application study of known and novel...
TRANSCRIPT
Application Study of Known and Novel Transforms
Ekapol Chuangsuwanich and Peter A. Milder
Electrical and Computer EngineeringCarnegie Mellon University
Generic next neighbor (GNN)DTTs
Chebyshevpolynomials Orthogonal Properties of DTTs in image processing applications are well-studied
polynomialsWe aim to study related GNN transforms in this domain
Introduction
Overview
Background
Approach
Experimental Results
Conclusions
Generic Next Neighbor Signal Model[Pueschel and Moura, 2006]
Generalized Shift Model
Associated Orthogonal Polynomial Model
(Visualization from [Pueschel and Moura, 2006])
GNN: Relationship with DTTs
GNNs:
DTTs:
(Visualizations and shift matrices from [Pueschel and Moura, 2006])
Overview
Background
Approach
Experimental Results
Conclusions
GNNs and Associated Polynomials
Recall: DTTs associated with Chebyshev polynomials
( expressed in terms of )
Idea: examine GNN transforms associated with other well-known polynomials
PolynomialsHermite:
Laguerre:
Polynomials
Legendre:
Image Processing
256x256 image with 16x16 blockingWhy blocking?
Used in JPEG (8x8 blocking)GNN limitations
Apply a 16x16 Transform on eachblock separately
256
256
116
16 2 3 1617 18 19 32
241 242 243 256
Energy Compaction & Reconstruction
Let EL,K be the energy ofthe first L x L values of the Kth transformed blockEnergy compaction of the whole image due to first L x L values is
Reconstruction is also done by keeping the first L x L values of each block.
Kth block
Transformed image
L x L first values kept
L
L
∑
∑
=
== 256
1,16
256
1,
KK
KKL
L
E
EE
Images used : Tree, Desk, Book, and …
Overview
Background
Approach
Experimental Results
Conclusions
Pure Frequencies : 1-D DCT DCs
0 50 100 150 200 2500.04
0.05
0.06
0.07
0.08DCT1
0 50 100 150 200 2500.04
0.05
0.06
0.07
0.08DCT2
0 50 100 150 200 2500
0.02
0.04
0.06
0.08
0.1DCT3
0 50 100 150 200 2500
0.02
0.04
0.06
0.08
0.1DCT4
0 50 100 150 200 2500.04
0.05
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0.08DCT5
0 50 100 150 200 2500.04
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0.06
0.07
0.08DCT6
0 50 100 150 200 2500
0.02
0.04
0.06
0.08
0.1DCT7
0 50 100 150 200 2500
0.02
0.04
0.06
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0.1DCT8
Pure Frequencies : 1-D DST DCs
0 50 100 150 200 2500
0.02
0.04
0.06
0.08
0.1DST1
0 50 100 150 200 2500
0.02
0.04
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0.1DST2
0 50 100 150 200 2500
0.02
0.04
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0.1DST3
0 50 100 150 200 2500
0.02
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0.1DST4
0 50 100 150 200 2500
0.02
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0.1DST5
0 50 100 150 200 2500
0.02
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0.1DST6
0 50 100 150 200 2500
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0.1DST7
0 50 100 150 200 2500
0.02
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0.1DST8
Pure Frequencies : 1-D GNN DCs
Hermite Laguerre
Legendre
Pure Frequencies : 2-D
DCT2 Hermite
Pure Frequencies : 2-D
LegendreLaguerre
Energy Compaction
New transforms provide less compaction than DCT-2, but compare with DST-7
Image Reconstruction
Quality of reconstruction matches energy compaction
Blocking has no effect on DCT-2DCT-2 gains resolution as more samples are kept
Overview
Background
Approach
Experimental Results
Conclusions
Conclusion
DCT-2 is still best suited for image compressionImmune to blocking effectsBest energy compaction
But…There are much more GNNs we have not tried.Other possible reconstruction schemes are possible. For example
Kth Block
L x L last values discarded
50 100 150 200 250
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New scheme with the same number of points kept
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DST7 with old scheme
Questions?
DST7
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DCT216
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