4 Experiments

The aim in this section is to explore the performance of the sigmoidal regularizer compared to that of the conventional Horn and Brooks algorithm. We start with comparison on noiseless, synthetic images, then consider the degradation of needle-map recovery with increasing image noise. Finally, we present a selection of real-world examples.

Our first test image is a Lambertian synthetic solid consisting of two back-to-back segments of a sphere. At the line of symmetry where the two segments meet, the tangent planes make an angle of . This image was generated to test the relative ability of the two schemes to recover the needle map of a piecewise-smooth object without smoothing out the discontinuity where the two segments meet. The resulting needle-maps and reconstructed images are shown in Figure 3. It is clear that the sigmoidal regularizer yields a significant improvement over the conventional Horn and Brooks algorithm, particularly in the region of the centre-line.

 
Figure 3: Comparison of Horn and Brooks with Sigmoidal Regularizer algorithm on synthetic image.

Part of the motivation for applying robust statistics to the shape-from-shading problem is to improve noise rejection. To investigate this, we use the same test image as previously, but with increasing levels of additive Gaussian noise. Having shown in Figure 1 that the sigmoidal regularizer yields better reconstruction results than Horn and Brooks in noiseless conditions, Figures 4 illustrate that this advantage is maintained on noisy images.

 
Figure 4: Comparison of reconstructed images for 5% (top row) and 10% (bottom row) added noise.

Figure 5 shows plots of normal error and ground-truth brightness error for the two schemes. The normal error is the magnitude of the vector difference between the recovered normal at a point, and the ground-truth normal. The ground-truth brightness error is the difference between the reconstructed image from noisy input and the noiseless test image of Figure 3.

In terms of normal error, the true measure of accurate needle-map recovery, the sigmoidal regularizer demonstrates significant improvements over the Horn and Brooks algorithm. Comparing the reconstructed images against ground-truth, we also see that both algorithms incorporate useful levels of noise rejection; the brightness error measured against ground-truth rises more slowly than the brightness error measured against the noisy input images, showing that the schemes converge towards the underlying surface structure and do not try to model noise artifacts. The sigmoidal regularizer demonstrates excellent rejection of noise up to noise levels of 8-10%, i.e. Gaussian noise of zero mean and unit variance with Signal-to-Noise Ratio of down to around 10.

 
Figure 5: Variation of normal error (left) and ground-truth brightness error (right) with increasing image noise. In both cases, the top curve corresponds to the Horn and Brooks algorithm, and the bottom curve is due to the Sigmoidal regularizer.

Finally, to illustrate the effectiveness on real-world images, Figure 6 shows a noisy IndyCam image of a blu-tac model of a torus impaled on a rod, together with the reconstructed images and recovered needle-maps for the two algorithms. From left to right, the columns in Figure 6 show the original image together with the reconstructed images obtained with the Horn and Brooks algorithm, and the result produced using the Sigmoidal regularizer. Shown below the reconstructed images are the recovered needle-maps. The sigmoidal regularizer recovers a needle-map containing finer curvature detail. Horn and Brooks's method invariably oversmooths the sharp surface detail. In particular, the robust method is much better at recovering the concavities and convexities of the surfaces.

 
Figure 6: Comparison of Horn and Brooks vs Sigmoidal Regularizer.