CS/EE 5590 / ENG 401 Special Topics (17804, 17815, 17803) Lec 07 Transforms and Quantization II Zhu Li Course Web: http://l.web.umkc.edu/lizhu/teaching/2016sp.video-communication/main.html Z. Li Multimedia Communciation, 2016 Spring p.1 Outline Lecture 06 Re-Cap Scalar Quantization Vector Quantization Z. Li Multimedia Communciation, 2016 Spring p.2 Unitary Transforms y=Ax, x,y in R d , A: dxd Unitary Transforms: A is unitary if: A -1 =A T , AA T = I d The basis of A is orthogonal to each other Examples: Z. Li Multimedia Communciation, 2016 Spring p.3 y x = A= [a 1 T , a 2 T ,…, a d T ] � = � � � � � � ��� Inner product<x, a k > � < ��, �� >=0 < � � , � � >=1 cos sin sin cos 2 1 2 1 2 1 2 1 2 1 3 2 Unitary Transform Properties Preserve Energy: Preserve Angles: DoF of Unitary Transforms k-dimension projections in d-dimensional space: kd – k 2 . Above example: 3x2-2x2 = 2; normal points to the unit sphere Z. Li Multimedia Communciation, 2016 Spring p.4 � = �� � � = �� � = �� � �� = � � � � �� = � � �� = � � � � � = � � � = � � the angles between vectors are preserved unitary transform: rotate a vector in R n , i.e., rotate the basis coordinates n
11
Embed
Lec 07 Transforms and Quantization II - University of …l.web.umkc.edu/.../notes/lec07.pdfLec 07 Transforms and Quantization II Zhu Li Course Web: Z. Li Multimedia Communciation,
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
CS/EE 5590 / ENG 401 Special Topics (17804, 17815, 17803)
DCT Example: y=DCT(x), Question: Is there an optimal transform that do best in this ?
display scale: log(1+abs(g))
linear display scale: g
x: columns of image pixels
��� = ��� � = �{ � − � � � − �{�} �}
x1,x2,…, x600Rxx y1,y2,…, y600
Ryy
Karhunen-Loève Transform (KLT)
a unitary transform with the basis vectors in A being the “orthonormalized” eigenvectors of Rxx
assume real input, write AT instead of AH
denote the inverse transform matrix as A, AAT=I Rx is symmetric for real input, Hermitian for complex input
i.e. RxT=Rx, Rx
H = Rx
Rx nonnegative definite, i.e. has real non-negative eigen values
Attributions Kari Karhunen 1947, Michel Loève 1948 a.k.a Hotelling transform (Harold Hotelling, discrete formulation 1933) a.k.a. Principle Component Analysis (PCA, estimate Rx from samples)
� = ��, � = ���AT = [a1, a2, …, ad]
����� = ���� , � = 1, 2, … ,�
Decorrelation by construction:
Minimizing Error under limited coefficients reconstruction
Properties of K-L Transform
Basis restriction: Keep only a subset of m transform coefficients and then perform inverse transform (1 m N)
Keep the coefficients w.r.t. the eigenvectors of the first m largest eigenvalues (indication of energy)
�� = �{���} = � ������ = ������=
����
��
�������� = �
0, ���! = ��� , ��� = �
Energy Compaction Comparison
Transform on 2D signals
Given a m x n image block, how to compute its 2D transform ? By applying 1D DCT to the rows and then by the columns. (Separable)
DCT transform matrix is a kronecker product of 1D DCT basis function
Z. Li Multimedia Communciation, 2016 Spring p.9
u0
u7
u1
N(=8)-pt 1D DCT basis 8-pt 2D DCT basis
Matlab Exercise: SVD, PCA, and DCT approximation
In compression: DCT: not data dependent, motivated by DFT, no need to signal basis PCA: data driven, obtained from a class of signals, need to signal per
class SVD: directly approx. from the signal, need to signal per image block
o Question: can we encode basis better ?
Z. Li Multimedia Communciation, 2016 Spring p.10
DNA Sequence Compression
Seq Data in real world:
Z. Li Multimedia Communciation, 2016 Spring p.11
Quality score
Many “Reads” that are aligned, with mutations/errors: 3.2 billion x 2bit each = 800 MB 200 reads + quality (confidence) score + labeling = 1.5 TB Question: how to compress sequence (lossless) and confidence (lossy)
Z. Li Multimedia Communciation, 2016 Spring p.12
reads
FastQ and SAM
Current solutions: Reminds of zigzag and run-level coding…
Transforms Unitary transform preserves energy, angle, limited DoF KLT/PCA: energy compaction and de-correlation DCT: a good KLT/PCA approximation A bit of intro to Genome Info Compression, more to come
Scalar Quantization: If signal is uniform, what is the expected quantization error ? Non-uniform signal distribution, optimal quantization design (Lloyd-
Max)
Vector Quantization: More efficient Fast algorithm exists like kd-tree based A special case of transform: over-complete basis, very sparse
coefficient (only 1 none zero entry) Shall revisit with coupled dictionary approach in super resolution