Geometric nonlinear diffusion filter and its application to X-ray imaging
© Michel-González et al; licensee BioMed Central Ltd. 2011
Received: 15 February 2011
Accepted: 5 June 2011
Published: 5 June 2011
Denoising with edge preservation is very important in digital x-ray imaging since it may allow us to reduce x-ray dose in human subjects without noticeable degradation of the image quality. In denoising filter design for x-ray imaging, edge preservation as well as noise reduction is of great concern not to lose detailed spatial information for accurate diagnosis. In addition to this, fast computation is also important since digital x-ray images are mostly comprised of large sized matrices.
We have developed a new denoising filter based on the nonlinear diffusion filter model. Rather than employing four directional gradients around the pixel of interest, we use geometric parameters derived from the local pixel intensity distribution in calculating the diffusion coefficients in the horizontal and vertical directions. We have tested the filter performance, including edge preservation and noise reduction, using low dose digital radiography and micro-CT images.
The proposed denoising filter shows performance similar to those of nonlinear anisotropic diffusion filters (ADFs), one Perona-Malik ADF and the other Weickert's ADF in terms of edge preservation and noise reduction. However, the computation time has been greatly reduced.
We expect the proposed denoising filter can be greatly used for fast noise reduction particularly in low-dose x-ray imaging.
Image denoising is very important in digital x-ray imaging since it may allow us to reduce x-ray dose in human subjects without noticeable degradation of the image quality. Owing to the advent of 2D flat-panel x-ray detector, digital x-ray imaging modalities like digital radiography (DR) or 3D cone-beam CT are now widely used for clinical imaging [1, 2]. Taking x-ray images with low x-ray dose is always desired due to the hazardous biological effects of x-rays. However, low dose x-ray images may suffer from high level noises stemming from randomness of x-ray photon fluence and detector noise. There have been many developments of denoising filters that can be used to improve signal-to-noise ratio (SNR) of x-ray images without compromising the image resolution to a noticeable extent [3–5]. In denoising x-ray images, it is critical not to lose detailed spatial information in the original images which is crucial for accurate diagnosis. Recent developments of non-linear iterative denoising filters make it possible to remove noise from the x-ray images while maintaining edge features, which is impossible with linear filtering techniques. In implementing non-linear iterative denoising filters in x-ray imaging, fast computation time is one of the critical factors since x-ray images usually have large sized image matrix. DR images have big matrix sizes as large as 3500 × 4300 for chest imaging and 3600 × 4800 for breast imaging. 3D cone-beam CTs based on a flat-panel detector often make images with the matrix size as large as 1024 × 1024 × 512. For x-ray images of this big matrix size, fast computation is essential for practical use of the iterative denoising filters.
Anisotropic diffusion filter (ADF) is one of the most widely used denoising techniques in medical imaging [6–9]. ADF is believed to have good performance in denoising while well preserving edge features, and it is also often used for edge detection [10–13]. Explicit schemes of ADF needs large amount of computation , mostly coming from calculation of diffusion coefficients and gradients at every pixel. To improve the performance of ADF particularly in terms of computation time, we propose an iterative denoising filter considering the local image topology in the nonlinear diffusion model. We have tested the filter performance using synthetically and experimentally obtained x-ray images. We present the performance of the proposed filter in comparison with the Perona-Malik ADF and Weickert ADF with experimental results obtained from low dose 3D micro-CT images and 2D DR images.
A. Modified nonlinear diffusion filter
where k is the noise threshold that controls the amount of diffusion to be applied in the gradient direction. Optimal choice of k is known to be very important for successful denoising with the ADF and many kinds of k estimators have been proposed for practical applications [15, 16].
where is the discrete image at time t, s and p denotes the pixel position in a discrete 2D grid, Δt is the time step size in PDE, is the spatial neighborhood of pixel s , is the number of pixels in the window, and . By implementing Eq. (4), calculation of the diffusion coefficient is most computationally intensive. In most discrete implementations of ADF, the number of neighborhood pixels is often chosen to be four, that is, east, west, north, and south pixels. Therefore, we need to calculate the diffusion coefficients at least four times at every pixel.
One of the limitations of this explicit discretization scheme is the stability requirement that limits the step size Δt to less than 1/2d, where d is the number of directions along which the gradients are calculated . Therefore it takes longer computation time if we choose more number of directions. Despite success of the Perona-Malik model for practical use, it is known to be an ill-posed problem [10, 11, 17]. To overcome this problem, incorporating regularization into the PDE has been proposed [13, 18]. In the regularized methods, the diffusion coefficients are derived from the regularized image rather than from the noisy image to avoid excessive noise effects on the gradient calculation. Other approaches, that modify the diffusion coefficients to take account of local image statistics, have been also proposed. Black and Saphiro have proposed a robust statistical estimation approach in which they used diffusion coefficients derived from the Tukey's biweight function, and they have shown better edge preservation after denoising . You et al. explored a variety of anisotropic diffusion equations and they reported some improved results . In all the aforementioned methods, the image features have been simply classified as ∇I < k for noise regions and ∇I < k for edge regions and its performance is well known to be successful. On the other hand, Weickert proposed a more strict anisotropic diffusion model where a diffusion tensor is used . The anisotropic model mitigates the limitation of the Perona-Malik model in which the diffusion process stops at edges leaving the noises near the edges. It is believed that the Weickert model outperforms other isotropic models but it takes more extensive computation time.
In this work, we concern about the computation time of ADF in denoising medical images. We have noticed that using only a global k in defining the diffusivity function works well in most cases, but more local image information must be incorporated into the diffusivity function for faster convergence. Therefore, instead of using only a global k and image gradient in the diffusivity function, we are to use the global noise behavior and the local topology of the pixel intensities in defining new diffusivity functions.
where ∇ p = I p - I S (p = E, W, N, and S) represents the difference between the central pixel and one of the east, west, north, and south pixels respectively. Since we use the same explicit scheme used by Perona and Malik for our numerical PDE solver, the stability will depend only on the step size Δt. Previous works [10, 17] have pointed that the solution always converges to a steady state under a small step size (normally) .
B. Metrics for filter performance evaluation
For quantitative evaluation of the filter performances in terms of noise reduction and edge preservation, we use the following metrics.
Normalized Mean Square Error (NMSE)
where I f and I n are the filtered and original images, respectively, and I T is the true image.
Noise Energy Reduction Ratio (NERR)
where u and v are the spatial frequencies in the x- and y-directions, D n is the Fourier transform of the difference of the original image from the true (high x-ray dose) image after the high-pass filtering. D f is the Fourier transform of the difference of the filtered image from the true image after the high-pass filtering. The cutoff frequency k c was chosen in each image so that most of the signal components are rejected in the calculation of NERR.
Beta metric (β)
where ΔI n and ΔI f represent the edge images of the original image I n and the filtered image I f , respectively. The edge images are calculated by applying the 3 × 3 Laplacian operator followed by thresholding to reject false edges stemming from noise. and are the mean intensities of ΔI n and ΔI f , respectively. The beta metric represents the resemblance of the edge images derived from the original and filtered images. If the edge is well preserved during the denoising process, the beta metric will be close to unity.
SNR gain (g)
where SNR f and SNR n are the average SNRs in the filtered and original images, respectively.
Computation time is evaluated on a personal computer with an Intel-Pentium® Dual-Core CPU E6500 @ 2.93GHz processor. The filters have been implemented using MATLAB™ (The Mathworks, Inc., Natic, MA.) and the computation time is evaluated excluding the data readout time from the disk to the PC memory.
Performance of the proposed denoising filter in comparison with other ADFs
Computation time (s)
512 × 512
2240 × 2240
512 × 512
1024 × 1024
512 × 512
3072 × 3072
To test the proposed filter with real x-ray images, we first obtained 3D micro-CT images of a sacrificed rat using a lab-built micro-CT system. During the micro-CT scan, the micro-CT acquires 2D projection images at every angle which are to be used for 3D image reconstruction. Therefore, any of the 2D projection images can be thought to be a DR image. We reconstructed 3D tomographic images using the modified Feldkamp algorithm . The micro-CT system consists of a micro-focus x-ray tube with a variable focal spot size of 5-50 μm (L8121-01, Hamamatsu, Japan), a CMOS flat-panel detector with a matrix size of 2240 × 2240 (C7942, Hamamatsu, Japan), a rotating subject holder for the cone-beam CT scan motion, and a data acquisition system. The pixel size of the flat-panel detector is 50 × 50 μm2. We performed the micro-CT scan with the tube voltage of 60kVp and the tube current of 340 μA which gave the nominal focal spot size of 50 μm. The number of projection views, which determines the angular spatial resolution of the micro-CT images, was 900. We reconstructed 3D micro-CT images with the matrix size of 512 × 512 × 512 with isotropic pixel size of 170 × 170 × 170 μm3 or 1024 × 1024 × 1024 with isotropic pixel size of 85 × 85 × 85 μm3. Since the tube current of the micro-CT is two orders of magnitude smaller than those of medical CTs, micro-CT images usually suffer from high level noise in spite of long scan time.
Table 1 summarizes the filter performances in all the cases shown in the previous figures and also in extra cases of an axial-view micro-CT image of the rat and a DR image of a human chest. In the case of human chest imaging, we took the true image with much higher tube current of 2.2mA as compared to 0.4 mA in normal imaging. The tube voltage was 104 kVp. As can be noticed from Table 1, the overall performance of the proposed filter is similar to those of the two ADFs in terms of noise reduction and edge preservation. But, the computation time of the proposed filter is much shorter than the two ADFs.
The proposed filter uses the same form of PDE equation as the Perona-Malik ADF but with a different way of diffusion coefficient calculation. In the original Perona-Malik ADF, four different diffusion coefficients are employed to a given pixel in four orthogonal directions to allow different amount of diffusion in each direction according to the gradient strength. Owing to the four directional diffusion processes, the Perona-Malik ADF has better edge preservation than a linear diffusion filter. Even though the proposed filter uses only two diffusion coefficients in solving the PDE, it turns out that the proposed filter has similar performance to the Perona-Malik ADF. In Perona-Malik ADF, regularization must be incorporated into the diffusion coefficient calculation to stabilize the diffusion process avoiding the influence of high noises. Since the proposed filter is not based on the gradient-dependent diffusivity function with a global noise threshold, high impulsive noises do not affect the performance of the proposed filter as can be seen from the salt-and-pepper noise case in Figure 11. Even without any regularization in the proposed filter, we have not observed any unstable operations either in denoising the images we have used so far. Filtering without regularization has also allowed further reduction of computation time of the proposed filter. It is believed that the Weickert's ADF better removes the noise at the edge region, but it takes much longer computation time. Although fast algorithms to solve the Weickert's ADF have been proposed, those schemes suffer from bigger errors and artifacts . There are also many other iterative non-linear denoising filters than ADFs, which are thought to outperform ADFs in many respects [20–22].
We have experimentally confirmed that the proposed filter and the two ADFs have similar filter performance in terms of noise reduction and edge preservation. However, the proposed filter takes much shorter computation time than the two ADFs do. We expect the proposed filter can be greatly used to mitigate the noise effects in low-dose x-ray imaging.
This work was supported by the National Research Foundation (NRF) grant funded by the Korean government (MEST) (No: 2009-0078310)
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