We present a method for lossy compression of three dimensional gray scale images that is based on a 3D linear spline approximation to the image. We have extended an approach that has previously been successfully applied in two dimensions. In our method, we first select significant points in the data, and use them to create a 3D tetrahedralization. The tetrahedrons of the tetrahedralization are used as cells for a linear interpolation spline that gives an approximation of the original image. The compression is done by storing the positions of the vertices of the tetrahedralization and the values there instead of the value of the approximation at each grid point. We introduce a novel concept of using a smoothed version of the original image to improve the quality of the approximating spline. To increase the efficiency of the algorithm, we combine it with a refinement/decimation technique. We compare our compression technique to JPG2000 3D. We show that our algorithm performs similarly to, and in some cases even outperforms it, for high compression ratios. Our approach gives images that have significantly different properties than ones created using wavelets, and have the potential of being more suitable for some applications. In addition, this type of compression is particularly suitable for visualization.