5.2. Real-world data set

We next conduct experiments on the isometric feature mapping face (ISOFACE) data set [1], which contains 698 images of a 3-D human head. The ISOFACE data set is collected under different poses and lighting directions. The resolution of each image is 64 · 64. The intrinsic degrees of freedom are the horizontal rotation, vertical rotation, and lighting direction. The 2- D embedding results of different algorithms and the corresponding GSCD results are shown in Figure 6. In the embedding, we randomly mark about 8% points with red circles and attach their corresponding training images. In the experiment, we fix the number of nearest neighbors to k ¼ 12 for all the algorithms. We empirically set r in FLM as 4. Figure 6 reveals the following interesting observations.


Figure 2. Embeddings of the synthetic manifold S-curve. The title of each subplot indicates the abbreviation of the manifold learning algorithm and the GSCD result. (a) Sample data. The title of subplots (b)-(f) indicates the abbreviation of the the manifold learning algorithm and the GSCD result.

A Fusion Scheme of Local Manifold Learning Methods http://dx.doi.org/10.5772/66303 145

Figure 3. Embeddings of the synthetic manifolds Swiss Hole. The title of each subplot indicates the abbreviation of the manifold learning algorithm and the GSCD result. (a) Sample data. The title of subplots (b)-(f) indicates the abbreviation of the the manifold learning algorithm and the GSCD result.

Figure 2. Embeddings of the synthetic manifold S-curve. The title of each subplot indicates the abbreviation of the manifold learning algorithm and the GSCD result. (a) Sample data. The title of subplots (b)-(f) indicates the abbreviation

of the the manifold learning algorithm and the GSCD result.

Figure 4. Embeddings of the synthetic manifolds Punctured Sphere. The title of each subplot indicates the abbreviation of the manifold learning algorithm and the GSCD result. (a) Sample data. The title of subplots (b)-(f) indicates the abbreviation of the the manifold learning algorithm and the GSCD result.

Figure 5. Embeddings of the synthetic manifolds Toroidal Helix. The title of each subplot indicates the abbreviation of the manifold learning algorithm and the GSCD result. (a) Sample data. The title of subplots (b)-(f) indicates the abbreviation of the the manifold learning algorithm and the GSCD result.

Figure 4. Embeddings of the synthetic manifolds Punctured Sphere. The title of each subplot indicates the abbreviation of the manifold learning algorithm and the GSCD result. (a) Sample data. The title of subplots (b)-(f) indicates the abbrevi-

ation of the the manifold learning algorithm and the GSCD result.

Figure 6. Embeddings of the ISOFACE data set. Subfigure (a) shows nine sample images, and subfigure (b) to subfigure (f) are the embedding results of different manifold learning algorithms. The title of each subplot indicates the abbreviation of the manifold learning algorithm and the GSCD result.

3. As we can observe from Figure 6f, orientations of the faces change smoothly from left to right along the horizontal direction, while the orientations of the faces change from down to up, and the light of the faces varies from bright to dark simultaneously along the vertical direction. These results illustrate that our FLM method successfully discovers the underlying manifold structure of the data set.

Our FLM performs the best on the ISOFACE data set, since our method makes full use of the complementary geometric information learned from different manifold learning methods. The corresponding GSCD results further verify the above visualization results in a quantitative way.
