Local Transform

1
Local Transform

• Apart from the square template, other template
  shapes may also be used.

Vertical 1x3 template
Horizontal 3x1 template
Cross template
2
Global Transform

• Block based transformations
• Discrete Cosine Transform
• Wavelet Transform
• Linear Geometric Transform

3
Global Transform

• Ideally, a transform should be applied to an
  image to fully exploit the spatial correlation
  among pixels
• Block based transform coding
• Divides an image into non-overlapping blocks
• Apply the transform over each block
• ? Reduce computational complexity

4
Global Transforms
image
5
Global Transforms
...
Divide into non-overlapping image blocks
image
6
Global Transforms
...
Transform each block into coefficients
image
7
Global Transform

• Block based transform coding
• ? Reduce computational complexity
• ? Discontinuity at the edge of blocks

Lena 100x100 block based Transform
8
Global Transform
Input samples
Output samples
Forward transform
Inverse transform
Transform coefficients
Transform coefficients
Quantizer
Dequantizer
Quantized coefficients
Quantized coefficients
Coded bit stream
Binary encoder
Binary decoder
Block diagram for the encoding and decoding of a
block-transform coding system
9
Global Transform

• An image block can be represented as a linear
  combination of a set of basic patterns (transform
  basis functions).
• The amount of contribution from each basic
  pattern is controlled by the transform
  coefficient corresponding to that transform basis
  function.

10
Global Transform

T1,1
T1,2
T2,1
T2,2
Transform basis basic block patterns to make up
the image
11
Global Transform

• A good transform design should
• Decorrelate the signal to be quantized
• Compact the energy into as few coefficients as
  possible
• E.g. Fourier Transform, DCT, wavelet

12
Fourier Transform

• Transform an image into spatial frequency
  coefficients
• Fourier functions exist in image processing
  software like MathCAD, MathLab, etc.
• 2D FT can be separated into a 1D FT followed by
  another 1D FT on the transformed coefficients.

13
Discrete Cosine Transform

• Adjacent pixels in an image are usually
  correlated, independent representation of each
  pixel value is inefficient
• Energy compact

14
Discrete Cosine Transform

• Program codes
• View source code in Excel
• Depth of loop
• X Y Index step by blocksize
• u v
• X1 Y1 index from 0 to N-1
• Refer to cells in worksheet
• Cosine function

15
Discrete Cosine Transform
For Y 1 To maxY Step blocksizeY For X 1 To
maxX Step blocksizeX For v 0 To blocksizeY -
1 For u 0 To blocksizeX - 1 If u 0 And v 0
Then For Y1 0 To blocksizeY - 1 For X1 0
To blocksizeX - 1 DCTRed DCTRed
Worksheets("Red").Cells(Y Y1, X X1).Value
same for green blue Next X1 Next Y1
DCTRed DCTRed / (blocksizeX blocksizeY) Else
...
16
Discrete Cosine Transform
For Y1 0 To blocksizeY - 1 For X1 0 To
blocksizeX - 1 DCTRed DCTRed
Worksheets("Red").Cells(Y Y1, X X1).Value
Cos((2 X1 1) u Pi / 2 / blocksizeX)
Cos((2 Y1 1) v Pi / 2 / blocksizeY)
Next X1 Next Y1 DCTRed sqrt2 DCTRed /
(blocksizeX blocksizeY) End If Worksheets("DCTRe
d").Cells(Y v, X u).Value DCTRed Next
u Next v Next X Next Y
17
Wavelet Transform

• Wavelet analysis was proposed as an alternative
  method of representing a function (a signal or an
  image) using frequency and spatially localized
  basis function functions that are finite over a
  restricted range of frequencies and locations.
• An infinite number of functions can be defined
  that have this property. The function F is
  usually called the Mother Wavelet and it is given
  as
• the index l defines the wavelets location and s
  its scale.

18
Mother Wavelet Transform

• The data domain is spanned at varying resolutions
  by using the mother wavelet in a scaling equation
• Four other conditions must be satisfied before
  the coefficient ck can be qualified as wavelet
  coefficients
• Even number coefficients must sum to one.
• Odd number coefficients must sum to one.
• The sum of the products of adjacent odd or even
  coefficients must be zero.
• The sum of the products of a coefficient and its
  complex conjugate must be two.

19
Haar Wavelet Transform

• Haar Wavelet Transform is given as

20
Haar Wavelet Transform

• f(x) and ?(x) are defined as non-zero in the
  range 0,1 and zero elsewhere.

?(x)
1
x
1
-1
21
Geometric Transform

• Geometric transformations
• An arbitrary geometric transformation moves a
  pixel at (x, y) to (x, y).
• Thus,

22
Geometric Transform

• For linear transformations, the two coordinates
  are related using linear polynomial

• In matrix terms, we have

23
Geometric Transform
Translation of an image pixel by (x0, y0)
is Thus, we have a0 b11, a1b00, a2x0 and
b2y0.
y
yy0
p
x
p
xx0
Original image
New image
24
Geometric Transform
To scale an image by s, we have Thus, we have
a0 b1 s, a1 b0 a2 b20.
25
Geometric Transform
To rotate an image by ?, we let the pixels
position at a distance of L and angle a from the
origin. We have
a
y
y
L
?
?
x
x
p
p
p
Original image
New image
26
Geometric Transform
The new position (x, y) of the pixel is
thus Expand both sin(a?) and cos(a?) and
substitute x and y into the equations, we
get Thus, we have a0 b1 cos?, a1-sin?, b0
sin?, a2 b20.
27
Geometric Transform
Using appropriate values for the six parameters,
the various linear transformations of
translation, rotation, scaling, and shearing can
be realised.
28
Geometric Transforms

• These linear transformations can be done one
  after another.
• We may compute a compound matrix by multiplying
  the individual matrices, and simply transform the
  image once.
• Non-integral coordinates may be found from the
  calculations. The pixels of the new image are
  found as mapping from interpolation between
  pixels in the original image.

29
Geometric Transforms

• Example
• Find the new coordinate of the pixel at (14, 20)
  if we translate it by (2, 4), rotate the image by
  p/6, and scale the image by 1.5.
• The translation moves (14, 20) to (142, 204)
  (16,24).
• The rotation moves the pixel to
  (16cos(p/6)-24sin(p/6),16sin(p/6)24cos(p/6))
  (1.856, 28.78)
• The scaling moves the pixel to (1.51.856,
  1.528.78) (2.784, 43.17)

30
Geometric Transforms

• Exercise
• Find the new coordinate of the pixel at (30, 20)
  if we translate it by (-5, 5), rotate the image
  by -p/6, and vertically shear the image by 0.8.
• A line with two end points at (30, 20) and (20,
  30) is translated by (40, 60) and scaled by 1.2.
  Find the new position of the two end points of
  the line.

31
Geometric Transforms

• Exercise solution
• Find the new coordinate of the pixel at (30, 20)
  if we translate it by (-5, 5), rotate the image
  by -p/6, and vertically shear the image by 0.8.
• The translation moves (30, 20) to (30-5,205)
  (25, 25).
• The rotation moves the pixel to
  (25cos(-p/6)-25sin(-p/6), 25sin(-p/6)25cos(-p/6))
  (21.6512.5, -12.521.65) (34.15, 9.15)
• The scaling moves the pixel to (34.150.8,
  9.150.8) (27.32, 7.32).

32
Geometric Transforms

• Exercise solution
• A line with two end points at (30, 20) and (20,
  30) is translated by (40, 60) and scaled by 1.2.
  Find the new position of the two end points of
  the line.
• The translation moves the end points of the line
  to (70, 80) and (60, 90).
• The scaling moves the end points to (84, 96) and
  (72, 108).