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### Virtual energy function is minimalpersonal makeover

Computer Vision: Stages

- Image formation
- Low-level
- Single image processing
- Multiple views

- Mid-level
- Grouping information
- Segmentation

- High-level
- Estimation,
- Recognition
- Classification

Why Model variations

Some objects have similar basic form but some variety in the contour shape as and perhaps also pixel values

Segmentation using snakes (from segmentation)

Modeling variations (PCA)

Eigen faces and active shape models

Combining shape and pixel values ( Active Appearance models)

TodayDeformable contours

a.k.a. active contours, snakes

Given: initial contour (model) near desired object

(Single frame)

[Snakes: Active contour models, Kass, Witkin, & Terzopoulos, ICCV1987]

Fig: Y. Boykov

Deformable contours

a.k.a. active contours, snakes

Given: initial contour (model) near desired object

Goal: evolve the contour to fit exact object boundary

(Single frame)

[Snakes: Active contour models, Kass, Witkin, & Terzopoulos, ICCV1987]

Fig: Y. Boykov

Deformable contours: intuition

Image from http://www.healthline.com/blogs/exercise_fitness/uploaded_images/HandBand2-795868.JPG

Figure from Shapiro & Stockman

final

intermediate

Deformable contoursa.k.a. active contours, snakes

- Initialize near contour of interest
- Iteratively refine: elastic band is adjusted so as to
- be near image positions with high gradients, and
- satisfy shape “preferences” or contour priors

Fig: Y. Boykov

final

Deformable contoursa.k.a. active contours, snakes

Like generalized Hough transform, useful for shape fitting; but

intermediate

Hough

Fixed model shape

Single voting pass can detect multiple instances

Snakes

Prior on shape types, but shape iteratively adjusted (deforms)

Requires initialization nearby

One optimization “pass” to fit a single contour

intermediate

final

Deformable contoursa.k.a. active contours, snakes

- How is the current contour adjusted to find the new contour at each iteration?
- Define a cost function (“energy” function) that says how good a possible configuration is.
- Seek next configuration that minimizes that cost function.

What are examples of problems with energy functions that we have seen previously?

Snakes energy function

The total energy (cost) of the current snake is defined as:

Internal energy: encourage prior shape preferences: e.g., smoothness, elasticity, particular known shape.

External energy (“image” energy): encourage contour to fit on places where image structures exist, e.g., edges.

A good fit between the current deformable contour and the target shape in the image will yield a low value for this cost function.

Parametric curve representation(discrete form)

- Represent the curve with a set of n points

External energy: intuition

- Measure how well the curve matches the image data
- “Attract” the curve toward different image features
- Edges, lines, etc.

External image energy

How do edges affect “snap” of rubber band?

Think of external energy from image as gravitational pull towards areas of high contrast

Magnitude of gradient

- (Magnitude of gradient)

External image energy

- Image I(x,y)
- Gradient images and
- External energy at a point v(s) on the curve is
- External energy for the whole curve:

Internal energy: intuition

A priori, we want to favor smooth shapes, contours with low curvature, contours similar to a known shape, etc. to balance what is actually observed (i.e., in the gradient image).

http://www3.imperial.ac.uk/pls/portallive/docs/1/52679.JPG

Internal energy

For a continuous curve, a common internal energy term is the “bending energy”.

At some point v(s) on the curve, this is:

The more the curve bends the larger this energy value is.

The weights α and β dictate how much influence each component has.

Elasticity,

Tension

Stiffness,

Curvature

Internal energy for whole curve:

Dealing with missing data

- The smoothness constraint can deal with missing data:

[Figure from Kass et al. 1987]

Discrete energy function:external term

- If the curve is represented by n points

Discrete image gradients

Discrete energy function:internal term

- Curve is represented by n points

Elasticity,

Tension

Stiffness

Curvature

Penalizing elasticity

- Current elastic energy definition uses a discrete estimate of the derivative, and can be re-written as:

Possible problem with this definition?

This encourages a closed curve to shrink to a cluster.

Penalizing elasticity

- To stop the curve from shrinking to a cluster of points, we can adjust the energy function to be:
- This encourages chains of equally spaced points.

Average distance between pairs of points – updated at each iteration

small

medium

weight controls the penalty for internal elasticity

Function of the weightsFig from Y. Boykov

Optional: specify shape prior

- If object is some smooth variation on a known shape, we can use a term that will penalize deviation from that shape (more about this later):
where are the points of the known shape.

Fig from Y. Boykov

Summary: elastic snake

- A simple elastic snake is defined by
- A set of n points,
- An internal elastic energy term
- An external edge based energy term

- To use this to locate the outline of an object
- Initialize in the vicinity of the object
- Modify the points to minimize the total energy

How should the weights in the energy function be chosen?

- For each point, search window around it and move to where energy function is minimal
- Typical window size, e.g., 5 x 5 pixels

- Stop when predefined number of points have not changed in last iteration, or after max number of iterations
- Note
- Convergence not guaranteed
- Need decent initialization

Shape energy function is minimal

How to describe the shape of the human face?

Face energy function is minimal database

Objective Formulation energy function is minimal

- Millions of pixels
- Transform into a few parameterse.g. Man / woman, fat / skinny etc.

Key Idea energy function is minimal

Images are points in a high dimensional space

Images in the possible set are highly correlated.

So, compress them to a low-dimensional subspace that

captures key appearance characteristics of the visual DOFs.

Today we will use PCA

Dimensionality Reduction energy function is minimal

The set of faces is a “subspace” of the set

of images

- Suppose it is K dimensional
- We can find the best subspace using PCA (see later)
- This is like fitting a “hyper-plane” to the set of faces

Any face is spanned by basis vectors:

Eigenfaces: the idea energy function is minimal

Think of a face as being a weighted combination of some “component” or “basis” faces. These basis faces are called eigenfaces

-8029 2900 1751 1445 4238 6193

…

Eigenfaces: representing faces energy function is minimal

The basis faces can be differently weighted to represent any face

-8029 -1183 2900 -2088 1751 -4336 1445 -669 4238 -4221 6193 10549

Learning energy function is minimal the basis images

Learn a set of basis faces which best represent the differences between the examples

Store each face as a set of weights for those basis faces

…

Eigenfaces energy function is minimal

Eigenfaces look somewhat like generic faces.

recognition energy function is minimal& reconstruction

Store and reconstruct a face from a set of weights

Recognise a new picture of a familiar face

Representation

Synthesis

Learning energy function is minimal Variations

Use Principle Components Analysis (PCA)

Need to understand

- What is an eigenvector
- What is covariance

Principal Component analysis energy function is minimal

A sample of nobservations in the 2-D space

Goal: to account for the variation in a sample

in as few variables as possible, to some accuracy

Subspaces energy function is minimal

Imagine that our face is simply a (high dimensional) vector of pixels

We can think more easily about 2d vectors

Here we have data in two dimensions

But we only really need one dimension to represent it

Finding Subspaces energy function is minimal

Suppose we take a line through the space

And then take the projection of each point onto that line

This could represent our data in “one” dimension

Finding Subspaces energy function is minimal

Some lines will represent the data in this way well, some badly

This is because the projection onto some lines separates the data well, while on others result in bad separation

Finding Subspaces energy function is minimal

Rather than a line we can perform roughly the same trick with a vector

Scale the vector to obtain any point on the line

Eigenvectors energy function is minimal

Aneigenvectorof a matrix A is a vector such that:

Where is a matrix, is a scalar (called the eigenvalue)

Example energy function is minimal

one eigenvector of A is

so for this eigenvector of this matrix the eigenvalue is 4

Matlab: [eigvecs, eigVals] = eigs(C);

Facts about Eigenvectors energy function is minimal

- The eigenvectors of a matrix are special vectors (for a given matrix) that are only scaled bythematrix
- Different matrices have different eigenvectors
- Only square (but not all) matrices have eigenvectors
- AnN xNmatrix has at mostNdistinct eigenvectors
- All the distinct eigenvectors of a matrix are orthogonal (ie perpendicular)

How to separate the 2D points? energy function is minimal

Covariance energy function is minimal

The covariance of two variables is:

The diagonal elements are the variances e.g. Var(x1)

For data that have been centred around the mean

Example energy function is minimal

Matlab: C= cov(X);

Principal Component analysis energy function is minimal

The 1stPC is a minimum distance fit to a line in spacealong the direction of most variance (eigenvalue=variance)

The 2nd PCa minimum distance fit to a line

in the plane perpendicular to the 1st PC

PCs are a series of linear least squares fits to a sample,

each orthogonal to all the previous. Combined they constitute a change of basis

A point’s position in this new coordinate system is what we earlier referred to as its “weight vector”

i energy function is minimal=

K

NM

Dimensionality Reductioneigenvalues

- We can represent the points with only their v1 coordinates
- since v2 coordinates are all essentially 0
- The eigenvalues are directly related to the variance along a particular direction (ignore directions with low eigenvalue).

- This makes it much cheaper to store and compare points

Eigenfaces energy function is minimal

- PCA extracts the eigenvectors of A
- Gives a set of vectors v1, v2, v3, ...
- Each one of these vectors is a direction in face space

Eigenfaces energy function is minimal - summary

- Treat images as points in a high-dimensional space
- Training:
- Calculate the covariance matrix of the face
- Calculate eigenvectors of the covariance matrix
- Eigenfaces with bigger eigenvalues will explain more of the variation in the set of faces, i.e. will be more distinguishing. Chose a subset of the eigenvalues (eg 95% of the total variation).

- These eigenvectors are the eigenfaces or basis faces

Eigenfaces energy function is minimal: image space to face space

Generative process:

When we see an image of a face we can transform it to face space

Subset of eigenvectors

Reconstruction energy function is minimal

- The more eigenfaces you have the better the reconstruction, but you can have high quality reconstruction even with a small number of eigenface

82 70 50

30 20 10

Recognition in face space energy function is minimal

Recognition could be done by calculating the

Euclidean distance d between our face and all the other stored faces in face space:

The closest face in face space is the chosen match

Summary energy function is minimal

- Statistical approach to visual recognition
- Also used for object recognition
- Reference: M. Turk and A. Pentland (1991). Eigenfaces for recognition, Journal of Cognitive Neuroscience, 3(1): 71–86.

Active Appearance Models energy function is minimal

- Appearance models capture variability of objects
- In terms of shape (landmark points) and texture (image intensities)

- Ingredients
- Statistical analyses of an annotated training set

- Result
- A generative model synthesising complete images of objects

- Registration
- Adjusting the model to fit an image

- If this adjustment done automatically and fast, we have an ActiveAppearance Model (AAM)[Edwards, Taylor & Cootes, AFGR 1998], [Cootes, Edwards & Taylor, ECCV 1998]

Model building energy function is minimal

- Training set
- 35 face images annotated with 58 landmarks

- Active Appearance Model
- RGB texture model sampled at 30.000 positions
- 28 model parameters span 95% variation

Training image

Annotation

Model mesh

Shape-compensation

Procrustes energy function is minimal analysis

One shape

Principal Component Analysis

Shape parameterisation

Modelling Shape II energy function is minimal

PC1 vs PC2 energy function is minimal

Modelling Shape III energy function is minimal

- The three first shape modes shown with a static texture

Mode 1 – 38%

Mode 2 – 13%

Mode 3 – 9%

Modelling texture I energy function is minimal

- Grey-level images
- Colorimages

Modelling Texture II energy function is minimal

- The three first texture modes shown with the mean static shape

Mode 1 – 21%

Mode 2 – 10%

Mode 3 – 8%

AAMExplorer Demo energy function is minimal

Model optimisation I energy function is minimal

- AAMs use a simple and efficient iterative scheme
- Synthesize
- Calculate difference between synthetic and real image
- Estimate parameter update
- Update
- Iterate until convergence

Model optimisation IV energy function is minimal

- We already know the optimal solution (annotations)
- Updates are based on differences in a shape-normalised frame(hence “similar” optimisation problems)
- Limited support
- Works well for “mild” texture changes
- Extensions for moresevere texture variationhave been proposed

Example energy function is minimalAAM search

reconstructed face

unknown face

detected facial features

Tracking example energy function is minimal

Applications of face image analysis energy function is minimal Lip-reading Eye-tracking Virtual characters Medical applications

- Biometric security
- Access systems

- Assisted speech recognition
- Automated lip-syncing in cartoons

- Human-computer interaction
- Attention analysis (maps, instruments, road signs, road cues)

- Hugo!

- Improve understanding of syndromic facial dysmorphologies, e.g. the Noonan syndrome

Eye energy function is minimaltracking

- Down-scaling of eye-tracking systems to consumer hardware, i.e. low-priced web-cameras

“A Brief Introduction to Statistical Shape energy function is minimalAnalysis”Mikkel B. Stegmann, David Delgado Gomez http://www2.imm.dtu.dk/pubdb/views/publication_details.php?id=403

Suggested readingEfficient Computation of Eigenvectors energy function is minimal

If B is MxN and M<<N then A=BTBis NxN >> MxM

- M number of images, N number of pixels
- use BBTinstead, eigenvector of BBTis easily
converted to that of BTB

(BBT) y = e y

=> BT(BBT) y = e (BTy)

=> (BTB)(BTy) = e (BTy)

=> BTyis the eigenvector ofBTB

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