This term I am particularly busy as I am writing a book about variational methods in imaging. The book will be a textbook and I am writing it parallel to a lecture I am currently teaching. (And it is planned that Kristian Bredies, the co-author, teaches the same stuff next term – then there will be another few month of editing, so it will be at least a year until publishing.)

In the book we will treat variational approaches to a variety of basic imaging problems. Of course we treat denoising and deblurring but there will also be sections about image interpolation, segmentation and optical flow. In the first part of the book, we present the variational problem and model them properly in Lebesgue and Sobolev spaces and of course in the space ${BV}$. Some effort goes into the analysis of the models and the first step is usually to establish existence of solutions, i.e. minimizers of the respective minimization problems. The work horse is the direct method in the calculus of variations and we mainly use the method for convex functionals in Banach spaces.

When I started the section on optical flow I noticed that I hadn’t thought about existence of minimizers before and moreover, most papers and books do not treat this issue. Let’s recall the method of Horn and Schunck to calculate the optical flow:

For two images ${u_0}$, ${u_1}$ defined on a domain ${\Omega\subset{\mathbb R}^2}$ one seeks a flow field ${V:\Omega\rightarrow{\mathbb R}^2}$ such that ${V}$ the describes the apparent motion that has happened between both images. Assuming that the points keep their gray value during motion (an assumption known as the brightness constancy constraint) and linearizing this assumption one arrives at the condition

$\displaystyle \frac{u_1-u_0}{dt} + V\cdot\nabla u_0 = 0$

(where ${dt}$ is the time between the images ${u_0}$ and ${u_1}$). First, this does not give enough equations to determine ${V}$ and secondly, points with ${\nabla u_0=0}$ are problematic.

Horn and Schunck proposed to loose the constraint and to enforce some smoothness of the flow field ${V}$: Their model was to minimize

$\displaystyle F(V) = \int_\Omega\big(\frac{u_1-u_0}{dt} + V\cdot\nabla u_0\big)^2{\mathrm d}{x} + \lambda\int_\Omega|\nabla V|^2{\mathrm d}{x}$

for some parameter ${\lambda}$ weighting smoothness of ${V}$ (large ${\lambda}$) against the brightness constancy constraint (small ${\lambda}$). A little bit more general one could choose exponents ${p}$ and ${q}$ and minimize

$\displaystyle F(V) = \int_\Omega\big|\frac{u_1-u_0}{dt} + V\cdot\nabla u_0\big|^q{\mathrm d}{x} + \lambda\int_\Omega|\nabla V|^p{\mathrm d}{x}.$

To apply the direct method to obtain existence of minimizers of ${F}$ one ensures

1. properness, i.e. there is some ${V}$ such that ${F}$ is finite,
2. convexity of ${F}$,
3. lower semi-continuiuty of ${F}$ and
4. coercivity of ${F}$.

To check these things one has to choose an appropriate space to work in. It seems reasonable to choose ${V\in L^{q}(\Omega,{\mathbb R}^2)}$. Then properness of ${F}$ is easy (consider ${V=0}$, of course assuming that ${u_1-u_0\in L^q(\Omega)}$). Convexity is also clear and for lower semi-continuity one has to work a little more, but that is possible if, e.g., ${\nabla u_0}$ is bounded. Coercivity is not that clear and in fact ${F}$ is not coercive in general.

Example 1 (Non-coercivity of the Horn-and-Schunck-model) Simply consider ${u_0(x,y) = ax + by}$ for some ${a,b\in{\mathbb R}}$. Then ${\nabla u(x,y) \equiv [a\ b]^T}$. Set ${V_n(x,y) \equiv [-nb\ na]^T}$ and note that ${\|V^n\|_q\rightarrow\infty}$ while ${F(V^n)}$ stays bounded (in fact constant).

I just checked the book “Mathematical problems in Imaging” by Gilles Aubert and Pierre Kornprobst and in Section 5.3.2 they mention that the Horn and Schunck model is not coercive. They add another term to ${F}$ which is roughly a weighted norm of ${V}$ which ensures coercivity. However, it turns out that coercivity of ${F}$ is true under a mild assumption of ${u_0}$. The idea can be found in a pretty old paper by Christoph Schnörr which is called “ Determining Optical Flow for Irregular Domains by Minimizing Quadratic Functionals of a Certain Class” (Int. J. of Comp. Vision, 6(1):25–38, 1991). His argument works for ${q=2}$:

Theorem 1 Let ${\Omega\subset{\mathbb R}^2}$ be a bounded Lipschitz domain, ${u_0,u_1\in L^2(\Omega)}$ with ${\nabla u_0\in L^\infty(\Omega)}$ such that ${\partial_x u_0}$ and ${\partial_y u_0}$ are linearly independent in ${L^2(\Omega)}$ and let ${1. Then it holds that ${F:L^2(\Omega)\rightarrow {\mathbb R}\cup\{\infty\}}$ defined by

$\displaystyle F(V) = \int_\Omega\big(\frac{u_1-u_0}{dt} + V\cdot\nabla u_0\big)^2{\mathrm d}{x} + \lambda\int_\Omega|\nabla V|^2{\mathrm d}{x}$

is coercive.

Proof: Now consider ${V^n}$ such that ${\|V^n\|_2\rightarrow\infty}$. Now we decompose the components of ${V}$ into the constant parts ${QV^n_x}$ and ${QV^n_y}$ and the “zero-mean”-part ${PV^n_x = V^n_x - QV^n_x}$ and ${PV^n_y = V^n_y - QV^n_y}$. First consider that ${PV^n}$ is unbounded, i.e. there is subsequence (also denoted by ${V^n}$) such that ${\|PV^n\|_2\rightarrow\infty}$. By Sobolev embedding and the \href{http://en.wikipedia.org/wiki/Poincar inequality}, we get that ${\int_\Omega|\nabla V^n|^p{\mathrm d}{x}\rightarrow\infty}$.

Now consider bounded ${PV^n}$ and hence, unbounded mean values ${QV^n}$. Using a subsequence, we assume that ${QV^n\rightarrow\infty}$. Now we use

$\displaystyle \Big\|\frac{u_1 - u_0}{\Delta t} + V\cdot \nabla u_0\Big\|_2 \geq \Big\|QV\cdot\nabla u_0\Big\|_2 - \Big\|\frac{u_1 - u_0}{\Delta t} + PV\cdot \nabla u_0\Big\|_2 \ \ \ \ \ (1)$

and estimate the first term from below, noticing that ${QV_x}$ and ${QV_y}$ are constants, by

$\displaystyle \begin{array}{rcl} \|QV\cdot\nabla u_0\|_2^2 & = &\|QV_x\,\partial_x u_0 + QV_y\,\partial_y u_0\|_2^2\\ & = & \|QV_x\,\partial_x u_0\|_2^2 + \|QV_y\,\partial_y u_0\|_2^2 + 2\langle QV_x\,\partial_x u_0,QV_y\,\partial_y u_0\rangle\\ & \geq &|QV_x|^2\|\partial_x u_0\|_2^2 + |QV_y|^2\|\partial_y u_0\|_2^2\\ &&\qquad - \|QV_x\,\partial_xu_0\|_2\|QV_y\,\partial_yu_0\|_2\,2\frac{|\langle \partial_x u_0,\partial_y u_0\rangle|}{\|\partial_xu_0\|_2\|\partial_y u_0\|_2}\\ & \geq &(|QV_x|^2\|\partial_x u_0\|_2^2 + |QV_y|^2\|\partial_y u_0\|_2^2) \Big(1 - \frac{|\langle \partial_x u_0,\partial_y u_0\rangle|}{\|\partial_xu_0\|_2\|\partial_y u_0\|_2}\Big). \end{array}$

Since ${\partial_x u_0}$ and ${\partial_y u_0}$ are linearly independent, it holds that ${1 - \frac{|\langle \partial_x u_0,\partial_y u_0\rangle|}{\|\partial_xu_0\|_2\|\partial_y u_0\|_2}>0}$ and we conclude that ${\|QV^{n_k}\|_2\rightarrow\infty}$ implies that ${\|QV^{n_k}\cdot\nabla u_0\|_2^2\rightarrow\infty}$. Together with~(1) and boundedness of ${PV^{n_k}}$ we obtain that ${F(V^{n_k})\rightarrow\infty}$. Since for every subsequence of ${V^n}$ we get another subsequence ${V^{n_k}}$ such that ${F(V^{n_k})\rightarrow\infty}$, the same conclusion holds for the whole sequence, showing coercivity of ${F}$. $\Box$

Basically the same arguments works for ${TV}$ optical flow, i.e. coercivity of

$\displaystyle F(V) = \int_\Omega\big(\frac{u_1-u_0}{dt} + V\cdot\nabla u_0\big)^2{\mathrm d}{x} + \lambda TV(V).$

However, I do not know yet what happens for ${q\neq 2}$ and if the result on coercivity is “sharp” in the sense that linear independence of ${\partial_x u_0}$ and ${\partial_y u_0}$ is necessary. Also, I don’t know yet what is true in dimensions higher than ${2}$.