#### 3.25.18 dgrid3d

The dgrid3d command enables, and can set parameters for, non-grid to grid data mapping. See ‘splot grid_data‘ for more details about the grid data structure.

Syntax:

          set dgrid3d {<rows>} {,{<cols>}}
                      { splines |
                        qnorm {<norm>} |
                        (gauss | cauchy | exp | box | hann)
                          {kdensity} {<dx>} {,<dy>} }
          unset dgrid3d
          show dgrid3d

By default dgrid3d is disabled. When enabled, 3D data read from a file are always treated as a scattered data set. A grid with dimensions derived from a bounding box of the scattered data and size as specified by the row/col_size parameters is created for plotting and contouring. The grid is equally spaced in x (rows) and in y (columns); the z values are computed as weighted averages or spline interpolations of the scattered points’ z values. In other words, a regularly spaced grid is created and the a smooth approximation to the raw data is evaluated for all grid points. This approximation is plotted in place of the raw data.

The number of columns defaults to the number of rows, which defaults to 10.

Several algorithms are available to calculate the approximation from the raw data. Some of these algorithms can take additional parameters. These interpolations are such the closer the data point is to a grid point, the more effect it has on that grid point.

The ‘splines‘ algorithm calculates an interpolation based on "thin plate splines". It does not take additional parameters.

The ‘qnorm‘ algorithm calculates a weighted average of the input data at each grid point. Each data point is weighted inversely by its distance from the grid point raised to the norm power. (Actually, the weights are given by the inverse of dx^norm + dy^norm, where dx and dy are the components of the separation of the grid point from each data point. For some norms that are powers of two, specifically 4, 8, and 16, the computation is optimized by using the Euclidean distance in the weight calculation, (dx^2+dy^2)^norm/2. However, any non-negative integer can be used.) The power of the norm can be specified as a single optional parameter. This algorithm is the default.

Finally, several smoothing kernels are available to calculate weighted averages: z = Sum_i w(d_i) * z_i / Sum_i w(d_i), where z_i is the value of the i-th data point and d_i is the distance between the current grid point and the location of the i-th data point. All kernels assign higher weights to data points that are close to the current grid point and lower weights to data points further away.

The following kernels are available:

          gauss :     w(d) = exp(-d*d)
          cauchy :    w(d) = 1/(1 + d*d)
          exp :       w(d) = exp(-d)
          box :       w(d) = 1                     if d<1
                           = 0                     otherwise
          hann :      w(d) = 0.5*(1-cos(2*pi*d))   if d<1
                      w(d) = 0                     otherwise

When using one of these five smoothing kernels, up to two additional numerical parameters can be specified: dx and dy. These are used to rescale the coordinate differences when calculating the distance: d_i = sqrt( ((x-x_i)/dx)**2 + ((y-y_i)/dy)**2 ), where x,y are the coordinates of the current grid point and x_i,y_i are the coordinates of the i-th data point. The value of dy defaults to the value of dx, which defaults to 1. The parameters dx and dy make it possible to control the radius over which data points contribute to a grid point IN THE UNITS OF THE DATA ITSELF.

The optional keyword ‘kdensity2d‘, which must come after the name of the kernel, but before the (optional) scale parameters, modifies the algorithm so that the values calculated for the grid points are not divided by the sum of the weights ( z = Sum_i w(d_i) * z_i ). If all z_i are constant, this effectively plots a bivariate kernel density estimate: a kernel function (one of the five defined above) is placed at each data point, the sum of these kernels is evaluated at every grid point, and this smooth surface is plotted instead of the original data. This is similar in principle to + what the ‘smooth kdensity‘ option does to 1D datasets. (See kdensity2d.dem for usage demo)

A slightly different syntax is also supported for reasons of backwards compatibility. If no interpolation algorithm has been explicitly selected, the ‘qnorm‘ algorithm is assumed. Up to three comma-separated, optional parameters can be specified, which are interpreted as the the number of rows, the number of columns, and the norm value, respectively.

The dgrid3d option is a simple scheme which replaces scattered data with weighted averages on a regular grid. More sophisticated approaches to this problem exist and should be used to preprocess the data outside ‘gnuplot‘ if this simple solution is found inadequate.

See also dgrid3d.dem: dgrid3d demo. and scatter.dem: dgrid3d demo.