transforms.425.09.4up - Block-based Transform Coding...

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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
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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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transforms.425.09.4up - Block-based Transform Coding...

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