Technische Universität München
Algorithms of Scientific Computing
The Quarter-Wave DFT and the (Quarter-Wave)
Discrete Cosine Transform (QW-DCT)
Michael Bader
Summer Term 2013
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
1
Technische Universität München
DFT and Symmetry
INPUT
TRANSFORM
real symmetry
fn ∈ R
→
Real DFT (RDFT)
even symmetry
fn = f−n
→
Discrete Cosine Transform (DCT)
odd symmetry
fn = −f−n
→
Discrete Sine Transform (DST)
“QUARTER-WAVE”
INPUT
TRANSFORM
even symmetry
fn = f−n−1
→
QW-DCT
odd symmetry
fn = −f−n−1
→
QW-DST
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
2
Technische Universität München
Motivation: Compression of Image Data (JPEG)
Compression steps of the JPEG method
1. Conversion into a suitable colour model (YCbCr, e.g.),
separation of brightness and colour information
2. Downsampling (in particular of the colour components)
3. blockwise “quarter-wave discrete cosine transform”
(blocks of size 8 × 8)
4. Quantisation of the coefficients (→ reduce information)
5. run-length encoding, Huffman/arithmetic coding
(loss-free compression of the quantified coefficients)
Example: jpeg on matlab central (see link on webpage)
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
3
Technische Universität München
Discrete Fourier Transform (DFT)
Definition:
T
For a vector of N complex numbers (f0 , . . . , fN−1 ) , the discrete
T
Fourier transform is given by the vector (F0 , . . . , FN−1 ) , where
Fk =
N−1
1 X
fn e−i2πnk /N .
N
n=0
Interpretation:
• as trigonometric interpolation/approxmiation
• as approximation of the coefficients of the Fourier series
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
4
Technische Universität München
Fourier Coefficients and Numerical Quadrature
For a 2π-periodic function f , the corresponding Fourier series is
defined as
f (x) ∼
∞
X
ikx
ck e ,
k=−∞
where
1
ck =
2π
Z2π
f (x)e−ikx dx
0
The ck are called (continuous) Fourier coefficients.
If f is piecewisely smooth, Fourier series converges pointwisely
(i.e. for each x) towards
+
−
1
2 (f (x ) + f (x )),
i.e. in particular towards f (x), if f is coninuously differentiable at x.
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
5
Technische Universität München
Computation of Fourier Coefficients ck
∞
X
Assume: f (x) given by Fourier series, then f (x) =
ck eikx
k =−∞
Multiply by e−inx and integrate:
Z2π
f (x)e
0
−inx
dx =
∞ Z2π
X
ikx −inx
ck e e
∞
X
dx =
k=−∞ 0
k=−∞
Z2π
ck
0
|
⇒ only term for k = n remains in the series, and
2π
R
ei(k−n)x dx
{z
=0, if k 6= n
}
ei(n−n)x dx = 2π
0
The Fourier coefficients ck thus need to be
1
ck =
2π
Z2π
f (x)e−ikx dx
0
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
6
Technische Universität München
Approximate Computation of ck
The continuous Fourier coefficients are given as
1
ck =
2π
Z2π
f (x)e−ikx dx
0
Steps to compute ck approximately:
• consider ck only for ±k = 0, . . . , K ; then: f (x) ≈
K
X
ck eikx
k =−K
Z2π
• compute numerical approximation of integral
f (x)e−ikx dx
0
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
7
Technische Universität München
Computation of ck via Trapezoidal Sum
Trapezoidal sum: for equidistant xn :=
Z2π
2π
g(x) dx ≈ TN {g} :=
N
2πn
N :
!
N−1
X
1
1
g(x0 ) +
g(xn ) + g(xN )
2
2
n=1
0
Use g(x) := f (x)e−ikx and fn := f (xn ), then:
1
ck ≈
TN {f (x)e−ikx }
2π
=
1
N
=
1
N
N−1
1 0 X
1
f0 e +
fn e−i2πnk/N + fN e−i2πNk/N
2
2
n=1
!
N−1
f0 X
fN
+
fn e−i2πnk/N +
2
2
!
n=1
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
8
Technische Universität München
Computation of ck via Trapezoidal Sum (2)
If f0 = fN (periodic data), we obtain
ck ≈ Fk =
N−1
1 X
fn e−i2πnk/N
N
n=0
⇒ Fk are approximations of ck
⇒ approximate computation leads to solution of the interpolation
problem
⇒ approximation error is of order O(N −2 )
For f0 6= fN , or for “discontinuities”, we get a recommendation:
Average Values at Endpoints and Discontinuities (AVED)
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
9
Technische Universität München
Computation of ck via Midpoint Rule
Midpoint rule: evaluate g(x) at midpoints xn :
Z2π
N−1
2π X
g(x) dx ≈
g(xn )
N
n=0
0
2π n +
with xn :=
N
1
2
.
With g(x) := f (x)e−ikx and fn := f (xn ), we obtain:
N−1
X
1
ek := 1
ck ≈ F
fn e−i2π(n+ 2 )k/N
N
n=0
“Quarter-Wave Discrete Fourier Transform”
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
10
Technische Universität München
Quarter-Wave Discrete Fourier Transform
• new variant of DFT:
N−1
X
1
ek := 1
F
fn e−i2π(n+ 2 )k/N
N
n=0
fn :=
N−1
X
ek ei2π(n+ 21 )k /N
F
k=0
• Comparison with coefficients Fk of the “usual” DFT:
ek eiπk/N = F
ek ω k/2
Fk = F
N
• Supporting points compared to “usual” DFT shifted by a “quarter
wave length” (midpoints of intervals).
• Derivation via midpoint rule motivates usage for piecewise
constant data
⇒ Transformation of image data
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
11
Technische Universität München
Quarter-Wave DFT on Symmetric Data
Given 2N real-valued input data f0 , . . . , f2N−1 with symmetry
f2N−n−1 = fn
Inserting the symmetric data in Quarter-Wave DFT results in
ek
F
=
2N−1
1 X
−k (n+ 21 )
fn ω2N
2N
n=0
=
=
N−1
1 X
−k (2N−n−1+ 12 )
f2N−n−1 ω2N
2N
n=0
n=0
!
N−1
N−1
X
1
πk n + 12
1
1 X
−k (n+ 2 )
−k (−n− 12 )
fn ω2N
+ ω2N
=
fn cos
.
2N
N
N
1
2N
N−1
X
−k (n+ 12 )
fn ω2N
+
n=0
n=0
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
12
Technische Universität München
Quarter-Wave DFT on Symmetric Data (2)
Quarter-Wave DFT of symmetric data results in real-valued
coefficients:
!
N−1
πk n + 21
1 X
e
Fk =
fn cos
for k = 0, . . . , 2N − 1
N
N
n=0
Additional symmetry:
e2N−k
F
=
N−1
1 X
fn cos
N
n=0
=
1
N
N−1
X
n=0
π(2N − k ) n +
N
1
2
!
πk n +
fn cos 2πn + π −
N
1
2
!
ek
= −F
⇒ again: only N independent coefficients
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
13
Technische Universität München
Quarter-Wave Even Discrete Cosine Transform
Backward transform:
fn :=
2N−1
X
ek e
F
i2π (n+ 12 )k /2N
e2N−k =−F
ek
F
−→
e0 +2
fn = F
k=0
N−1
X
ek cos
F
k=1
Definition of the quarter-wave even DCT:
!
N−1
N−1
1
X
X
n
+
πk
1
2
ek =
e0 +2
ek cos
F
fn cos
fn = F
F
N
N
n=0
k =1
πk n +
N
πk n +
N
1
2
1
2
!
!
N real values ←→ N real-valued coefficients
(no symmetry any more in data/coefficients!)
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
14
Technische Universität München
2D Cosine Transform
Definition of the 2D-DCT:
ekl
F
=
N−1 M−1
1 XX
fnm cos
N ·M
πk n +
N
n=0 m=0
fnm
= 4
N−1
X M−1
X
0
0
k =0
πk n +
N
ekl cos
F
l=0
shortened notation:
N−1
P0
k=0
xk :=
x0
2
+
N−1
P
1
2
1
2
!
cos
!
cos
πl m +
M
πl m +
M
1
2
1
2
!
!
xk
k=1
Application: blockwise 2D-DCT in JPEG/MPEG compression
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
15
Technische Universität München
Reduction of the 2D-FCT to 1D-FCTs
In the 2D cosine transform, we can rearrange:
!
!
N−1 N−1
πl m + 12
πk n + 12
1 XX
e
Fkl =
cos
fnm cos
N2
N
N
n=0 m=0
!!
!
N−1
N−1
πk n + 12
πl m + 21
1 X 1 X
cos
=
fnm cos
.
N
N
N
N
n=0
m=0
|
{z
}
b
:= Fnl
bnl are computed via N 1D transforms
• For each n, F
• we may first 1D-transform all rows and then all columns to get
the 2D-transform
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
16
Technische Universität München
Application Example: Compression of Image Data
(JPEG)
Compression steps of the JPEG method
1. Conversion into a suitable colour model (YCbCr, e.g.),
separation of brightness and colour information
2. Downsampling (in particular of the colour components)
3. blockwise “quarter-wave discrete cosine transform”
(blocks of size 8 × 8)
4. Quantisation of the coefficients (→ reduce information)
5. run-length encoding, Huffman/arithmetic coding
(loss-free compression of the quantified coefficients)
Example: jpeg on matlab central (see link on webpage)
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
17
Technische Universität München
QW-DCT – Algorithm
Reduce to Real FFT:
(1) for n = 0, . . . , N − 1:
gn = fn
g2N−n−1 = fn
(2) 2N-Real-FFT: compute Gk from gn (for k = 0, . . . , N)
(3) for k = 0, . . . , N − 1:
ek = Gk e−iπk/2N
F
Further optimisations:
• substitute real 2N-FFT by complex N-FFT
• compact (divide-and-conquer) real FFT
• compact Fast (QW-)DCT −→ paper Swarztrauber
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
18
Technische Universität München
Compact Fast DCT
Consider QW-DCT: with symmetry f2N−n−1 = fn
2N−1
X
−k (n+ 21 )
ek = 1
F
fn ω2N
2N
n=0
−→
N−1
X
ek = 1
fn cos
F
N
n=0
πk n +
N
1
2
!
.
Split into even and odd indices: gn := f2n and hn := f2n+1
(as in FFT)
• gn := f2n :
gn = f2n = f2N−2n−1 = f2(N−n)−1 = f2(N−n−1)+1 = hN−n−1
• hn := f2n+1 :
hn = f2n+1 = f2N−(2n+1)−1 = f2(N−n−1) = gN−n−1
• thus: two real DFTs with symmetric data sets
see exercises: reversed-data DFT easily obtained from DFT
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
19
Technische Universität München
Compact Fast Inverse QW-DCT
e2N−k = −F
ek
Consider backward transform: with symmetry F
fn :=
2N−1
X
ek e
F
i2π (n+ 12 )k /2N
k=0
−→
e0 +2
fn = F
N−1
X
ek cos
F
k =1
πk n +
N
1
2
!
Split into even and odd indices: (as in FFT)
e2k : again leads to Inverse QW-DCT
• Gk := F
e2k = F
e2N−2k = F
e2(N−k ) = GN−k
−Gk = −F
e2k+1 : leads to new kind of inverse DCT
• Hk := F
e2k+1 = F
e2N−(2k +1) = F
e2(N−k )−1 = F
e2(N−k−1)+1 = HN−k −1
−Hk = −F
(next even/odd split leads to two real DFTs with symm. data sets)
Michael Bader: Algorithms of Scientific Computing
Quarter-Wave DFT and (Quarter-Wave) Discrete Cosine Transform (QW-DCT), Summer Term 2013
20