Methods, apparatuses, and computer program products for rendering a piece-wise smooth image are disclosed. The image is specified by image values along opposite sides of paths. The paths are converted to chains of straight line segments, and image values are set for each side of each segment. Coefficients are determined for each line segment. A first coefficient for each segment is based on a difference between specified image values on opposite sides of the segment. A second coefficient for each segment is obtained using an average of image values on opposite sides of the segment and a system of linear equations. Each image value is calculated as the addition of a sum over all segments of the product of first coefficient for a segment with a dipole field value for the segment and a sum over all segments of the product of second coefficient for a segment with a logarithmic field value for the segment.