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Hemami/ECE 425-1-11/17/09
Block-based Transform Coding
•
General properties
•
Efficient implementation
•
Transform code goodness measures
•
The discrete cosine transform
•
The Karhunen-Loeve/Hotelling transform
•
Other transforms
•
Some more about the DCT
Hemami/ECE 425-2-11/17/09
Block-based Transform Coding
Each transform coefficient is formed as the inner
product of the block
and a basis vector
:
In matrix form,
, where
is the
transform matrix,
c
&
p
are
coefficient
and
signal
vectors, respectively.
n
pn
()
block
Segment original signal into blocks of length
N
.
p
p
0
p
1
…
p
N
1
–
′
=
p
φ
c
i
φ
i
p
,
⟨⟩
=
c
Φ′
p
=
Φ
φ
0
…φ
N
1
–
=
NN
×
N
1
×
Hemami/ECE 425-3-11/17/09
Transform Coding Properties
•
is orthogonal (
).
•
Forward transform
.
•
Inverse transform
. If
, then
. If the coefficients
are lossily coded, this is not true.
•
preserves Euclidean distances (
).
Another way of saying this is the transform
conserves energy. (Useful for computing MSE.)
•
preserves the determinant of the pixel
autocorrelation matrix.
Φ
Φ′Φ
ΦΦ′
I
==
c
p
=
p
ˆ
Φ
c
ˆ
=
c
ˆ
c
=
p
ˆ
Φ
c
pI
pp
=
=
Φ
c
′
cp
′
p
=
Φ
Ec
c
′
{}
E
′Φ
Φ
′
Ep
p
′
Φ
p
′
===
Hemami/ECE 425-4-11/17/09
Extension to 2-D Transform Coding
•
Basis functions are now 2-D; formed as outer
products of 1-D basis functions.
•
basis functions & corresponding transform
coefficients.
•
Each basis function is a mini-image.
=
φ
ij
φ
i
φ
j
′
=
×
N
1
×
1
N
×
N
2

This
** preview**
has intentionally

Hemami/ECE 425-5-11/17/09
•
Apply transform as 1-D inner product: vectorize
the basis function matrix and the signal matrix.
multiplications,
additions
•
Better: apply transform independently in vertical
and horizontal directions.
multiplications;
additions.
cT
′
p
;
=
cp
,
N
2
1
×
N
2
N
2
×
T
ΦΦ
⊗
=
N
4
N
4
C
˜
Φ′
P
=
C
˜
Φ
P
,,
NN
×
CC
˜
′
P
Φ
==
2
N
3
2
N
3
Hemami/ECE 425-6-11/17/09
The concept of Energy Compaction
•
“Pack” as much energy as possible into as few
pieces of information as possible.
•
Example — the DFT. For a DC signal, all the
energy has been packed into a single coefficient.
Time domain
DC signal
Equal energy at each
time index
n
.
Frequency domain
DC signal
Non-zero energy at only
frequency index 0.
Hemami/ECE 425-7-11/17/09
Transform Code Goodness Measures
1. Transform efficiency:
is the variance of the
i
th
transform coefficient.
The higher
TE
M
is, the higher the percentage of
energy packed into the first
M
coefficients.
σ
i
2
TE
M
σ
i
2
i
0
=
M
1
–
∑
σ
i
2
i
0
=
N
1
–
∑
----------------
=
energy in first
M
coefficients
energy in all
N
coefficients
=
Hemami/ECE 425-8-11/17/09
2. Transform coding gain
If
TCG = 1
, then we haven’t gained anything from
transform coding. The higher
TCG
is, the better
our transform at decorrelation.
TCG
comes from high-rate low-distortion
quantization theory.
Note that if the transform is the identity matrix,
then
TCG = 1.

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