1. Amir Yavariabdi Introduction to the Calculus of Variations and Optical Flow

2. Agenda 1. CALCULUS OF VARIATIONS  Introduction  Strong and Weak Extrema  Mixed Partial Derivatives  Chain Rule for Several Variables  Statement of Problems  Euler-Lagrange and Beltrami Identity 2. Optical Flow  Estimating the Partial Derivatives  Estimating the Laplacian of the Flow Velocities  Minimization

3. Introduction Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers and often formed as definite integrals involving unknown functions and their derivatives.

4. Weak and Strong Extrema In order to define local optimality, we must select a norm. There are two natural candidates for the norms: Extrema of J with respect to the 0-norm are called strong extrema and those with respect to the 1-norm are called weak extrema. J: real value of functional defined on v v: vector space of functions equipped with a norm

5. Mixed Partial Derivatives Derivative of f(x) at x = a For a function of several variable the total derivative of a function is: Mixed Partial Derivative

6. The Chain Rule Suppose f(x) is a differentiable function of x and x=x(t) is differentiable of t. By the chain rule theorem, the composite function z(t) is differentiable function of t By a chain rule theorem for functions of several variables The hypothesis for the chain rule theorem require the function z = f(x; y) to have continuous partial derivatives and for x(t) and y(t) to be differentiable.

7. Statement of Problem The arc length of the curve y(x) is given by integral If there are no constraint, the curve which minimizes the arc-length is the straight line. However, if the curve is constrained to lie on a surface, then the solution is less obvious (geodesics). It is not obvious what choice of y(x) will result in minimizing this integral.

8. Statement of Problem the integrals can be viewed as special kinds of functions, functions whose inputs are functions and whose outputs are real numbers. F is actually called a functional and f is an ordinary function of the variables x, y and y’. Suppose the functional F obtains a minimum (or maximum) value. How do we determine the curve y(x) which produces such a minimum (maximum) value for F?

9. Euler- Lagrange Equation If y(x) is a curve which minimizes the functional. then the following differential equation must be satisfied: The proof of above equation relies on three things: 1) The Leibniz rule 2) Integration by parts 3) Lemma 1 Let M(x) be a continuous function on the interval [a; b] Suppose that for any continuous function h(x) with h(a) = h(b) = 0 we have Then M(x) is identically zero3 on [a; b].

10. Beltrami Identity Often in applications, the function f which appears in the integrand does not depend directly on the variable x Beltrami Identity:

11. Optical Flow The Horn–Schunck method of estimating optical flow is a global method which introduces a global constraint of smoothness to solve the aperture problem. Steps: 1. Estimating the Partial Derivatives 2. Estimating the Laplacian of the Flow Velocities 3. Minimization

12. Estimating Partial Derivatives While there are many formulas for approximate differentiation. They used a set which gives them an estimate of Ex, Ey and Et at a point in the centre of a cube formed by eight measure.

13. Estimating the Laplacian of the Flow Velocities The local average of u and v are defined as follows

14. Minimizing The flow is formulated as a global energy functional which is then sought to be minimized. This function is given for two-dimensional image streams as: V [u(x,y), v(x,y)] is the optical flow vector, and the parameter α is a regularization constant. Larger values of α lead to a smoother flow. This functional can be minimized by solving the associated Euler–Lagrange equations. where L is the integrand of the energy expression

15. Minimizing where subscripts again denote partial differentiation and denotes the Laplace operator. where is a weighted average of u calculated in a neighbourhood around the pixel at location (x,y).

16. Minimizing which is linear in u and v and may be solved for each pixel in the image. However, since the solution depends on the neighbouring values of the flow field, it must be repeated once the neighbours have been updated. The following iterative scheme is derived: where the superscript k+1 denotes the next iteration, which is to be calculated and k is the last calculated result.

17. Advantage and Disadvantage the Horn–Schunck algorithm include that it yields a high density of flow vectors. it is more sensitive to noise than local methods.