Sand faces and Eigenmaps

FACE envelope-05

Each sunrise in a person’s life brings changes into the face in subtle ways. The changes are apparent on FaceBook watching pictures of children reaching different thresholds. The minds eye captures only the last image of a face. If a year or two suddenly passes,  the innocent picture taken at breakfast of a proud Mother beside her son, catches the observer, remarking with, ‘His face is fuller, he’s becoming an adolescent, way past that baby face stage.’

FACE aging OBAMA Each progressive decade has a criteria performing these dynamic changes sweeping over our faces. We scratch then pull on our face, smiling at our reflection in the morning mirror gazing back at our reflections. New lines, new wrinkles, new stress lines around the eyes, the nose and mouth appear as if an artist has painted in new features.  But is it all just skin sagging by drooping with the tug of constant gravity? William Shakespeare repeatedly called the face a mask of mobile motion revealing the angst inside,

Your face, my thane, is as a book where men
May read strange matters. To beguile the time,
Look like the time; bear welcome in your eye,
Your hand, your tongue: look like the innocent flower,
But be the serpent under’t. ”


But all mere mortals suffer the motion within the bones of the face, which underlies facial aging. A digital recreation of Shakespeare’s face stares at the reader. But Shakespeare  did not know the underlying shifting bony dimensions that lay as the cause for facial features to be read like a book. The face mask appears as all of skin,  muscle and fat. Those Hollywood starlets seeking facial rejuvenation at one point are told by their plastic surgeon, ‘I can’t pull things any tighter, you won’t have any face mobility, you’ll only have a mask of expressionless features.’ Behind this assessment are recent  studies suggesting that the bony aging of the face is primarily a process of contraction plus morphologic change within the very bony density of the facial bones.

Think of the face as if designed by an artist who has cleverly draped over a scaffolding this elastic soft tissue envelope. The scaffolding strength is determined by measuring the bone density within the facial bones using dual-energy X-ray absorptiometry (DXA) scans. The earliest suggestion of an association between osteoporosis with facial bone loss was made in 1060 by Groen, Duyvensz and Halsted. I will be quoting from The Aesthetic Surgery Journal on Facial Bone Density: Effects of Aging and Impact on Facial Rejuvenation authored by Robert Shaw, Evan  Katzel, Peter Koltz, David Khan, Edward Puzas and Howard Langstein: 2012, 32:937-942.

“The bones of the face are formed by intramembranous  ossification without cartilaginous precursors, which differs from the rest of the axial skeleton and long bones. Thus, the
growth and bony resorption of the face may be regulated by different factors. This has led many to believe that the facial bones and long bones age differently. Deguchi et al, however, analyzed this question by studying 134 subjects in 3 separate age categories based on mandibular cortex erosions and the lab values of serum bone-specific alkaline
phosphatase (S-BAP) and urinary N-telopeptide cross-links of type 1 collagen (U-NTX). He found that mandibular inferior cortical erosion on radiographs was associated with increased
levels of S-BAP and U-NTX and that there was a strong association between mandible and general bone metabolism.”

“It is well known that subjects with tooth loss undergo significant alveolar bone loss, but decreased mandibular bone density has also been found in multiple studies independent of
dental status. D’Amelio et al analyzed the mandibles of 15  men (ages 34-85 years) and 16 women (ages 23-82 years) with an X-ray densitometer. He found a significant bone
density decrease in the ramus for both sexes with increasing age.”

RAMUS Gray176

“Eighteen  postmenopausal women over 2 years  showed more bone loss in the mandible by DXA compared with the femur trochanter and phalanges. Thinning of the mandibular cortices of ❤ mm has also been associated  with low skeletal bone mass.16 In this study, we hoped to expand upon the previous research by including both sexes  and by analyzing the largest number of subjects to date. Various imaging modalities are utilized to measure BMD. In this study, we used DXA imaging, as it has been shown  to best predict patients who are at risk of osteoporosis”

FACE FRACTAL code 7046728-0-large

A fractal mask can be superimposed over the face mask creating the mapping points to build the surface features that make up the envelope of skin. We do not think of the shape of the bone as determining the shape of the face but that is the dynamic motion as if the tissue is a stretched mobile fabric membrane of flesh across the bone scaffolding beneath the surface. Facial recognition algorithms can now employ a rapid scanning capacity toward the creation of a individual facial biomarker unique as a fingerprint. But the use of the face shape itself as its own biomarker of aging is only just beginning to be considered.

Eigenfaces The very edges of shape features termed eigenmaps are able to be grasped by the brain as recognizable faces like ghosts of dreams on the cusp of memory. These mathematical mapping treatments come from an authored paper by  Si Si, Dacheng Tao and Kwok-Ping Chan 2010 IEEE entitled : Discriminative Hessian Eigenmaps for Face Recognition

“A key role for face recognition is the distance or similarity
between face images which can be solved via dimension
reduction, as dimension reduction performs the recognition
by enlarging the similarity among the intra-class samples
and maximizing the difference among the inter-class samples in a subspace rather than the original feature space. A dimension reduction algorithm projects the original high-dimensional feature space to a low-dimensional subspace, where specific statistical properties can be well preserved. For example, principle component analysis (PCA) [1], one of the most popular unsupervised dimension reduction algorithms, maximizes the variance of the data in the projected subspace; Fisher’s linear discriminative analysis (FLDA) [2], the most traditional supervised dimension reduction algorithm, minimizes the trace ratio between the within class scatter and the between class scatter so that the Gaussian distributed samples can be well separated in the selected subspace; locality preserving projections (LPP) [4] preserves the local geometry of samples by processing an undirected weighted graph that represents the neighbourhood relations of pairwise samples; Marginal Fisher analysis (MFA) [12] considers both the
intra-class geometry and interaction of samples from different classes; Discriminative locality alignment (DLA) [5] preserves the discriminative information by maximizing the distance among the inter-class samples and minimizing the distance among the intra-class samples over the local patch of each sample. However the geometric and discriminative information in these dimension reduction algorithms are not well modeled, e.g., LDA does not consider the geometric information; MFA ignores the discriminative information of non-marginal samples from different classes. By using the patch alignment framework [6], we can model both the intra-class local geometry and the inter-class discriminative information conveniently. In particular, for each sample and its associated patch (neighbours of the sample), it is important to consider the following two properties: 1) the intra-class local geometry can be represented by the local tangent space, which is locally isometric to the manifold of the intra-class nearest samples of the patch; and 2) the inter-class discriminative information can be represented by the margin between the intra-class neighbor samples and the inter-class nearest samples of the patch. Because the method used for local geometry representation is similar to Hessian Eigenmaps [7], the proposed dimension reduction algorithm is termed the Discriminative Hessian Eigenmaps or DHE for short.” (The interested reader may consult the equations pertinent to the facial algorithms from their paper.)

sand-ds FRACTAL PATTERNAs Nature carves fractal patterns into the sand the image is essentially amorphous in terms of a lack of connection to the layers beneath the surface features. Yet faces can be carved into the sand loam.

FACE OF SAND 121123496_640But their unattached dimensional platform- the structure of an unseen scaffolding separates in any strong wind gusting dissolving into antishape.

tensegrity SURFACE -floating-tension-webNature always designs using the integrity of both shape holding tension within its stretched connectedness both locally into distance connections. It is all about attachment in a flexible stretched membrane that is a Snelson floating tension/compression sequence. So when we speak of the human face we are not used to seeing the elements of features which Shakespeare said are carved into the mobile motions of emotions that play across the face surface. The face is much more than that stretched skin that we pay so much attention to. The face is attached within itself tethered beneath to the bony structures. As the facial bones age they change contours, they deform hence the aging face reflects this as a shadow reflects a building’s shape. It is the unfolding shapes skirting over time that determine how the face ages.


Which can now be mathematically modelled as a tension net respecting local neighbourhood geometry of positions relative to each other so that the statistical coherence of this intactness can now be blended into mathematical compressed forms that are just barely recognizable as edges of coherence that determine our recognition of a frowning face shape or a smiling face shape or any of the immense subtleties in between a la Shakespeare’s poetic descriptions.

The combined skin envelope with the connected integrated structure beneath are dynamically linked in time as the bone changes shape with aging. The skin reflects this shifting tension displacement across the surface, into what we call the wrinkled face.

Facial depressions with aging-front

The skin is the stretched Snelson floating tension envelope across the surface that we watch so intently for expressions of conversation of understanding of compassion of hatred. In the diagram above the zones of shape change happening as we age are highlighted at the blue arrows.

TENSEGRITY ENVELOPE 46486_470232419687350_1351397705_nAs you study a face think at the same time of a stretched envelope over the surface that at the same time reflects the integrity of the supporting scaffolding beneath the surface that changes shape in measured time as the face bones evolve in their shape distorting the Snelson floating tension/compression skin envelope.

FACE pinart-toy-profile-of-human-faceLike a pin art, the tension rises from the displacing Snelson floating/compression from below hidden from sight. As we age our face is the quality, the integrity of our bone density. As we age we lose this inherent strength this ability of bone density to reveal healthy facial features. As we age our face is our compass of health. If astronauts prematurely age and concurrently suffer accelerated bone loss at the same time, you will see their changing facial features as aging lines drawn at the stress points onto their stretched Snelson tension/compression face membranes. If concussions are also severely affecting the aging process then those afflicted  with multiple concussions will have that change from impacts recorded directly written into their facial structure.


About cerebrovortex

Montreal Grandmother, Agnes Kent was saved by Raul Wallenberg from certain death, when he provided papers for her and her Mom to escape away from the Nazis. Today when asked what that escape meant, she replied,"Remind people, that while statesmen and whole countries remained silent and did nothing, a single individual chose to act, with ramifications that proved enormous. Similar choices confront us today. Write that simple truth she said, it can never be repeated often enough because the world keeps forgetting it."
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One Response to Sand faces and Eigenmaps

  1. LA says:

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