This paper investigates map-making using sonar sensors.  The technique is to mark regions of the map as empty, occupied, or unknown.  Multiple sensors are placed on the mobile robot and measurements are taken from different positions which are progressively added into the map and enhance its detail.  Three steps are used to remove some of the error before measurements are processed.  These steps are thresholding, averaging, and clustering.  Probability density functions are then used to reduce uncertainty [actually, to represent the information from a sonar reading]. These functions are projected onto a horizontal plane to generate map information.  The paper includes a brief mathematical description of the probability density functions used.  A sonar map is represented as a 2D array of cells containing a value which represents its occupancy and degree of uncertainty.  The values are between -1 and 1 with 0 meaning unknown. Every new reading for a cell will increase or decrease its certainty of being occupied/empty.  Also, empty and occupied probabilities are treated differently.  [Explain why!]  This paper also considers a procedure for matching two maps.  A slow method for doing this is to compute the sum of products of corresponding cells.  The paper does not mention what a reasonable sum would be for this method.  [This is correlation --- you look for the maximum.]  A quicker method uses a transformation to match cells in the maps.  [They are doing matching to _determine_ the transformation between maps!]  A sum of products is taken in this method as well.  This method can be sped up further by creating cells corresponding to a 2x2 grouping of cells in the map.  The authors found that in practice, blurring was necessary to account for noise in the measurements.