+void print_3x3_matrix(const Eigen::Matrix3d &m);
+
+// Output a GLSL 3x3 matrix declaration.
+std::string output_glsl_mat3(const std::string &name, const Eigen::Matrix3d &m);
+
+// Calculate where to sample, and with what weight, if one wants to use
+// the GPU's bilinear hardware to sample w1 * x[0] + w2 * x[1].
+//
+// Note that since the GPU might have limited precision in its linear
+// interpolation, the effective weights might be different from the ones you
+// asked for. sum_sq_error, if not NULL, will contain the sum of the
+// (estimated) squared errors of the two weights.
+void combine_two_samples(float w1, float w2, float *offset, float *total_weight, float *sum_sq_error);