2.2. Variational Approaches

with a constraint that the mean of the input image I0 (I0=Ic+N) is the same as the noise free signal Ic, and the standard deviation of I0 is the same as the noise signal N. With the calculus of variations, the denoising process was implemented by a PDE as:

The first component corresponds to the level set evolution [49] with the local curvature

term to prevent the final result deviating far from the input image. This original TV model has shortcomings to preserve fine details like edges and textures. More general assumptions of the noise models (e.g. additive and multiplicative noise) and blurring process have been investigated in [45]. Later approaches with the similar idea to apply the curvature term in image denoising have been inspired, including the min/max curvature flow [7] and modified curvature flows [14,15]. In [14,15], an image is modeled/mapped as an embedding manifold (i.e., harmonic mapping) that flows (Beltrami flow) toward the minimal surface in a higher dimensional space, e.g. in the case of color images a 2D manifold in a 5D space. Based on the early works in [12,44,45], a discrete low pass TV filter has been implemented in [43], and color and texture applications can be seen in [13,46]. Particularly, to overcome the staircase effect, a dynamically changed weighting function has been created to combine the TV component with a higher (2nd) order functional [32], which reserves the good properties of both models and alleviates their difficulties, i.e., TV model is good at preserving image features (jumps and discontinuities) and the second-order functional is good at recovering the smooth regions. Another major category of variational approach is related to Mumford-Shah functional [51,52]. Since the proposition of the Mumford-Shah functional for image segmentation, many variations of the original functional have been proposed for image denoising [16,17,47,48]. For example, a single framework [16] combining AmbrosioTortorelli functional [53] (an elliptic approximation of the Mumford-Shah functional) and the geodesic active contour model [54] is applied for both segmentation and denoising, and it is implemented by iterating two coupled PDEs. Similar work can be seen in [17,47,48], which all use coupled PDEs for several objectives, including curve evolution for image segmentation, diffusion for noise elimination and image magnification [17].

2.3. Statistics-Based Anisotropic Diffusion

The nonlinear techniques introduced above do not explicitly account for the noise statistics in the diffusion PDEs. Robust statistical techniques have been widely used in image processing and computer vision [55], and their early applications in image restoration can be seen in [56-58]. In [24,59,60], image noise statistics has been embedded into the diffusion PDEs for image denoising. The approach in [24] interprets the PM equation as a robust statistical estimation that is to minimize the image intensity difference of local neighborhood,

I=0 (Black)

Figure 1. Relationship of the RGB and HSI spaces.

2.5. Gradient Vector Flow Field

The GVF field was first presented in the snake model for image segmentation [31], in which it was used as an external force field. The GVF field is computed as the diffusion of an image edge map (e.g. gradient magnitude map). For example, given an image edge map as f(x) = | (Gσ*I) |2, x 2 and Gσ is a Gaussian smoothing filter, the GVF field v(x)=(u(x), v(x)) can be obtained as the solution minimizing the energy functional

where μ is a regularization parameter. With a variational approach, the GVF field can be derived by solving the following diffusion equation:

with the initial condition v(x, t=0)= f [31] and v is considered as a function of time. The first component in the diffusion equation is a smoothing term and it produces a slowly varying field when there is no data (edge). The second component is a data term that ensures the GVF field close to f at edges. In other words, the diffusion equation produces a vector field that is close to the gradient of the edge map at boundaries, and is slowly varying in smooth regions. A more general functional (Generalized GVF) [61] has been proposed with the parameters μ and | f|2 being replaced by weighting functions.

Compared with the gradient field, the GVF field has a wider boundary capture range and is more robust to image noise and small boundary gaps. Figure 2(b) and (c) compares the gradient field and the GVF field of an artificial image in Figure 2(a), which illustrates that the GVF field outperforms the gradient field on the noise sensitivity and gap problems. With these favorable features, the GVF direction is selected in our diffusion algorithm, instead of the traditional η-ξ direction. Experiments show that it produces a more stable and accurate result than η-ξ based diffusions. A recent GVF-based image denoising approach [33] has been proposed to apply the GVF field to replace the high-order derivative terms in several popular frameworks, including the original PM equation, shock filters and the mean curvature flow.

Figure 2. Comparison of image gradient and GVF.