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Multiresolution Image Processing: Image Pyramids and Subband Coding - Prof. S. Todorovic, Study notes of Electrical and Electronics Engineering

Oregon State University (OSU)Electrical and Electronics Engineering

Prof. S. Todorovic

Multiresolution image processing, specifically image pyramids and subband coding. The motivation for multiresolution analysis, the construction of image pyramids, typical filters used, and perfect reconstruction. Subband coding is also explained, including the use of biorthogonal filters and orthonormal filter banks.

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Uploaded on 08/30/2009

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ECE 468: Digital Image Processing

Lecture 9

Prof. Sinisa Todorovic

[email protected]

Multiresolution Image Processing

•Informal motivation:

•Images may show both very large and very small objects.

•It may be useful to process the images at different resolutions.

Discover Study notes of Electrical and Electronics Engineering Oregon State University (OSU)

Partial preview of the text

Download Multiresolution Image Processing: Image Pyramids and Subband Coding - Prof. S. Todorovic and more Study notes Electrical and Electronics Engineering in PDF only on Docsity!

ECE 468: Digital Image Processing

Lecture 9

Prof. Sinisa Todorovic

[email protected]

Multiresolution Image Processing

• Informal motivation:

• Images may show both very large and very small objects.

It may be useful to process the images at different resolutions.

Multiresolution Image Processing

A more formal motivation:

An image is a 2D random process with locally varying

statistics of pixel intensities

Analysis of statistical properties of pixel neighborhoods of

varying sizes may be useful

Histogram of Small Pixel Neighborhoods

Steps to Construct the Image Pyramid

1. Given an image at level j

2. Filter the input and and downsample the filtered result by

a factor of 2; This gives the image at level j-

3. Goto 1

4. Upsample and filter the image at level j-1; this gives an

approximation of the image at level j

5. Subtract this result from the image at level j; this give the

prediction residual at level j

6. Goto 1

Typical Filters

For the multiresolution pyramid, we use spatial filters:
- Neighborhood averaging
- Lowpass Gaussian filter
For the residual pyramid, we use interpolation filters:
- bilinear
- bicubic

Upsampling/Downsampling

Upsampling = Inserting zeros
Downsampling = Discarding pixels

f 2 ↑

(x, y) =

{

f (x/ 2 , y/2) , x, y are even

0 , o.w.

f 2 ↓

(x, y) = f (2x, 2 y)

Subband Image Coding

g 0 (n) = (−1)

h 1 (n)

g 1

(n) = (−1)

n+ h 0

(n)

for perfect reconstruction:

Subband Image Coding

g 0 (n) = (−1)

h 1 (n)

g 1

(n) = (−1)

n+ h 0

(n)

for perfect reconstruction:

flp(n) =

f (2n)! h 0

(2n) , 2 n

0 , 2 n + 1

f hp

(n) =

f (2n + 1)! h 1

(2n + 1) , 2 n + 1

0 , 2 n

Subband Image Coding

g 0 (n) = (−1)

h 1 (n)

g 1

(n) = (−1)

n+ h 0

(n)

for perfect reconstruction:

flp(n) =

f (2n)! h 0

(2n) , 2 n

0 , 2 n + 1

f hp

(n) =

f (2n + 1)! h 1

(2n + 1) , 2 n + 1

0 , 2 n

f = f (2n)! h 0

(2n)! g 0

(2n) + f (2n + 1)! h 1

(2n + 1)! g 1

(2n + 1)

Subband Image Coding

g 0 (n) = (−1)

h 1 (n)

g 1

(n) = (−1)

n+ h 0

(n)

for perfect reconstruction:

flp(n) =

f (2n)! h 0

(2n) , 2 n

0 , 2 n + 1

f hp

(n) =

f (2n + 1)! h 1

(2n + 1) , 2 n + 1

0 , 2 n

f = f (2n)! h 0

(2n)! g 0

(2n) + f (2n + 1)! h 1

(2n + 1)! g 1

(2n + 1)

f = f (2n)! h 0 (2n)! h 1 (2n) + f (2n + 1)! h 1 (2n + 1)! h 0 (2n + 1)

Vector Inner Product

Given sequences

f 1

(n), f 2

(n)

〈f 1

, f 2

〉 =

∑

∗

(n)f 2

(n)

Subband Image Coding

g 0

(n) = (−1)

h 1

(n)

g 1

(n) = (−1)

h 0

(n)

h 0 , h 1 , g 0 , g 1 are biorthogonal

〈h i

(2n − k), g j

(k)〉 = δ(i − j)δ(n), i, j = { 0 , 1 }

Subband Image Coding

g

(n) = (−1)

h

(n)

g

(n) = (−1)

h

(n)

h 0 , h 1 , g 0 , g 1 are biorthogonal

〈h

(2n − k), g

(k)〉 = δ(i − j)δ(n), i, j = { 0 , 1 }

Example:

〈h

(2n − k), g

(k)〉 = 0

Subband Image Coding

g

(n) = (−1)

h

(n)

g

(n) = (−1)

h

(n)

h 0 , h 1 , g 0 , g 1 are orthonormal

〈h

(2n − k), g

(k)〉 = δ(i − j)δ(n), i, j = { 0 , 1 }

〈g

(n), g

(n + 2m)〉 = δ(i − j)δ(m), i, j = { 0 , 1 }

Example: Orthonormal Filter Bank

g 1

(n) = (−1)

g 0

(K even

− 1 − n)

h

(n) = g

(K

even

− 1 − n), i = { 0 , 1 }

= g 0

(n)

Example: Orthonormal Filter Bank

g 1

(n) = (−1)

g 0

(K even

− 1 − n)

h

(n) = g

(K

even

− 1 − n), i = { 0 , 1 }

= g 0

(n)

= g 1

(n)

Example: Orthonormal Filter Bank

g 1

(n) = (−1)

g 0

(K even

− 1 − n)

h

(n) = g

(K

even

− 1 − n), i = { 0 , 1 }

= g 0

(n)

= g 1

= h (n) 0

(n)

Example: Orthonormal Filter Bank

g 1

(n) = (−1)

g 0

(K even

− 1 − n)

h

(n) = g

(K

even

− 1 − n), i = { 0 , 1 }

= g 0

(n)

= g 1

= h (n) 0

(n) = h 1

(n)

Multiresolution Image Processing: Image Pyramids and Subband Coding - Prof. S. Todorovic, Study notes of Electrical and Electronics Engineering

Related documents

Partial preview of the text

Download Multiresolution Image Processing: Image Pyramids and Subband Coding - Prof. S. Todorovic and more Study notes Electrical and Electronics Engineering in PDF only on Docsity!

ECE 468: Digital Image Processing

Lecture 9

Prof. Sinisa Todorovic

[email protected]

Multiresolution Image Processing

• Informal motivation:

• Images may show both very large and very small objects.

It may be useful to process the images at different resolutions.

An image is a 2D random process with locally varying

statistics of pixel intensities

Analysis of statistical properties of pixel neighborhoods of

varying sizes may be useful

1. Given an image at level j

2. Filter the input and and downsample the filtered result by

a factor of 2; This gives the image at level j-

3. Goto 1

4. Upsample and filter the image at level j-1; this gives an

approximation of the image at level j

5. Subtract this result from the image at level j; this give the

prediction residual at level j

6. Goto 1

g 0 (n) = (−1)

h 1 (n)

for perfect reconstruction:

g 0 (n) = (−1)

h 1 (n)

for perfect reconstruction:

g 0 (n) = (−1)

h 1 (n)

for perfect reconstruction:

g 0 (n) = (−1)

h 1 (n)

for perfect reconstruction:

Subband Image Coding

g

(n) = (−1)

h

(n)

g

(n) = (−1)

h

(n)

h 0 , h 1 , g 0 , g 1 are biorthogonal

〈h

(2n − k), g

(k)〉 = δ(i − j)δ(n), i, j = { 0 , 1 }

Example:

〈h

(2n − k), g

(k)〉 = 0

Subband Image Coding

g

(n) = (−1)

h

(n)

g

(n) = (−1)

h

(n)

h 0 , h 1 , g 0 , g 1 are orthonormal

〈h

(2n − k), g

(k)〉 = δ(i − j)δ(n), i, j = { 0 , 1 }

〈g

(n), g

(n + 2m)〉 = δ(i − j)δ(m), i, j = { 0 , 1 }

h

(n) = g

(K

− 1 − n), i = { 0 , 1 }

h

(n) = g

(K

− 1 − n), i = { 0 , 1 }

h

(n) = g