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On a spatial-temporal decomposition of optical flow

  • * Corresponding author: Aniello Raffaele Patrone

    * Corresponding author: Aniello Raffaele Patrone 
The first author is supported by WWTF.
Abstract Full Text(HTML) Figure(8) / Table(3) Related Papers Cited by
  • In this paper we present a decomposition algorithm for computation of the spatial-temporal optical flow of a dynamic image sequence. We consider several applications, such as the extraction of temporal motion features and motion detection in dynamic sequences under varying illumination conditions, such as they appear for instance in psychological flickering experiments. For the numerical implementation we are solving an integro-differential equation by a fixed point iteration. For comparison purposes we use a standard time dependent optical flow algorithm, which in contrast to our method, constitutes in solving a spatial-temporal differential equation.

    Mathematics Subject Classification: Primary: 65F22, 35A15, 68U10.

    Citation:

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  • Figure 1.  $f(x,t)=x(1-x)(1-t)$ from (7). Level lines of $f$ are parametrized by $(\Psi(x,t),t)$

    Figure 2.  $g(t)=\exp \left\{-\frac{1}{\beta}(1-t)^\beta\right\}$

    Figure 3.  Color Wheel

    Figure 4.  ${\vec u^{\left( 2 \right)}}$ at different frequencies of rotations: $2$, $4$ and $8 \times$ faster than the original motion frequency. $\alpha^{(1)}=1$, $\alpha^{(2)}=\frac{1}{4}$. The intensity of ${\vec u^{\left( 2 \right)}}$ increases when the frequency of rotation is increased

    Figure 6.  ${\vec u^{\left( 1 \right)}}$: Movement of a Ferris wheel and people walking in the foreground (top left). ${\vec u^{\left( 2 \right)}}$ consists of blinking lights and the reflections of the wheel (top right). The third image (bottom) is a reference frame

    Figure 5.  The dynamic sequence consists of the smooth (translation like) motion of a cube and an oscillating background. The oscillation has a periodicity of four frames and takes place along the diagonal direction from the bottom left to the top right, moving at a rate of 5% of the frame size in each frame. The proposed model decomposes the motion, obtaining the global movement of the cube in ${\vec u^{\left( 1 \right)}}$ (left) and the background movement exclusively in ${\vec u^{\left( 2 \right)}}$ (right).

    Figure 8.  The two frames of the flickering sequence containing information (top), the difference between these two frames (down left), and the ${\vec u^{\left( 2 \right)}}$ flow field resulting from the proposed approach (down right). As predicted in Section 3 and Appendix A the ${\vec u^{\left( 1 \right)}}$ component is negligible, instead ${\vec u^{\left( 2 \right)}}$ detects the change of intensity across the blank sheet.

    Figure 7.  Result with Horn-Schunck

    Table 1.  Continuous notation

    $\vec x = (x_1,x_2)$vector in two-dimensional Euclidean space
    $\partial_k = \frac{\partial}{\partial x_k}$differentiation with respect to spatial variable $x_k$
    $\partial_t = \frac{\partial}{\partial t}$differentiation with respect to time
    $\nabla = (\partial_1, \partial_2)^T$gradient in space
    $\nabla_3 = (\partial_1, \partial_2, \partial_t)^T$gradient in space and time
    $\nabla \cdot = \partial_1 + \partial_2$divergence in space
    $\nabla_3 \cdot = \partial_1 + \partial_2 + \partial_t$divergence in space and time
    $\vec{n}$outward pointing normal vector to $\Omega$
    $f$input sequence
    $f(\cdot,t)$movie frame
    ${\vec u^{\left( i \right)}}$optical flow module, $i=1,2$
    $\vec u = {\vec u^{\left( 1 \right)}} + {\vec u^{\left( 2 \right)}}$optical flow
    $u_j^{\left( i \right)}$$j$-th optical flow component of the $i$-th module
    $\widehat{u}(\cdot,t) = \int_0^t u(\cdot,\tau)\,{\rm{d}} \tau$primitive of $u$
    $\widehat{\widehat{u}}(\cdot,t) = -\int_t^1 \widehat{u}(\cdot,\tau)\,{\rm{d}} \tau$2nd primitive of $u$ -note that $\partial_t \widehat{\widehat{u}}(\cdot,t)=\widehat{u}(\cdot,t)$
     | Show Table
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    Table 2.  Discrete Notation

    $f= f(r,s,t) \in \mathbb{R}^{M \times N \times T}$input sequence
    ${\vec u^{\left( i \right)}} = {\vec u^{\left( i \right)}}(r,s,t;k) \in \mathbb{R}^{M \times N \times T \times K \times 2}$discrete optical flow approximating the
    continuous flow ${\vec u^{\left( i \right)}}$ at $(\frac{r}{M-1},\frac{s}{N-1},\frac{t}{T-1})$
    $\partial_k^h$finite difference approximation in direction $x_k$
    $\partial_t^h$finite difference approximation in direction $t$
    $\Delta_x=\frac{1}{M-1}$, $\Delta_y=\frac{1}{N-1}$ and $\Delta_t=\frac{1}{T-1}$Discretization
    $ \hat u_j^{\left( 2 \right)}(r,s,t;k) = \Delta_t \sum_{\tau=1}^t u_j^{\left( 2 \right)}(r,s,\tau;k)$, $j=1,2$finite difference approximation of $\widehat{u}(\cdot,t)$
    $ \hat {\hat {u}}_j^{\left( 2 \right)}(r,s,t;k)= - \Delta_t \sum_{\tau = t}^T \hat u_j^{\left( 2 \right)}(r,s,\tau;k)$finite difference approximation of $\widehat{\widehat{u}}(\cdot,t)$
     | Show Table
    DownLoad: CSV

    Table 3.  Comparison of squared residuals over space and time $\mathcal{E}$ between Weickert-Schnörr and the proposed method

    Weickert-SchnörrProposed model
    Hamburg Taxi1374.91021
    RubberWhale4459.73046.8
    Hydrangea8533.37647.2
    DogDance9995.48217.6
    Walking8077.55944.3
     | Show Table
    DownLoad: CSV
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