The EM Method in a Probabilistic Wavelet-Based MRI Denoising

Abstract

Human body heat emission and others external causes can interfere in magnetic resonance image acquisition and produce noise. In this kind of images, the noise, when no signal is present, is Rayleigh distributed and its wavelet coefficients can be approximately modeled by a Gaussian distribution. Noiseless magnetic resonance images can be modeled by a Laplacian distribution in the wavelet domain. This paper proposes a new magnetic resonance image denoising method to solve this fact. This method performs shrinkage of wavelet coefficients based on the conditioned probability of being noise or detail. The parameters involved in this filtering approach are calculated by means of the expectation maximization (EM) method, which avoids the need to use an estimator of noise variance. The efficiency of the proposed filter is studied and compared with other important filtering techniques, such as Nowak's, Donoho-Johnstone's, Awate-Whitaker's, and nonlocal means filters, in different 2D and 3D images.

Figure 1

Comparative images: (a) without shift-invariant filtering; (b) with shift-invariant filtering (K = 2). Saw tooth (artifacts) can be seen in the case (a).

Figure 2

Noiseless image/volume wavelet coefficients (blue bar chart) against approximated Laplacian distribution (red graph).

Figure 3

Rayleigh noise image/volume wavelet coefficients (blue bar chart) against approximated Gaussian distribution (red graph).

Figure 4

Mixture approximated model distributions (Detail/Laplace = green graph, Noise/Gauss = red graph) superimposed on real image wavelet coefficients histogram (blue bar chart) at level/orientation ∣ NEX: (a) 1/horizontal ∣ 1; (b) 1/horizontal ∣ 10; (c) 1/horizontal ∣ 20; (d) 2/vertical ∣ 1; (e) 2/vertical ∣ 10; (f) 2/vertical ∣ 20.

Figure 5

Comparative profiles of 1D section of noiseless, noisy, and the different filtered 2D images. (a) 1D section of Section 4.2.1 data set image with parameter σ = 10 from pixel (50,126) to pixel (99,126); (b) detail of profile (a) in coordinates [27,28,…, 34].

Figure 6

Comparative results for the different filtering methods in experiment 1 with parameter σ ∈ {5,6,…, 20}.

Figure 7

Example of experiment 1 with parameter σ = 15. (a) Noiseless image; (b) noisy image; (c) noisy image filtered by Villullas-Martin's method; (d) noisy image filtered by Nowak's method; (e) noisy image filtered by Donoho-Johnstone's method; (f) noisy image filtered by Awate-Whitaker's method with Gaussian model; (g) noisy image filtered by Awate-Whitaker's method with Rician model; (h) noisy image filtered by nonlocal means method.

Figure 8

Comparative results for the different filtering methods in experiment 2 with number of averaged images NEX ∈{4,8, 16}.

Figure 9

Example of experiment 2 with number of averaged images NEX = 8. (a) Noiseless image; (b) noisy image; (c) noisy image filtered by Villullas-Martin's method; (d) noisy image filtered by Nowak's method; (e) noisy image filtered by Donoho-Johnstone's method; (f) noisy image filtered by Awate-Whitaker's method with Gaussian model; (g) noisy image filtered by Awate-Whitaker's method with Rician model; (h) noisy image filtered by nonlocal means method.

Figure 10

Comparative results for the different filtering methods in experiment 3 with parameter σ ∈ {5,6,…, 20}.

Figure 11

Experiment 4. (a) Noiseless image; (b) noisy image for NEX = 2; (c) noisy image for NEX = 3; (d) noisy image for NEX = 4; (e) noisy image for NEX = 2 filtered by Villullas-Martin method.

Figure 12

Examples subset of experiment 5. (a) Noisy images; (b) noisy images filtered by Villullas-Martin's method; (c) noisy images filtered by Nowak's method; (d) noisy images filtered by Donoho-Johnstone's method.