Technische Universit¨ at M¨ unchen Algorithms of Scientific Computing Discrete Cosine Transform (DCT) Michael Bader Summer Term 2012 Michael Bader: Algorithms of Scientific Computing Discrete Cosine Transform (DCT), Summer Term 2012 1
Technische Universit¨ at M¨ unchen DFT and Symmetry INPUT TRANSFORM real symmetry f n ∈ R → Real DFT (RDFT) even symmetry f n = f − n → Discrete Cosine Transform (DCT) odd symmetry f n = − f − n → Discrete Sine Transform (DST) “QUARTER-WAVE” INPUT TRANSFORM even symmetry f n = f − n − 1 → QW-DCT odd symmetry f n = − f − n − 1 → QW-DST Michael Bader: Algorithms of Scientific Computing Discrete Cosine Transform (DCT), Summer Term 2012 2
Technische Universit¨ at M¨ unchen 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. Quantification 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 Discrete Cosine Transform (DCT), Summer Term 2012 3
Technische Universit¨ at M¨ unchen Discrete Fourier Transform (DFT) Definition: For a vector of N complex numbers ( f 0 , . . . , f N − 1 ) , the discrete Fourier transform is given by the vector ( F 0 , . . . , F N − 1 ) , where N − 1 � F k = 1 f n e − i 2 π nk / N . N n = 0 Interpretation: • as trigonometric interpolation/approxmiation • as approximation of the coefficients of the Fourier series Michael Bader: Algorithms of Scientific Computing Discrete Cosine Transform (DCT), Summer Term 2012 4
Technische Universit¨ at M¨ unchen Fourier Coefficients and Numerical Quadrature For a 2 π -periodic function f , the corresponding Fourier series is defined as � 2 π � ∞ c k = 1 f ( x ) e − ikx dx c k e ikx , f ( x ) ∼ mit 2 π k = −∞ 0 The c k 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 Discrete Cosine Transform (DCT), Summer Term 2012 5
Technische Universit¨ at M¨ unchen Approximate Computation of c k The continuous Fourier coefficients are given as � 2 π c k = 1 f ( x ) e − ikx dx 2 π 0 Approaches to compute c k approximately: K � c k e ikx • compute c k only for ± k = 0 , . . . , K ; then: f ( x ) ≈ k = − K � 2 π f ( x ) e − ikx dx numerically • compute integral 0 Michael Bader: Algorithms of Scientific Computing Discrete Cosine Transform (DCT), Summer Term 2012 6
Technische Universit¨ at M¨ unchen Computation of c k via Trapezoidal Sum Trapezoidal sum : for equidistant x n := 2 π n N : � � � 2 π N − 1 � g ( x ) dx ≈ T N { g } := 2 π 1 g ( x n ) + 1 2 g ( x 0 ) + 2 g ( x N ) N n = 1 0 Use g ( x ) := f ( x ) e − ikx and f n := f ( x n ) ; hence: � � N − 1 � c k ≈ 1 1 1 f n e − i 2 π nk / N + 1 2 f 0 e 0 + 2 π T N { f ( x ) e − ikx } 2 f N e − i 2 π Nk / N = N n = 1 � � N − 1 � 1 f 0 f n e − i 2 π nk / N + f N = 2 + N 2 n = 1 Michael Bader: Algorithms of Scientific Computing Discrete Cosine Transform (DCT), Summer Term 2012 7
Technische Universit¨ at M¨ unchen Computation of c k via Trapezoidal Sum (2) If f 0 = f N (periodic data), we obtain N − 1 � c k ≈ F k = 1 f n e − i 2 π nk / N N n = 0 ⇒ F k are approximations of c k ⇒ approximate computation leads to solution of the interpolation problem ⇒ approximation error is of order O ( N − 2 ) For f 0 � = f N , or for “discontinuities”, we get a recommendation: Average Values at Endpoints and Discontinuities (AVED) Michael Bader: Algorithms of Scientific Computing Discrete Cosine Transform (DCT), Summer Term 2012 8
Technische Universit¨ at M¨ unchen Computation of c k via Midpoint Rule Midpoint rule: evaluate g ( x ) at midpoints x n : � � � 2 π N − 1 � n + 1 x n := 2 π g ( x ) dx ≈ 2 π 2 g ( x n ) with . N N n = 0 0 With g ( x ) := f ( x ) e − ikx and f n := f ( x n ) , we obtain: N − 1 � F k := 1 f n e − i 2 π ( n + 1 2 ) k / N c k ≈ � N n = 0 “Quarter-Wave Discrete Fourier Transform” Michael Bader: Algorithms of Scientific Computing Discrete Cosine Transform (DCT), Summer Term 2012 9
Technische Universit¨ at M¨ unchen Quarter-Wave Discrete Fourier Transform • new variant of DFT: N − 1 N − 1 � � F k := 1 � f n e − i 2 π ( n + 1 2 ) k / N F k e i 2 π ( n + 1 � 2 ) k / N f n := N n = 0 k = 0 • Comparison with coefficients F k of the “usual” DFT: F k e i π k / N = � F k = � F k ω k / 2 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 Discrete Cosine Transform (DCT), Summer Term 2012 10
Technische Universit¨ at M¨ unchen Quarter-Wave DFT on Symmetric Data Given 2 N real-valued input data f 0 , . . . , f 2 N − 1 with symmetry f 2 N − n − 1 = f n Inserting the symmetric data in Quarter-Wave DFT results in 2 N − 1 � 1 − k ( n + 1 2 ) � F k = f n ω 2 N 2 N n = 0 N − 1 � N − 1 � 1 + 1 − k ( n + 1 2 ) − k ( 2 N − n − 1 + 1 2 ) = f n ω f 2 N − n − 1 ω 2 N 2 N 2 N 2 N n = 0 n = 0 � � � � � � N − 1 N − 1 � � n + 1 π k 1 = 1 − k ( n + 1 2 ) − k ( − n − 1 2 ) 2 = + ω f n f n cos ω . 2 N 2 N 2 N N N n = 0 n = 0 Michael Bader: Algorithms of Scientific Computing Discrete Cosine Transform (DCT), Summer Term 2012 11
Technische Universit¨ at M¨ unchen Quarter-Wave DFT on Symmetric Data (2) Quarter-Wave DFT of symmetric data results in real-valued coefficients: � � � � N − 1 � n + 1 π k F k = 1 � 2 for k = 0 , . . . , 2 N − 1 f n cos N N n = 0 Additional symmetry: � � � � N − 1 � n + 1 π ( 2 N − k ) 1 � 2 F 2 N − k = f n cos N N n = 0 � � � � N − 1 � n + 1 1 2 π n + π − π k = − � 2 = f n cos F k N N n = 0 ⇒ again: only N independent coefficients Michael Bader: Algorithms of Scientific Computing Discrete Cosine Transform (DCT), Summer Term 2012 12
Technische Universit¨ at M¨ unchen Quarter-Wave Even Discrete Cosine Transform backward transform: � � � � 2 N − 1 N − 1 � � n + 1 π k F 2 N − k = − � � F k F k e i 2 π ( n + 1 � 2 ) k / 2 N f n = � � 2 f n := − → F 0 + 2 F k cos N k = 0 k = 1 Definition of the quarter-wave even DCT: � � � � � � � � N − 1 N − 1 � n + 1 � n + 1 π k π k F k = 1 � f n = � � 2 2 F 0 + 2 f n cos F k cos N N N n = 0 k = 1 N real-valued data ← → N real-valued coefficients (no symmetry any more in data/coefficients!) Michael Bader: Algorithms of Scientific Computing Discrete Cosine Transform (DCT), Summer Term 2012 13
Technische Universit¨ at M¨ unchen 2D Cosine Transform Definition of the 2D-DCT: � � � � � � � � N − 1 M − 1 � � n + 1 m + 1 π k π l 1 � 2 2 F kl = f nm cos cos N · M N M n = 0 m = 0 � � � � � � � � N − 1 � M − 1 � n + 1 m + 1 π k π l ′ � 2 2 ′ f nm = 4 F kl cos cos N M k = 0 l = 0 N − 1 � N − 1 � ′ x k := x 0 shortened notation: 2 + x k k = 0 k = 1 Application: blockwise 2D-DCT in JPEG/MPEG compression Michael Bader: Algorithms of Scientific Computing Discrete Cosine Transform (DCT), Summer Term 2012 14
Technische Universit¨ at M¨ unchen Reduction of the 2D-FCT to 1D-FCTs In the 2D cosine transform, we can rearrange: � � � � � � � � N − 1 N − 1 � � n + 1 m + 1 π k π l 1 � 2 2 F kl = f nm cos cos N 2 N N n = 0 m = 0 � � �� � � � � � � N − 1 N − 1 � � m + 1 n + 1 1 1 π l π k 2 2 = f nm cos cos . N N N N n = 0 m = 0 � �� � := � F nl • For each n , � F nl are computed via N 1D transforms • we may first 1D-transform all rows and then all columns to get the 2D-transform Michael Bader: Algorithms of Scientific Computing Discrete Cosine Transform (DCT), Summer Term 2012 15
Technische Universit¨ at M¨ unchen 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. Quantification 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 Discrete Cosine Transform (DCT), Summer Term 2012 16
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