An Overflow-Free, Fixed-point based Singular Value Decomposition Algorithm for Dimensionality Reduction of Hyperspectral Images Bibek Kabi, Anand S. Sahadevan, Ramanarayan Mohanty Aurobinda Routray, Bhabani. S. Das Anmol Mohanty (Me!) Indian Institute of Technology, Kharagpur HyspIRI symposium-2015 NASA Goddard Spaceflight Center
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An Overflow-Free, Fixed-point basedSingular Value Decomposition Algorithm
for
Dimensionality Reduction of Hyperspectral Images
Bibek Kabi, Anand S. Sahadevan, Ramanarayan Mohanty Aurobinda Routray, Bhabani. S. Das
Anmol Mohanty (Me!)
Indian Institute of Technology, Kharagpur
HyspIRI symposium-2015NASA Goddard Spaceflight Center
OverflowingMy Cost
Hyperspectral ImageProcessed Image
Motivation
Hardware PricePower
Consumption
Floating-Point Processor
Fixed-Point Processor
Fixed- Point ASIC
Conversion to Fixed-Point
WordlengthOptimization
HyperspectralAlgorithms
Uniform Wordlength
Optimized Wordlength
Floating-point Program
FPGA: Field Programmable Gate ArrayASIC: Application Specific Integrated Circuit
Jerez et al., 2015
Previous Works on Linear Algebra based on fixed point
*SVD – Singular Value Decomposition
Hyperspectral Images for Validation
• Hyperion (Space-borne): Hyperion imagecontains the Chilika Lake site, India.
IWLs - Integer Wordlengths SQNR - Signal-to-quantization-noise-ratio PCs - Principal Components MSE - Mean Squared Error
Proposal - Scaling Method!
Scaling
If each element of a matrix is divided by thesquare root of the product of its one-norm andinfinity-norm or Frobenius norm then all thevariables generated during the computation ofSVD will have tight analytical ranges
AAm
1=
=F
m A
Proof: Using vector and matrix norm properties, theranges of the variables can be derived. We start bybounding the elements of the input matrix as
2
ˆ ˆ| | 1max xyxy
A A
Derivation in brief
U is the left singular vector matrix, which isorthogonal and each column of U has unitynorm. Hence all elements of are in the range[-1,1] following (*).
2(:, ) (:, ) =1 (*)U i U i
Similar is the case for right singular vectors V.
PROOF
SQNR and MSE (scaling vs without scaling) with double precision floating-point as the reference