Fig. 1. The problem of non-linear dimensionality reduction, as illustrated for a three-dimensional model of a face. The 2D map illustrates the neighborhood-preserving mappings discovered by the LLE algorithm, by finding the internal coordinates of the manifold without indicating how the data should be embedded in two dimensions.

Fig. 2. Summary of the main steps for the LLE algorithm, which maps high-dimensional inputs X_i to low-dimensional outputs Y_i, using local weights W_ij obtained by computing the nearest neighbors of X_i.

Fig. 3. Schematic description of the training procedure for the proposed LLE face detection system.

Fig. 4. ROC curves indicating the performance of the proposed LLE face detector, using three different pre-processing methods for illumination compensation.

Fig. 5. LLE embeddings of the frontal database (DB1) with d = 2 for different values of D. Note that in some of the above cases embeddings had to be rotated either by 90° or by 180°, and sometimes flipped symmetrically in order to correctly evaluate the correlation. See [8].

Fig. 6. Evolution of the percentage (%) increase I(K) (number of significant weights) with respect to the number of K neighbors for: (a) DB1, (b) DB2, (c) DB3, DB4, DB5, and (d) DB6.

Fig. 7. Evolution of the residual variance ρ with respect to the value of the dimension d for: (a) DB1, (b) DB2, (c) DB3, DB4, DB5, and (d) DB6.

Fig. 8. Cross-validation accuracy with respect to the C and γ parameters of the RBF kernel.

Fig. 9. Embedding non-face images ( points) using SVR analytical mapping onto the LLE map of training facial images (+ points) from the head pose database with d = 3. (For interpretation of the references in colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 10. Testing six different non-face image sets for training DB1 (frontal faces in ambient illumination). (a) ROC curves indicating the performance of the proposed face detector, (b) ROC curves indicating the performance using the bootstrap method.

Fig. 11. ROC curves indicating the performance of the proposed face detector using three different fusion strategies.

Fig. 12. ROC curves for the proposed method ( ) and PCA ( ) using the testing images from: (a) DB1 (#Eigenfaces = 30), (b) DB2 (#Eigenfaces = 52), (c) DB3 (#Eigenfaces = 35), (d) DB4 (#Eigenfaces = 38), (e) DB5 (#Eigenfaces = 47), and (f) DB6 (#Eigenfaces = 30). The blue curves are for the PCA method. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)

Fig. 13. Schematic representation for the proposed LLE face detector.

Fig. 14. Sample (a) face and (b) non-face images from the CBCL database [32], which includes facial images under a wide variety of conditions, and non-face images.

Fig. 15. Sample images from the three subsets of the MIT-CMU image database [27], consisting of 130 images containing 510 face images. (a) Sample images from Test Set A, (b) sample image from Test Set B and (c) sample image from Test Set C.

Fig. 16. Distribution of cropped images in the CBCL database identified as faces, using the proposed LLE method, which combines six classifiers trained in the LLE domain, corresponding to the six challenges.

Fig. 17. ROC curves of different systems (proposed LLE method ( ), Viola–Jones ( ), SnOW ( ), SVM ( ), FDCM ( ), tested on images from the CBCL database. (For interpretation of the references in colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 18. ROC curves (using a log scale on the horizontal axis) for the proposed LLE face detector, the Rowley–Kanade system [27] and the Viola–Jones method [21], each tested on the MIT-CMU database.

Fig. 19. Results obtained from the proposed LLE face detector on a number of test images from the MIT-CMU database.

Databases used in this research

Non-face image sets used to train the proposed LLE face detector

Optimal detection results for all six facial databases corresponding to the six feature challenges

Comparison of detection rates for various face detection techniques, tested on the MIT-CMU database [27]