NL-Means Method: NL-Means Method: Buades (2005) Buades (2005) q - - PowerPoint PPT Presentation
New Idea: New Idea: NL-Means Filter (Buades 2005) NL-Means Filter (Buades 2005) Same goals: Smooth within Similar Regions KEY INSIGHT : Generalize, extend Similarity Bilateral: Averages neighbors with similar intensities
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The basic idea is simple: Embed the original image in a family of Derived images I(x,y,t) obtained by convolving the original image I_0(x,y) with a Gaussian kernel G(x,y;t) of variance t: I(x,t) = I_0(x,y)*G(x,y;t) This one parameter family of derived images may equivalently be viewed as the solution of the heat conduction, or diffusion, equation: With the initial condition I(x,y,0) = I_0(x,y), the original image.
We would like diffusion to satisfy two criteria: 1) Causality: Any feature at a coarse level of resolution is required to possess a (not necessarily unique) “cause” at a finer level of resolution although the reverse need not be true. In other words, no spurious detail should be generated when the resolution is diminshed 2) Homogeneity and Isotropy: The blurring is required to be space invariant. The second criteria is not necessary, and as we shall see, using something else is a good idea!
divergence gradient Laplacian
This reduces to the isotropic heat diffusion equation if c(x,y,t)=const. Suppose we knew at time (scale) t, the location of region bounaries appropriate for that scale. We would want to encourage smoothing within a region in preference to smoothing across boundaries. This could be achieved by setting the conduction coefficient to be 1 in the interior of each region and 0 at the boundary. The problem: We don’t know the boundaries!
We would like diffusion to satisfy three criteria: 1) Causality: Any feature at a coarse level of resolution is required to possess a (not necessarily unique) “cause” at a finer level of resolution although the reverse need not be true. In other words, no spurious detail should be generated when the resolution is diminshed 2) Immediate Localization. The boundaries should remain sharp and stable at all scales 3) Piecewise smooth: At all scales, intra-region smoothing should occur before inter-region smoothing. The second criteria is not necessary, and as we shall see, using something else is a good idea!
Let E(x,y,t) be an estimate of edge locations. It should ideally have the following properties: 1)E(x,y,t) = 0 in the interior of each region 2)E(x,y,t) = Ke(x,y,t) at each edge point, where e is a unit vector normal to the edge at the point and K is the local contrast (difference in the image intensities on the left and right) of the edge If an estimate E(x,y,t) is available, then the conduction coefficient c(x,y,t) can be chosen to be a function c = g(||E||). According to the previous discussion, g(.) has to be nonnegative monotonically decreasing function with g(0)=1
1) AD maintains the causality principle (proof omitted) 2) AD enhances edges Proof: w.l.o.g assume the edge is aligned with y axis: And we choose c to be a function of the gradient of I: c(x,y,t) = g(I_x(x,y,t)). Let (I_x) = g(I_x) I_x denote c I_x (known as flux) Then, the 1D version of equation (3) becomes: We are interested in the variation in time of the slope of the edge: We want it to increase for strong (real) edge and decrease for weak (probably noise) edges.
Given that I is smooth, we can invert the order of differentiation
Suppose the edge is oriented in such a way that I_x>0. At the point of inflection I_xx=0, and I_xxx<<0 since the point of inflection corresponds to the point with maximum slope. Then in a neighborhood of the point of inflection has sign opposite to ’(I_x) That’s because ’’(I_x)=0 at the point of inflection. So, if ’(I_x)>0 the slope of the edge will decrease in time; if ’(I_x)<0 the slope will increase with time
Discretizing equation: Leads to (a discrete Laplacian equation): And the conduction coefficients are:
Anisotropic Diffusion: By defining: we get equivalence between the two approaches. In the discrete case we have: Robust Statistics Objective Function Gradient Descent with
Perona-Malik suggest: For a positive constant K. We want to find a \rho() function such that the iterative solution of the diffusion equation and the robust statistics equation are equivalent. Letting we have:
Why settle for the Lorenzian? Maybe we can choose a more robust error measure?
Huber’s minimax norm is equivalent to the L_1 norm for large values. But, for normally distributed data, the L_1 norm produces estimates with higher variance than the optimal L_2 (quadratic) norm, so Huber’s minmax norm is designed to be quadratic for small values
Now we can compare the three error norms directly. The modified L_1 norm gives all outliers a constant weight of one while the Tukey norm gives zero weight to outliers whose magnitude is above a certain value. The Lorentzian (or Perona-Malik) norm is in between the other two. Based
Tukey normproduces sharper boundaries than diffusing with the Lorentzian (standard Perona-Malik) norm, and that both produce sharper boundaries than the modified L1 norm
Robust estimation minimizes: Where Equivalently, we can formulate the following line process minimization problem:
where
One benefit of the line-process approach is that the “outliers” are made explicit and therefore can be manipulated. As we will see shortly
Line Process of Lorenzian (Perona-Malik)
Our goal is to take a function (x) and construct a new function, E(x;l)=(x^2l+P(l)), such that the solution at the minimum is unchanged In the case of the Lorentzian norm, it can be shown that P(l) = l-1-log(l) And the line equation becomes: Differentiating with respect to I_s and l gives the following iterative equations for minimizing:
We consider two kinds of interaction terms, hysteresis and non-maximum suppression. Other common types of interactions (for example corners) can be modeled in a similar way. The hysteresis term assists in the formation of unbroken contours while the non-maximum suppression term inhibits multiple responses to a single edge present in the data. Hysteresis lowers the penalty for creating extended edges and non-maximum suppression increases the penalty for creating edges that are more that one pixel wide.
We define a new term that penalizes the configurations on the right of the figure and rewards those on the left. This term, E_I, encodes our prior assumptions about the organization of spatial discontinuities where the parameters \epsilon_1 and \epsilon_2 assume values in the interval [0;1] and \alpha controls the importance of the spatial interaction term.
Starting with the Lorentzian norm, the new error term with constraints on the line processes becomes Where Differentiating this equation with respect to I and l gives the following update equations: The value of the line process at each point is taken to be the product, l_{s,h}*l_{s,v} of the horizontal and vertical line processes.
Effect of Line process priors
Example on real image