Biomedical Image Analysis. Homomorphic Filtering and applications to PET
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1 Biomedical Image Analysis Homomorphic Filtering and applications to PET Contents: Image enhancement and correction Brightness normalization and contrast enhancement Applications to SPECT BMIA 15 V. Roth & P. Cattin 97
2 Homomorphic filtering Homomorphic filtering is a generalized technique for image enhancement and/or correction. It simultaneously normalizes the brightness across an image and increases contrast An image can be expressed as the product of illumination and reflectance: f(x, y) = i(x, y) r(x, y) Now define g = ln f = ln i + ln r. Then: F{g(x, y)} = F{ln i(x, y)} + F{ln r(x, y)} G(u, v) = I l (u, v) + R l (u, v). BMIA 15 V. Roth & P. Cattin 98
3 Illumination and reflectance Concept is also applicable for transmission images like X-ray: replace reflectivity by transmissivity. BMIA 15 V. Roth & P. Cattin 99
4 Homomorphic filtering (2) We then apply a filter to G: S(u, v) = H(u, v)g(u, v) = H(u, v)(i l (u, v) + R l (u, v)). In the spatial domain: s(x, y) = F 1 {S(u, v)} = F 1 {H(u, v)i l (u, v)} + F 1 {H(u, v)r l (u, v)} = i (x, y) + r (x, y) BMIA 15 V. Roth & P. Cattin 100
5 Homomorphic filtering (3) We then exponentiate s(x, y) to get the enhanced image: s (x, y) = exp(s(x, y)) = exp(i (x, y)) exp(r (x, y)) = i (x, y) r (x, y) Now i (x, y) and r (x, y) are the illumination and reflectance of the enhanced image. Illumination component tends to vary slowly across the image ( low frequencies). Reflectance tends to vary rapidly, particularly at junctions of dissimilar objects ( high frequencies). BMIA 15 V. Roth & P. Cattin 101
6 Homomorphic filtering (4) Therefore, by applying a frequency domain filter of the form H(u, v) = (γ H γ L ) [ 1 exp[ c(d 2 (u, v)/d 2 0)] ] + γ L, we can reduce intensity variation across the image while highlighting detail. BMIA 15 V. Roth & P. Cattin 102
7 Homomorphic filtering (5) BMIA 15 V. Roth & P. Cattin 103
8 Homomorphic filtering: PET example Image is blurred, many low-frequency features are obscured by hot spots dominating the dynamic range. Homomorphic filtering reduces the effect of dominant illumination components and sharpens the reflectance components (edge information) by enhancing the high frequencies. BMIA 15 V. Roth & P. Cattin 104
9 Biomedical Image Analysis Image Restoration and Reconstruction Contents: Linear Systems Theory Estimating the degradation function Inverse Filtering BMIA 15 V. Roth & P. Cattin 105
10 Linear, position-invariant degradations Input-output relationship: g(x, y) = H[f(x, y)] + n(x, y). Assume for the moment that n = 0 (noise-free case) BMIA 15 V. Roth & P. Cattin 106
11 Linear, position-invariant degradations (2) Operator H is linear if H[af 1 + bf 2 ] = ah[f 1 ] + bh[f 2 ], a, b R. Two special cases: 1. a = b = 1 operator H is additive. Can be extended to integrals: H[ f(x, y, α, β) dα dβ] = H[f(x, y, α, β)] dα dβ 2. f 2 = 0 operator H is homogeneous. BMIA 15 V. Roth & P. Cattin 107
12 Linear, position-invariant degradations (3) Operator H with g(x, y) = H[f(x, y)] is position invariant if H[f(x α, y β)] = g(x α, y β). Interpretation: Response at any point in the image depends only on the value of the input at that point, not on its position. BMIA 15 V. Roth & P. Cattin 108
13 Linear, position-invariant degradations (4) Rewrite f(x, y) = f(α, β)δ(x α, y β) dα dβ. and substitute in g(x, y) = H[f(x, y)]: [ ] g(x, y) = H f(α, β)δ(x α, y β) dα dβ = H [f(α, β)δ(x α, y β)] dα dβ = f(α, β)h } [δ(x {{ α, y β)] } dα dβ. impulse response h(x,α,y,β) BMIA 15 V. Roth & P. Cattin 109
14 Linear, position-invariant degradations (5) Impulse response is also called point spread function (PSF). Name arises from the fact that all physical optical systems blur (spread) a point of light. Fundamental result of linear system theory: If response of H to an impulse is known, the response to any other input can be calculated. A linear system is completely characterized by the impulse response. BMIA 15 V. Roth & P. Cattin 110
15 Linear, position-invariant degradations (6) If H is position-invariant, H [δ(x α, y β)] = h(x α, y β) and g(x, y) = f(α, β)h(x α, y β) dα dβ = h(x, y) f(x, y). In the presence of noise, we have g(x, y) = h(x, y) f(x, y) + n(x, y) G(u, v) = H(u, v)f (u, v) + N(u, v). BMIA 15 V. Roth & P. Cattin 111
16 Estimating the degradation function Estimation by modeling: Domain specific. Examples: Physical characteristics of atmospheric turbulences [Hufnagel & Stanley, 1964]: H(u, v) = exp[ k(u 2 + v 2 ) 5/6 )] Motion blur: uniform linear motion (total distances (a, b) in (x, y)-direction) during acquisition time T : H(u, v) = T π(ua + vb) sin[π(ua + vb)] exp[ iπ(ua + vb)] BMIA 15 V. Roth & P. Cattin 112
17 Motion blur example BMIA 15 V. Roth & P. Cattin 113
18 Estimating the degradation function (2) SPECT system: total resolution dominated by resolution of collimator. Parallel circular holes of length L, radius R, distance (source,collimator front) Z, distance (collimator, optical plane) B [Metz et al.,1980]: H(u, v) ( J 1 (2παR ) 2 u 2 + v 2 ) παr, u 2 + v 2 with α = 1 + [(Z + B)/L], where J 1 is the 1st-order Bessel function of the first kind. BMIA 15 V. Roth & P. Cattin 114
19 SPECT PSF BMIA 15 V. Roth & P. Cattin 115
20 Inverse Filtering Compute estimate ˆF (u, v) as G(u,v) H(u,v). Substituting into G = HF + N ˆF = F + N H Problem: even if we know H, we cannot reconstruct F due to the noise! If H has zero or small values (often the case for high frequencies), error term N/H dominates! Possible solution: limit filter frequencies to values near the origin (e.g. by high-order low-pass filter) BMIA 15 V. Roth & P. Cattin 116
21 Atmospheric blur example BMIA 15 V. Roth & P. Cattin 117
22 Atmospheric blur example (2) BMIA 15 V. Roth & P. Cattin 118
23 Motion blur example BMIA 15 V. Roth & P. Cattin 119
24 Biomedical Image Analysis Wiener Filtering and Applications to SPECT Contents: Optimal filtering Statistics of Natural Images Geometric mean filter Applications to SPECT BMIA 15 V. Roth & P. Cattin 120
25 Wiener (or minimum squared error) filtering Recall: degradation in Fourier space: G(u, v) = H(u, v)f (u, v) }{{} Convolved image C(u,v) + N(u, v). Uncorrelated noise How should we measure the quality of the reconstruction? One possibility: averaged squared reconstruction error e 2 = f(x, y) ˆf(x, y) 2 dx dy = F (u, v) ˆF (u, v) 2 du dv. BMIA 15 V. Roth & P. Cattin 121
26 Wiener filter: problem statement Goal: find real function φ(x, y) which, when applied to measurement g and deconvolved by h, produces signal ˆf that is as close as possible to uncorrupted signal f. Substituting ˆF (u, v) = Φ(u,v)G(u,v) H(u,v) yields = F (u,v) G(u,v) {}}{{}}{ 2 C(u, v) Φ(u, v) (C(u, v) + N(u, v)) H(u, v) H(u, v) du dv [ ] H(u, v) 2 C(u, v) 2 (1 Φ(u, v)) 2 + N(u, v) 2 Φ 2 (u, v) du dv Note that C and N are assumed uncorrelated C(u, v)n(u, v) du dv = 0. BMIA 15 V. Roth & P. Cattin 122
27 Wiener filter: derivation Strategy: Minimize integrand w.r.t. Φ for all possible (u, v). Derivative w.r.t. Φ: 2 C(u, v) 2 (1 Φ(u, v)) + 2 N(u, v) 2 Φ(u, v)! = 0 Optimal function ˆΦ: ˆΦ(u, v) = C(u, v) 2 C(u, v) 2 + N(u, v) 2 BMIA 15 V. Roth & P. Cattin 123
28 Wiener filter: derivation (2) Recall: ˆF (u, v) = Φ(u, v)g(u, v) H(u, v) = [ ] Φ(u, v)) G(u, v) } H(u, {{ v) } Filter in frequency domain ˆF (u, v) = = [ C(u, v) 2 ( C(u, v) 2 + N(u, v) 2 )H(u, v) H(u, v) 2 ) ( H(u, v) 2 + N(u,v) 2 F (u,v) 2 ] G(u, v) G(u, v) H(u, v) BMIA 15 V. Roth & P. Cattin 124
29 Wiener filter: derivation (3) Further variants of the Wiener filter equation are common in the literature. For instance, using H(u, v) 2 = H (u, v)h(u, v) we derive H (u, v) ) ( H(u, v) 2 + N(u,v) 2 F (u,v) 2 G(u, v). BMIA 15 V. Roth & P. Cattin 125
30 Wiener filter: interpretation N(u,v) 2 F (u,v) 2 is inverse of signal-to-noise ratio (SNR). Where the signal is very strong relative to the noise, N(u,v) 2 F (u,v) 2 0 Φ(u, v) 1, and the Wiener filter becomes H 1 (u, v) (i.e. the inverse filter). Where the signal is very weak, N(u,v) 2 F (u,v) 2 H(u, v) 2 and F (u,v) 2 WF H (u, v) = H (u, v)snr(u, v). N(u,v) 2 (Interpretation: assume H is real LPF smoothing). Open problems: estimate SNR. BMIA 15 V. Roth & P. Cattin 126
31 Statistics of Natural Images Observation [Fields, 1987]: the power spectrum of natural images f(x, y) decays as F (u, v) 2 = F (r, φ) 2 1 r 2 = 1 u 2 + v 2 Assumption: noise is constant and spatially uncorrelated: n n = η 0 δ(x, y) In Fourier space: N(u, v) 2 = η 0 (white noise). Plausible strategy: build filter with N(u,v) 2 F (u,v) 2 = η 0 (u 2 + v 2 ) Even stronger assumption: N(u, v) 2 / F (u, v) 2 = const. BMIA 15 V. Roth & P. Cattin 127
32 Inverse vs. Wiener BMIA 15 V. Roth & P. Cattin 128
33 Geometric mean filter Idea: geometric mean of inverse- and Wiener filter: [ ] α H(u, v) ˆF (u, v) = H(u, v) 2 H(u, v) ( H(u, ) v) 2 + β N(u,v) 2 F (u,v) 2 1 α G(u, v). α = 0: parametric Wiener filter (SNR 1 weighted by β) α = 1: inverse filter α = 1/2, β = 1: spectrum equalization filter BMIA 15 V. Roth & P. Cattin 129
34 SPECT example: phantom and reconstruction BMIA 15 V. Roth & P. Cattin 130
35 SPECT example: FT of collimator PSF BMIA 15 V. Roth & P. Cattin 131
36 SPECT example: Wiener filter BMIA 15 V. Roth & P. Cattin 132
37 SPECT example: spine image BMIA 15 V. Roth & P. Cattin 133
38 SPECT example: Wiener filter (spine) BMIA 15 V. Roth & P. Cattin 134
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