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12 | 12 | fine structure of the image and thereby changes subtle aspects of the |
13 | 13 | anatomical shapes in question. |
14 | 14 |
|
15 | | -Perona and Malik \cite{Perona1990} introduced an alternative to |
| 15 | +Perona and Malik~\cite{Perona1990} introduced an alternative to |
16 | 16 | linear-filtering that they called \emph{anisotropic diffusion}. Anisotropic |
17 | | -diffusion is closely related to the earlier work of Grossberg |
18 | | -\cite{Grossberg1984}, who used similar nonlinear diffusion processes to model |
| 17 | +diffusion is closely related to the earlier work of Grossberg~\cite{Grossberg1984}, |
| 18 | +who used similar nonlinear diffusion processes to model |
19 | 19 | human vision. The motivation for anisotropic diffusion (also called |
20 | 20 | \emph{nonuniform} or \emph{variable conductance} diffusion) is that a Gaussian |
21 | 21 | smoothed image is a single time slice of the solution to the heat equation, |
|
41 | 41 | conductance parameter, $k$, and the time parameter, $t$, that is analogous to |
42 | 42 | $\sigma$, the effective width of the filter when using Gaussian kernels. |
43 | 43 |
|
44 | | -Equation \ref{eq:aniso} is a nonlinear partial differential equation that can |
| 44 | +Equation~\ref{eq:aniso} is a nonlinear partial differential equation that can |
45 | 45 | be solved on a discrete grid using finite forward differences. Thus, the |
46 | 46 | smoothed image is obtained only by an iterative process, not a convolution or |
47 | 47 | non-stationary, linear filter. Typically, the number of iterations required |
|
50 | 50 | purpose, single-processor computers. The technique applies readily and |
51 | 51 | effectively to 3D images, but requires more processing time. |
52 | 52 |
|
53 | | -In the early 1990's several research groups \cite{Gerig1991,Whitaker1993d} |
| 53 | +In the early 1990's several research groups~\cite{Gerig1991,Whitaker1993d} |
54 | 54 | demonstrated the effectiveness of anisotropic diffusion on medical images. In |
55 | | -a series of papers on the subject |
56 | | -\cite{Whitaker1993,Whitaker1993b,Whitaker1993c,Whitaker1993d,Whitaker-thesis,Whitaker1994}, |
| 55 | +a series of papers on the subject~\cite{Whitaker1993,Whitaker1993b,Whitaker1993c,Whitaker1993d,Whitaker-thesis,Whitaker1994}, |
57 | 56 | Whitaker described a detailed analytical and empirical analysis, introduced a |
58 | 57 | smoothing term in the conductance that made the process more robust, invented a |
59 | 58 | numerical scheme that virtually eliminated directional artifacts in the |
|
78 | 77 | separately, but linked through the conductance term. Vector-valued diffusion |
79 | 78 | is also useful for processing registered data from different devices or for |
80 | 79 | denoising higher-order geometric or statistical features from scalar-valued |
81 | | -images \cite{Whitaker1994,Yoo1993}. |
| 80 | +images~\cite{Whitaker1994,Yoo1993}. |
82 | 81 |
|
83 | 82 | The output of anisotropic diffusion is an image or set of images that |
84 | 83 | demonstrates reduced noise and texture but preserves, and can also enhance, |
85 | 84 | edges. Such images are useful for a variety of processes including |
86 | 85 | statistical classification, visualization, and geometric feature extraction. |
87 | | -Previous work has shown \cite{Whitaker-thesis} that anisotropic diffusion, over |
| 86 | +Previous work has shown~\cite{Whitaker-thesis} that anisotropic diffusion, over |
88 | 87 | a wide range of conductance parameters, offers quantifiable advantages over |
89 | 88 | linear filtering for edge detection in medical images. |
90 | 89 |
|
91 | 90 | Since the effectiveness of nonlinear diffusion was first demonstrated, numerous |
92 | | -variations of this approach have surfaced in the literature \cite{Romeny1994}. |
93 | | -These include alternatives for constructing dissimilarity measures |
94 | | -\cite{Sapiro1996}, directional (i.e., tensor-valued) conductance terms |
95 | | -\cite{Weickert1996,Alvarez1994} and level set interpretations |
96 | | -\cite{Whitaker2001}. |
| 91 | +variations of this approach have surfaced in the literature~\cite{Romeny1994}. |
| 92 | +These include alternatives for constructing dissimilarity measures~\cite{Sapiro1996}, |
| 93 | +directional (i.e., tensor-valued) conductance terms~\cite{Weickert1996,Alvarez1994} |
| 94 | +and level set interpretations~\cite{Whitaker2001}. |
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