Abstract

In this paper equations for the output mean and variance of the intensity-dependent spread (IDS) filter and its reconstruction are derived for images corrupted by white Gaussian and white Poisson noise. These equations are then used to compute the signal-to-noise ratio (SNR) in both the filtered image and the reconstructed image. We show that for additive Gaussian noise the SNR in the reconstructed image varies as the square root of the mean intensity and for additive Poisson noise the SNR is constant. We compare the SNR in the reconstructed image with that of the linear-shift-invariant Gaussian filter, and we show that the IDS filter can achieve a specified minimum SNR over the image with substantially less blur than a Gaussian. We also derive a formula to trade off blur for SNR by adjusting the spread factor of the IDS kernel.

© 1998 Optical Society of America

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References

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  1. T. N. Cornsweet, J. I. Yellot, “Intensity-dependent spatial summation,” J. Opt. Soc. Am. A 2, 1769–1786 (1985).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  4. M. Vaezi, B. Bavarian, G. Healey, “Image reconstruction of IDS filter response,” in Visual Communications and Image Processing, ’91: Visual Communication, T. Koga, K. Tzou, eds., Proc. SPIE1606, 803–809 (1991).
    [CrossRef]
  5. S. Najand, D. Blough, G. Healey, “Forward and inverse model for the intensity-dependent spread filter,” J. Opt. Soc. Am. A 13, 1305–1314 (1996).
    [CrossRef]

1996 (1)

1990 (1)

1987 (1)

1985 (1)

Alter-Gartenberg, R.

Bavarian, B.

M. Vaezi, B. Bavarian, G. Healey, “Image reconstruction of IDS filter response,” in Visual Communications and Image Processing, ’91: Visual Communication, T. Koga, K. Tzou, eds., Proc. SPIE1606, 803–809 (1991).
[CrossRef]

Blough, D.

Cornsweet, T. N.

Healey, G.

S. Najand, D. Blough, G. Healey, “Forward and inverse model for the intensity-dependent spread filter,” J. Opt. Soc. Am. A 13, 1305–1314 (1996).
[CrossRef]

M. Vaezi, B. Bavarian, G. Healey, “Image reconstruction of IDS filter response,” in Visual Communications and Image Processing, ’91: Visual Communication, T. Koga, K. Tzou, eds., Proc. SPIE1606, 803–809 (1991).
[CrossRef]

Huck, F. O.

Najand, S.

Narayanswamy, R.

Vaezi, M.

M. Vaezi, B. Bavarian, G. Healey, “Image reconstruction of IDS filter response,” in Visual Communications and Image Processing, ’91: Visual Communication, T. Koga, K. Tzou, eds., Proc. SPIE1606, 803–809 (1991).
[CrossRef]

Yellot, J. I.

Yellott, J.

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Figures (13)

Fig. 1
Fig. 1

Forward model for the IDS filter.

Fig. 2
Fig. 2

Inverse model for the IDS filter.

Fig. 3
Fig. 3

Noise filtering model for the IDS reconstruction.

Fig. 4
Fig. 4

(a) SNR of the IDS filter (curve) and Gaussian filter (straight lines with I0=50, 150, 250 from top down) for additive white Gaussian noise, (b) Standard deviation of the IDS kernel (curve) and Gaussian kernel (straight lines with I0=50, 150, 250 from top down).

Fig. 5
Fig. 5

Experimental and theoretical SNR in the IDS reconstruction of constant image with additive Gaussian noise. The SNR was generated for various image intensities and various amounts of noise. The four sets of curves correspond to noise standard deviation of 4, 6, 8, and 10 from top to bottom. The experimental and theoretical curves match to within 7%.

Fig. 6
Fig. 6

(a) Multiple constant patch image; (b) with added white Gaussian noise of σn=10; (c) IDS reconstruction of (b).

Fig. 7
Fig. 7

Gaussian response of 6b with (a) I0=50, (b) I0=150, (c) I0=250.

Fig. 8
Fig. 8

(a) Original image of Lena, (b) with added white Gaussian noise of σn=10; (c) IDS reconstruction of (b).

Fig. 9
Fig. 9

Gaussian response of Fig. 8(b) with (a) I0=50, (b) I0=150, (c) I0=250.

Fig. 10
Fig. 10

(a) Original image of the noise with σn=10 amplified by a factor of 3.5, (b) noise in IDS reconstruction of the multiple constant patch image, (c) noise in IDS reconstruction of Lena. K=20. For display the reconstructed noise is amplified by a factor of 15 and the mean is changed from 0 to 128.

Fig. 11
Fig. 11

Gaussian response to noise with (a) I0=50, (b) I0=150, (c) I0=250. σn=10, K=20, the output noise is amplified by a factor of 15, and the mean is changed from 0 to 128.

Fig. 12
Fig. 12

Noise in the IDS reconstruction of Lena for the intensity ranges (a) 56–96, (b) 97–137, (c) 138–178, (d) 179–219. The noise is scaled up by a factor of 15 and has a mean of 128.

Fig. 13
Fig. 13

Calculated SNR in the IDS reconstruction of Lena and the theoretical SNR curve.

Tables (1)

Tables Icon

Table 1 Theoretical and Experimental SNR Values for the Gaussian and IDS Responses to the Multipatch Image with Additive White Gaussian Noise.a

Equations (77)

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σo2=σn212π202π0exp-k2ρ22I02ρdρdϕ
=σn22π0 exp-k2ρ2I0ρdρ=σn2I04πk2.
Ro=2kIπ/σnI0.
Ro=2kπI/I0.
μo=1-exp(-I),
σo2=[1-exp(-2I)]/8πk2.
Ro=μoσo=2k2π 1-exp(-I)1-exp(-2I)
2k2π.
o(r, θ)-1=2i=1gi(r)fi(r, θ),
μo=E[o(r, θ)]=E[o(0, 0)]=E02π0hi(r)rdrdθ
=i=1p(i)02π0 i2πk2exp-ir22k2rdrdθ
=i=1p(i)=1-i=-0p(i)1--0p(i)di=1-N-Iσn,
σo2=Var[o(r, θ)]=Var[o(0, 0)]=Var02π0hi(r)rdrdθ.
σo2=02π0Var[hi(r)]rdrdθ
=02π0{E[hi2(r)]-E2[hi(r)]}rdrdθ.
A=02π0i=1i=hi2(r)p(i)rdrdθ
=12πk4i=1i=i2p(i)0 exp-ir2k2rdr
=I4πk21--0ip(i)I4πk2.
B=02π0i=1hi(r)p(i)2rdrdθ
=12πk40i=1j=1ij exp-(i+j)r22k2p(i)p(j)rdr
=12πk4i=1j=1ijp(i)p(j)0 exp-(i+j)r22k2rdr
=12πk2i=1j=1 iji+jp(i)p(j).
B12πk2i=1ip(i)j=1j1i+I-j-I(i+I)2+(j-I)2(i+I)3p(j)
=12πk2i=1Ii+I-σn2(i+I)2+σn2I(i+I)3ip(i)
12πk2i=112-i-I4I+(i-I)28I2-σn214I2-i-I4I3+3(i-I)216I4+σn2I18I3-3(i-I)16I4+3(i-I)216I5ip(i)
=12πk2I2-σn24I+σn416I3=I4πk2-σn28πk2I1-σn24I2.
σo2=A-B=σn28πk2I1-σn24I2σn28πk2I.
Ro=μoσo2k2πIσn.
μf=E[fb(r, θ)]=02π0i=1 k22igi(r)p(i)rdrdθ
=k22i=1 1ip(i)02π0gi(r)rdrdθ
=k22i=1 1ip(i)
k22i=11I-i-II2+(i-I)2I3p(i)
=k22I1+σn2I2=k22I1+1Ri2,
σf2=Var[fb(r, θ)]
=02π0i=1k22igi(r)2p(i)rdrdθ-02π0i=1 k22igi(r)p(i)2rdrdθ
σn2k216πI31-32Ri2.
ib=k22 fbk22μf-k22μf2(fb-μf)=-k22μf2(fb-2μf).
μo=E{ib}=i=1ibp(i)=-k22μf2i=1(fb-2μf)p(i)=k22μf
=I1+1Ri2,
σo2=Var{ib}=i=1(ib-μo)2p(i)=i=1-k2fb2μf2+k22μf2p(i)
=k22μf22i=1(fb-μf)2p(i)=k22μf22σf2
=σn2I4πk2(1-3/2Ri2).
Ro=μoσo=2kπI(1-3/2Ri2)σn(1+1/Ri2).
Ro2kπI/σn.
μok22E[fb(r, θ)]=I1+1/I,
σo22I2k22 Var[fb(r, θ)]=I24πk2,
Ro2kπ.
kmin=σnRmin/2πI0.
kmax=σmaxI0,
σnRmin/2πI0kσmaxI0.
σnRmin/2πI0=k=σmaxI0.
σmax=σnRmin/2I0π.
Rmin/2πkσmaxI0.
σmax=Rmin/2πI0.
σf2=Var[fb(r, θ)]
=02π0i=1k22igi(r)2p(i)rdrdθ-02π0i=1 k22igi(r)p(i)2rdrdθ.
A=k4402π0i=11igi(r)2p(i)rdrdθ.
A=k48π0i=11iGi(ρ)2p(i)ρdρ
=k48πi=1 1i2p(i)0Gi2(ρ)ρdρ
=k48πi=1 1i2p(i)0exp-k2ρ22i-1-k2ρ22i2ρdρ
=k48πi=1 1ip(i)0exp-k2ρ22-1-k2ρ222ρdρ
=k48πi=1 1ip(i)0G12(ρ)ρdρ.
B=k4402π0i=1 1igi(r)p(i)2rdrdθ
=k48π0i=1 1iGi(ρ)p(i)2ρdρ
=k48π0i=1j=1 1ijGi(ρ)Gj(ρ)p(i)p(j)ρdρ
=k48πi=1j=1 1ijp(i)p(j)0Gi(ρ)Gj(ρ)ρdρ
=k48πi=1j=1 1ijp(i)p(j)0exp-k2ρ22i-1-k2ρ22i  ×exp-k2ρ22j-1-k2ρ22jρdρ.
B=k48πi=1j=1p(i)p(j)0 exp-k2ρ241i+1j-12-exp-k2ρ24i-exp-k2ρ24j2-k2ρ222ρdρ
=k48πi=1 1ip(i)0G12(ρ)ρdρ-k48πi=1j=1p(i)p(j)  ×0exp-k2ρ24i-exp-k2ρ24j-k2ρ222ρdρ.
σf2=A-B
=k48πi=1j=1p(i)p(j)  ×0exp-k2ρ24i-exp-k2ρ24j-k2ρ222ρdρ
=k28πi=1j=1p(i)p(j)  ×0 exp-ρi1-exp-ρ21j-1i-ρ2dρ.
σf2k232πi=1j=1p(i)p(j)1j-1i2  ×0exp-ρi1-ρ21j-1idρ
=k264πi=1j=1p(i)p(j)5ij2-4j-i2j3
=k264πj=1p(j)5Ij2-4j-I2+σn2j3.
σf2k264πj=1p(j)5I1I2-2(j-I)I3+3(j-I)2I4-41I-j-II2+(j-I)2I3-(I2+σn2)  ×1I3-3(j-I)I4+6(j-I)2I5
=k264π4σn2I3-6σn4I5=σn2k216πI31-3σn22I2=σn2k216πI31-32Ri2,

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