A survey of photon mapping state-of-the-art research and future challenges <s> Filters <s> Density estimation techniques such as the photon map method rely on a particle transport simulation to reconstruct indirect illumination, which is proportional to the particle density. In the photon map framework, particles are usually located using nearest-neighbour methods due to their generality. However, these methods have an inherent tradeoff between local bias and noise in the reconstructed illumination, which depends on the density estimate bandwidth. This paper presents a bias compensating operator for nearest-neighbour density estimation which adapts the bandwidth according to the estimated bias in the reconstructed illumination. <s> BIB001 </s> A survey of photon mapping state-of-the-art research and future challenges <s> Filters <s> This paper presents an analysis of the overestimation bias in common used filtering kernels in the context of photon mapping density estimation. We use the joint distribution of order statistics to calculate the expected value of the estimators of irradiance, and show that the estimator provided by the cone filter is not consistent unless the slope is one (yielding the triangular kernel), and that the Epanechnikov and Silverman kernels are consistent. The Gaussian filter has two different estimation biases: the original normalization constant α underestimates radiance by 46.9 percent, and the use of the kth nearest photon reduces this underestimation slightly. We also show that a new normalization constant for the Gaussian filter together with discarding the contribution of the kth nearest photon in the Gaussian and cone filter estimators produces new, consistent estimators. The specialized differential filter also benefits from the new estimate. <s> BIB002
The size of the bandwidth controls the balance between variance and bias. A small bandwidth reduces the bias, but it increases the variance of the estimate, and a large bandwidth results in an increase in bias and a decrease in variance. The selection of k for the nearest neighbor searching is important. In  , a filter was added to weigh each photon according to its distance to the shading point. By the filter, Eq. (1) can be formulated as where x i is the position of the ith photon. Denoting d = x − x i (the distance between the photon and shading point) and r an abbreviation of r(x), K(d/r) is a function of the filter that computes the weight according to the distance and can be expressed as the cone filter: The normalization of the filter based on a 2D distribution of the photons is 3. This filter is used to reduce the bias at the edges and obvious feature structures. In a set of photons around the shading point, photons near the point can get higher weights than those farther away. Another useful filter is the Epanechnikov kernel, which can reduce the mean integrated square error of the kernel density estimation. In photon mapping, BIB001 employed it and achieved smooth results. In addition, there are many other filtering methods such as the quartic filter and the Gaussian filter. All of these filtering methods can be used in photon mapping. We summarize some filtering methods which can be used in photon mapping generally (Table 1) . et al. (2012) presented an analysis of the nearest neighbour density estimator commonly used in the photon mapping algorithm. They found that the overestimated correct value is related to the number of photons in a query, and proposed a correct way to perform photon density estimation. Later, BIB002 analyzed different filtering kernels in the context of photon mapping density estimation by using the distribution of order statistics and produced new and consistent estimators through the improved filtering kernels. (1995) proposed an extended method to improve the balance between bias and variance called differential checking. They pointed out that the bias comes mainly from the bandwidth which may cross boundaries of distinct lighting features. If the radiance varies severely as more photons are used, then the iterative process stops and an optimal bandwidth is chosen.