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.

The majority of Computer Based Assessment (CBA) environments have been designed for fixed-response questions. This feature is not enough for the university context since not all the skills can be reduced to this typology of questions and free response exercises are required. In this paper, we present a web-based tool designed to automatically correct exercises that require a diagram or a graph to be solved. The main novelty of this tool is the strategy used for the correction. The tool starts without the knowledge about correct and incorrect solutions and then learns about the solutions provided by teachers and students. It automatically records the information of all entered solutions, the teacher correction and the corresponding feedback in a system database. When a new student solution is entered the information of the database is used for correction. The main components of the tool are the graph editor module, designed to support diagram drawing, and the correction module, which corrects the solutions entered by the students. These modules are integrated in a more general e-learning platform denoted ACME. ACME provides teachers all functionalities required for student work tracking, assessment, personalized attention, etc. The proposed tool has been evaluated on experimental group.