tradeoff between compression and
  image quality.
• Techniques are typically more comples and
  require more
• computations.
• Data can be remove that the human eye wouldnt
  notice. This
• is OK if the results are meant to be viewed
  by humans.
• If the images are to be analyzed by machine,
  lossy methods may
• not be appropriate.
• The goal is for the final decompressed images
  tobe visually lossless.

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Introduction to Image Compression (Cont)

• 2. Lossy (Cont).
• In addition to compression, the entire imaging
  operation is lossy
• scanning and digitizing, displaying, and/or
  printing.
• The goal is to keep the losses
  indistinguishable.
• The choice of technique depends on the image
  data. Some
• images, especially those used for medical
  diagnosis cannot afford
• to lose any data. Computer generated graphics
  with large areas of
• the same color compress well with simple lossless
  schemes, like
• run length encoding. Continous tone images with
  complex shapes
• and shading with a high degree of detail that
  cant be lost, such as
• detailed CAD drawings, cannot be compressed with
  lossy algorithms.

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Introduction to Image Compression (Cont)

• When choosing a compression technique, you must
  look at
• more than the achievable compression ratio. The
  CR doesnt tell
• anything about the quality of the resulting
  image. Other
• considerations include
• Compression/decompression time.
• Algorithm complexity
• Cost
• Availability of computational resources
• Standards. A terrific compression algorithm with
  one user has
• limited applications. If your receiver could
  be any hospital in the
• world, then you better use a standard
  compression technique and
• file format.
• If the C/D is limited to a small set of
  systems, then a uniquely
• developed algorithm might be appropriate.

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Introduction to Image Compression (Cont)
Definitions Character- A fundmental data
element in the input stream. It may be a single
letter or a pixel in an image file. Strings-
Sequences of characters. Input Stream- The
source of the uncompressed data. It may ba a
data file or some communication medium. Code
Words- The data elements used to represent the
input characters or strings. Encoding-
Compressing. Decoding (Decompression)- The
inverse of compressing.
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A Structured Compression System
Output Bit-stream
Input Data Stream
Reconstruction
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Coding

• Exploit statistical redundancy
• Ideally, the underlying random variables are
  statistically independent.
• Exploit any non-uniformity in the probability
  distributions.

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Coding (Cont)
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Coding (Cont)
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Quantization
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Quantization (Cont)
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Quantization (Cont)
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Transforms
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Transforms (Cont)
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Transforms (Cont)
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Transforms (Cont)
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Transforms (Cont)
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Transforms (Cont)
Original Image x 20 20 20 20
20 60 20 20 20 20 20 20
20 60 20 20 50 50 50 50
50 60 50 50 20 20 20 20
20 60 20 20 20 20 20 20
20 60 20 20 20 20 20 20
20 60 20 20 20 20 20 20
20 60 20 20 20 20 20 20
20 60 20 20
Wavelet Transform of Original Image y 20
0 20 0 40 20
20 0 0 0 0
0 -20 20 0 0 35
15 35 15 47.5 12.5 35
15 -15 -15 -15 -15 -27
12.5 -15 -15 20 0 20
0 40 20 20 0
0 0 0 0 -20
20 0 0 20 0
20 0 40 20 20
0 0 0 0 0 -20
20 0 0
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Transforms (Cont)
No Threshold
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Transforms (Cont)
Wavelet Transform of Original Image Threshold13 y
20 0 20 0
40 20 20 0 0 0
0 0 -20 20 0
0 35 15 35 15 47.5
0 35 15 -15 -15 -15
-15 -27 0 -15 -15 20
0 20 0 40 20
20 0 0 0 0
0 -20 20 0 0
20 0 20 0 40 20
20 0 0 0 0
0 -20 20 0 0
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Transforms (Cont)
Threshold13
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Transforms (Cont)
Wavelet Transform of Original Image Threshold16 y
20 0 20 0
40 20 20 0 0 0
0 0 -20 20 0
0 35 15 35 0 47.5
0 35 0 0 0 0
0 -27 0 0 0
20 0 20 0 40
20 20 0 0 0
0 0 -20 20 0
0 20 0 20 0 40
20 20 0 0 0
0 0 -20 20 0 0
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Transforms (Cont)
Threshold16
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The Wavelet Transform
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The Wavelet Transform (Cont)
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The Wavelet Transform (Cont)
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The Wavelet Transform (Cont)
Original 56 40 8 24 48
48 40 16 Thresholded 59 43 11
27 45 45 37 13
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The Wavelet Transform (Cont)
Example
Threshold4
Amplitude
Original
Sample
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The Wavelet Transform (Cont)
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The Wavelet Transform (Cont)
Original 56 40 8 24 48
48 40 16 Thresholded 51 51 19
19 45 45 37 13
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The Wavelet Transform (Cont)
Example
Threshold9
Amplitude
Original
Sample
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The Wavelet Transform (Cont)
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The Wavelet Transform (Cont)
xnlt68 52 48 36 54 25 63
81 37 66 74 19 22 60 43 62gt
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The Discrete Wavelet Transform
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The Discrete Wavelet Transform
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The Discrete Wavelet Transform
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The Discrete Wavelet Transform
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The DWT (Cont)
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The DWT (Cont)
56 40 8 24 48 48 40 16
48 16 48 28 -16 16 0 -24
32 38 -32 -20 -16 16 0 -24 35
6 -32 -20 -16 16 0 -24
56 40 8 24 48 48 40 16 48
16 48 28 8 -8 0 12 32 38
16 10 8 -8 0 12 35 -3
16 10 8 -8 0 12
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The DWT (Cont)
59 43 11 27 45 45 37 13
51 19 45 25 -16 16 0 -24
35 35 -32 -20 -16 16 0 -24
35 0 -32 -20 -16 16 0 -24
Original 56 40 8 24 48
48 40 16 Reconstruction 59 43
11 27 45 45 37 13
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The DWT (Cont)
59 43 11 27 45 45 37 13
51 19 45 25 -16 16 0 -24
35 35 -32 -20 -16 16 0 -24
35 0 -32 -20 -16 16 0 -24
Original 56 40 8 24 48
48 40 16 Reconstruction 59 43
11 27 45 45 37 13
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The DWT (Cont)
Standard Difference DWT
MSE9
Threshold9
Amplitude
Original
Sample
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The DWT (Cont)
59 43 11 27 45 45 37 13
51 19 45 25 0 0 0 -24
35 35 -32 -20 0 0 0 -24
35 0 -32 -20 0 0 0 -24
Original 56 40 8 24 48
48 40 16 Reconstruction 51 51
19 19 45 45 37 13
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The DWT (Cont)
Standard Difference DWT
MSE41
Threshold19
Amplitude
Original
Sample
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Lifting
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Lifting (Cont)
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Lifting (Cont)
P
split
U

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Lifting (Cont)
59
Lifting (Cont)
60
Lifting (Cont)
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Lifting (Cont)
split
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Lifting (Cont)
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Lifting (Cont)
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Lifting (Cont)
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Lifting (Cont)
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Example 2
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Example 2 (Cont)
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Example 2
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Example 2
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Example 2 (Cont)
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Example 2 (Cont)
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Example 2 (Cont)
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Lifting in General
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Lifting in General (Cont))
P
split
U
U
P
merge

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Lifting in General (Cont))
P1
split
P2
P3
U1
U2
U3
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Lifting in General (Cont))
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Lifting in General (Cont))
P1
split
U1
U2
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Lifting in General (Cont))
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Lifting in General (Cont))
U
U
P
merge
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Lifting in General (Cont))
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Lifting in General (Cont))
P1
split
U1
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Lifting in General (Cont))
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Lifting in General (Cont))
U
P
merge
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The DWT in General
Ts
Ta
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The DWT in General
Ta
Ta
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The DWT in General (cont)
Ta
Ta
Ta
Ta
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The DWT in General (cont)
Ts
Ts
Ts
Ts
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The DWT in General (Cont)
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The DWT in General (Cont)
90
The DWT in General (Cont)
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The DWT in General (Cont)
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The DWT in General (Cont)