### Images (Set 2), No. 3: Poissons dor

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Then the output image that corresponds to the targeting gradients is generated. In this process, since the gradient is usually non-integrable, the output cannot be obtained by the direct integration of gradients.

### Images from the First Colour Publication on Fish (1754)

Instead, an image whose gradient is close to the targeting gradient is obtained. Recently in [ 10 ], an energy function in the form of data term plus gradient term is also considered, i. This article considers some image processing problems that can be benefited by applying the spatially varying weights on the data constraint of above problem, i. Open image in new window. Note that the linear equation in 6 is reduced to the screened Poisson equation if the data weight is a constant.

To emphasize the inhomogeneous design of the data weight, we call this equation inhomogeneously screened Poisson equation. Loop : 1. For the fast convergence of PCG, it is important to find the optimal preconditioner M. But there is no general way to design the optimal M , i. Though, there are two well known requirements to be a good preconditioner. First, the preconditioning matrix should easily be inverted. In other words, a linear system involving M step 4 of Algorithm 1 should be easily solved, because it is the main problem in the loop.

It is easy to show that A s is non-singular because all the eigenvalues of A s is larger than 1. First, we build a simulation problem for evaluating the convergence rate: we assume that an arbitrary image is the optimal solution u o , and measure how fast the u i in Algorithm 1 converges to u o.

It can be seen that the proposed preconditioning method has the smallest condition number. Note that all the steps in Algorithm 1, except for the step 4, are equally implemented for all the methods.

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The step 4 in our problem is solved by using FFTW as stated previously. Table 2 shows the result that the amount of required memory space and CPU time for the proposed solver are less than those of others. It is found that the elapsed time of MILUC is much larger than others because it spends much time on the incomplete factorization of denser nonzero pattern. The ratio of the number of nonzero elements of MILUC to that of original matrix is also given in the parenthesis. Second, the convergence rate is measured with two real problems: image sharpening and gradient domain exposure fusion which will be addressed in the following section.

The comparison for the image sharpening problem is given in Table 3 , which shows that the proposed method shows better performance than the others. As shown in Table 4 , the proposed method also shows better performance than the others. The convergence error curve for this problem is plotted in Figure 3. Figure 2 Log RMS error curves for the image sharpening problem. Figure 3 Log RMS error curves for the exposure fusion problem.

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5. In the gradient domain processing, a similar operation to Laplacian subtraction has been developed in [ 10 ]. Consider a sequence of negative binomial random variables where the stopping parameter r goes to infinity, whereas the probability of success in each trial, p , goes to zero in such a way as to keep the mean of the distribution constant.

In other words, the alternatively parameterized negative binomial distribution converges to the Poisson distribution and r controls the deviation from the Poisson. This makes the negative binomial distribution suitable as a robust alternative to the Poisson, which approaches the Poisson for large r , but which has larger variance than the Poisson for small r.

## Work: Images, Set 2 no. 3: Poissons d'or, for piano

The negative binomial distribution also arises as a continuous mixture of Poisson distributions i. Together, the Success and Failure processes are equivalent to a single Poisson process of intensity 1, where an occurrence of the process is a success if a corresponding independent coin toss comes up heads with probability p ; otherwise, it is a failure. If r is a counting number, the coin tosses show that the count of successes before the r th failure follows a negative binomial distribution with parameters r and p. The count is also, however, the count of the Success Poisson process at the random time T of the r th occurrence in the Failure Poisson process.

The following formal derivation which does not depend on r being a counting number confirms the intuition. Because of this, the negative binomial distribution is also known as the gamma—Poisson mixture distribution. Note: The negative binomial distribution was originally derived as a limiting case of the gamma-Poisson distribution.

The negative binomial distribution is infinitely divisible , i. Then the random sum. To prove this, we calculate the probability generating function G X of X , which is the composition of the probability generating functions G N and G Y 1. The following table describes four distributions related to the number of successes in a sequence of draws:. The cumulative distribution function can be expressed in terms of the regularized incomplete beta function :.

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Suppose p is unknown and an experiment is conducted where it is decided ahead of time that sampling will continue until r successes are found. Details if other :. Thanks for telling us about the problem. Return to Book Page.

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