Optimal Quantization (Lloyd-Max) ti ti 2
Throwing out DFT coefficients
1
1
si
si
Original
2:1
3:1
4:1
s t i p s ds 0
si
Transform Based Compression
Minimize mean square quantization error: E s q s Consider only even PDF of data, p(s) = p(-s) Leads to odd quantizer function, q(s) = -q(-s) si – decision levels
2
ti – reconstruction levels Optimum decision levels lie halfway between the optimum reconstruction levels, which, in turn, lie at the centroid of the PDF in between the decision levels. 1
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Image Transforms
Image Transforms g(x,y,u,v) form an orthonormal basis:
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g2 x , u
g1 x , u
Symmetric transform if:
x , y ,u ,v
h
g1 x , u g 2 y , v
g x , y ,u ,v
g x , y ,u ,v
Unitary transform if:
if u u ' and v v ' otherwise
T x , y h x , y ,u ,v
u 0 v 0
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1 0
Separable transform if:
N 1 N 1
x 0 y 0
General reverse transform:
*
g x , y , u , v g x , y , u' , v '
x 0 y 0
f u ,v
f x , y g x , y ,u,v
T u ,v
N 1 N 1
N 1 N 1
General forward transform:
4
Discrete Fourier Transform (DFT)
Walsh-Hadamard Transform (WHT)
DFT:
Sequency
0 1
1 ux vy WN WN 2 N
W
vy N
2
h x , y ,u ,v
W
g x , y ,u ,v
ux N
Unitary DFT:
3
1 ux vy W WN N N
5
1 ux vy W WN N N
h x , y ,u ,v
4
g x , y ,u ,v
6 5
7
cos
$ %
2y 1 v 2N
2x 1 u 2N #
v cos
!
u
!
h x , y ,u ,v
for u 0 N
1
,
for u 1
+
Summation in modulo-2 arithmetic bk(z) is kth bit of z = (b7b6b5b4b3b2b1b0)
*
(
u
1 N 2 N
m
&'
*)
g x , y ,u ,v
#
"
bi y pi v
bi x pi u
i 0
m 1
1
2
N
1 N
h x , y ,u ,v
Discrete Cosine Transform (DCT) "
Walsh-Hadamard Transform (WHT) g x , y ,u ,v
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$ %
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WHT and DCT are both fast transforms – FFT-like algorithms exist. DCT is not the real part of the DFT
pk(z):
convert z to Gray code reverse bits of the Gray code pk(z) is the kth bit of the result
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0
Basis Images and Energy Compaction
-
Basis Images (
1
T u , v H uv 2
5
(
/ ,
u 0 v 0
4
8
1 N
8
N
F
0 1
if T u , v is neglected otherwise
(
1, u , v
5
4
9
10
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@
@
@
2 5
log2 MN log2 mn
MN mn
log2 512
2
log2 16
2
9 4
5
@
mnlog2 mn
MN log2 MN
4
The Karhunen-Loève transform (KLT) has optimal energy compaction properties – KLT depends upon the image data. DCT usually is the best practical transform in terms of energy compaction and minimization of blocking artifacts.
Image is subdivided into typical block sizes of 8x8 or 16x16 before transform encoding. Provides (some) gain in computational efficiency:
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Different transforms achieve a different distribution of coefficient variance, σ2T(u,v) , over transform space (u,v).
@
@
Sub Image Size
@
Energy Compaction
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1 T u,v
44
2 T u,v
u 0 v 0
2
1
3
1 N
2
N
ems
@
> 3
3
12
u u0 , v v 0
? 8
v0
8
0
2
4
1
Trans
5
Hu
F
3
F
5
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