Abstract
Frames have been used to capture significant signal characteristics, provide numerical stability of reconstruction, and enhance resilience to additive noise. This paper places frames in a new setting, where some of the elements are deleted. Since proper subsets of frames are sometimes themselves frames, a quantized frame expansion can be a useful representation even when some transform coefficients are lost in transmission. This yields robustness to losses in packet networks such as the Internet. With a simple model for quantization error, it is shown that a normalized frame minimizes mean-squared error if and only if it is tight. With one coefficient erased, a tight frame is again optimal among normalized frames, both in average and worst-case scenarios. For more erasures, a general analysis indicates some optimal designs. Being left with a tight frame after erasures minimizes distortion, but considering also the transmission rate and possible erasure events complicates optimizations greatly.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 203-233 |
| Number of pages | 31 |
| Journal | Applied and Computational Harmonic Analysis |
| Volume | 10 |
| Issue number | 3 |
| DOIs | |
| State | Published - May 1 2001 |
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ASJC Scopus subject areas
- Applied Mathematics
Cite this
Quantized Frame Expansions with Erasures. / Goyal, Vivek K.; Kovacevic, Jelena; Kelner, Jonathan A.
In: Applied and Computational Harmonic Analysis, Vol. 10, No. 3, 01.05.2001, p. 203-233.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Quantized Frame Expansions with Erasures
AU - Goyal, Vivek K.
AU - Kovacevic, Jelena
AU - Kelner, Jonathan A.
PY - 2001/5/1
Y1 - 2001/5/1
N2 - Frames have been used to capture significant signal characteristics, provide numerical stability of reconstruction, and enhance resilience to additive noise. This paper places frames in a new setting, where some of the elements are deleted. Since proper subsets of frames are sometimes themselves frames, a quantized frame expansion can be a useful representation even when some transform coefficients are lost in transmission. This yields robustness to losses in packet networks such as the Internet. With a simple model for quantization error, it is shown that a normalized frame minimizes mean-squared error if and only if it is tight. With one coefficient erased, a tight frame is again optimal among normalized frames, both in average and worst-case scenarios. For more erasures, a general analysis indicates some optimal designs. Being left with a tight frame after erasures minimizes distortion, but considering also the transmission rate and possible erasure events complicates optimizations greatly.
AB - Frames have been used to capture significant signal characteristics, provide numerical stability of reconstruction, and enhance resilience to additive noise. This paper places frames in a new setting, where some of the elements are deleted. Since proper subsets of frames are sometimes themselves frames, a quantized frame expansion can be a useful representation even when some transform coefficients are lost in transmission. This yields robustness to losses in packet networks such as the Internet. With a simple model for quantization error, it is shown that a normalized frame minimizes mean-squared error if and only if it is tight. With one coefficient erased, a tight frame is again optimal among normalized frames, both in average and worst-case scenarios. For more erasures, a general analysis indicates some optimal designs. Being left with a tight frame after erasures minimizes distortion, but considering also the transmission rate and possible erasure events complicates optimizations greatly.
UR - http://www.scopus.com/inward/record.url?scp=0035332731&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0035332731&partnerID=8YFLogxK
U2 - 10.1006/acha.2000.0340
DO - 10.1006/acha.2000.0340
M3 - Article
AN - SCOPUS:0035332731
VL - 10
SP - 203
EP - 233
JO - Applied and Computational Harmonic Analysis
JF - Applied and Computational Harmonic Analysis
SN - 1063-5203
IS - 3
ER -