Fig.1 Sampling into the reference frame, then applying local models to compute response images R(x)
The best fit is found by finding the shape and pose parameters to minimise:
The term "Constrained Local Model" originally referred to a particular type of model, in which the response images were generated by applying normalised correlation with a local patch, where the model patches are modified to fit the current face but constrained by a global texture model [1,2].However, the term has come to mean any method in which a set of local models are used to generate response images, then a shape model is used to search for the best combined response - thus earlier work by Cristinacce would also come under this revised definition [3,4].
Example from BioID, using models trained on AFLW:
 D. Cristinacce and T.F.Cootes, "Feature Detection and Tracking with Constrained Local Models", Proc. British Machine Vision Conference, Vol. 3, pp.929-938, 2006 (PDF)
 D.Cristinacce and T.F.Cootes, "Automatic Feature Localisation with Constrained Local Models", Pattern Recognition Vol.41, No.10, pp.3054-3067
 D.Cristinacce and T.F.Cootes, "A comparison of shape constrained facial feature detectors", Proc. Int.Conf on Face and Gesture Recognition, 2004, pp.375-380. (PDF)
 D. Cristinacce and T.F. Cootes, "Facial Feature Detection and Tracking with Automatic Template Selection", Proc. 7th IEEE International Conference on Automatic Face and Gesture Recognition 2006, pp. 429-434. (PDF)
 J.M.Saragih and S.Lucey and J.F.Cohn, "Deformable Model Fitting by Regularized Mean-Shifts", International Journal of Computer Vision, pp.200-215, 2011.
 T.F.Cootes, M.Ionita, C.Lindner and P.Sauer, "Robust and Accurate Shape Model Fitting using Random Forest Regression Voting", ECCV 2012 (PDF)
 C.Lindner, P.A.Bromiley, M.C.Ionita and T.F. Cootes,"Robust and Accurate Shape Model Matching using Random Forest Regression-Voting", IEEE Trans. PAMI, Vol.37, No.9, pp.1862-1874, 2015 (here)
 C. Lindner, S. Thiagarajah, J.M.Wilkinson, The arcOGEN Consortium, G.A. Wallis and T.F.Cootes, "Fully Automatic Segmentation of the Proximal Femur Using Random Forest Regression Voting",IEEE Trans. Medical Imaging, Vol. 32, No. 8, pages 1462-1472, 2013. (doi) , 2013