Improved optimization of soft-partition-weighted-sum filters and their application to image restoration

Yong Lin; Russell C. Hardie; Qin Sheng; Min Shao; Kenneth E. Barner

doi:10.1364/AO.45.002697

Applied Optics
Vol. 45,
Issue 12,
pp. 2697-2706
(2006)
•https://doi.org/10.1364/AO.45.002697

Improved optimization of soft-partition-weighted-sum filters and their application to image restoration

Yong Lin, Russell C. Hardie, Qin Sheng, Min Shao, and Kenneth E. Barner

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Yong Lin, Russell C. Hardie, Qin Sheng, Min Shao, and Kenneth E. Barner, "Improved optimization of soft-partition-weighted-sum filters and their application to image restoration," Appl. Opt. 45, 2697-2706 (2006)

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Abstract

Soft-partition-weighted-sum (Soft-PWS) filters are a class of spatially adaptive moving-window filters for signal and image restoration. Their performance is shown to be promising. However, optimization of the Soft-PWS filters has received only limited attention. Earlier work focused on a stochastic-gradient method that is computationally prohibitive in many applications. We describe a novel radial basis function interpretation of the Soft-PWS filters and present an efficient optimization procedure. We apply the filters to the problem of noise reduction. The experimental results show that the Soft-PWS filter outperforms the standard partition-weighted-sum filter and the Wiener filter.

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Step 1	Use the LBG algorithm to initialize codebook c.
	Use the sample variance of observation vectors in each partition to initialize s.
Step 2	Perform optimization with respect to w with c and s fixed.
Step 3	Perform the optimization with respect to c with w and s fixed.
Step 4	Perform the optimization with respect to s with w and c fixed.
Step 5	Go to Step 2 if the ending condition is not met.

Step 1	Set initial values of c ⁽⁰⁾.
Step 2	Set initial value of $H^{(0)} = I . H^{(k)}$ is a symmetric positive definite matrix and is used to approximate the second derivative function. $H^{(k)}$ is updated by iterations.

Step 3	Determine the direction of descent $δ^{(k)}$ by $δ^{(k)} = - H^{(k)} \nabla_{c} J [w, c^{(k)}, s]$ at the point $c^{(k)}$ .
Step 4	Determine the best step size α by using a line search method to minimize $J (w, c^{(k)} + {α δ}^{(k)}, s)$ .
Step 5	Set $c^{(k + 1)} = c^{(k)} + {α δ}^{(k)}$ .
Step 6	Update $θ^{(k)}$ and $γ^{(k)} .$ $θ^{(k)}$ and $γ^{(k)}$ are used for updating $H^{(k + 1)} . θ^{(k)} = α^{k} δ^{(k)} = c^{(k + 1)} - c^{(k)}$ , and $γ^{(k)} = ▿_{c} J (w, c^{(k + 1)}, s) - ▿_{c} J (w, c^{(k)} s)$ . If $∥ γ^{(k)} ∥ \leq ϵ,$ then stop.
Step 7	Update $H^{(k + 1)}$ by using $H^{(k + 1)} = H^{(k)} + \frac{(θ^{(k)} - H^{(k)} γ^{(k)}) (θ^{(k)} - H^{(k)} γ^{(k)})^{T}}{(θ^{(k)} - H^{(k)} γ^{(k)})^{T} γ^{(k)}}$ .

Step 1	Set initial values of c ⁽⁰⁾.
Step 2	Determine the direction of descent $δ^{(k)}$ by calculating the negative gradient $δ^{(k)} = - \nabla_{c} J (w, c^{(k)}, s)$ at point $c^{(k)}$ .
Step 3	Find the best step size α by using a line search algorithm to minimize the cost function $J ({w, c}^{(k)}, δ^{(k)}, S)$ .
Step 4	Set $c^{(k + 1)} = c^{(k)} + {α δ}^{(k)}$ . If $‖ J^{(k)} - J^{(k + 1)} ‖ \leq ε$ , then stop. Otherwise go to Step 2.

Method	MSE	Improvement over Wiener (%)	Improvement over PWS (%)
Wiener filter	234.72	0.0	−22.9
PWS	190.97	18.6	0.0
Soft-PWS (updated w only)	166.44	29.1	12.8
Soft-PWS (updated w and c only)	164.32	30.0	14.0
Soft-PWS (updated w, c, and s)	163.90	30.2	14.2

Method	MSE	Improvement over Wiener (%)	Improvement over PWS (%)
Wiener filter	326.98	0.0	−28.5
PWS	254.36	22.2	0.0
Soft-PWS (updated w only)	223.08	31.8	12.3
Soft-PWS (updated w and c only)	219.34	32.9	13.8
Soft-PWS (updated w, c, and s)	218.95	33.0	13.9
Soft-PWS (updated w one more time after updated w, c, and s)	221.05	32.4	13.1

Method	MSE	Improvement over Wiener (%)	Improvement over PWS (%)
Wiener filter	412.86	0.0	−25.6
PWS	328.65	20.4	0.0
Soft-PWS (updated w only)	271.95	34.1	17.3
Soft-PWS (updated w and c only)	267.92	35.1	18.5
Soft-PWS (updated w, c, and s)	267.83	35.1	18.5
Soft-PWS (updated w one more time after updated w, c, and s)	275.62	33.2	16.1

Method	MSE	Improvement over Wiener (%)	Improvement over PWS (%)
Wiener filter	220.90	0.0	−8.9
PWS	202.79	8.2	0.0
Soft-PWS (updated w only)	193.83	12.3	4.4
Soft-PWS (updated w and c only)	192.74	12.7	5.0
Soft-PWS (updated w, c, and s)	192.69	12.8	5.0

Method	MSE	Training Time
PWS	251.82	6.62 min
Previous Soft-PWS method	242.90	99 h
Compound solution Soft-PWS	236.83	8.14 min

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Applied Optics