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Yes, you are absolutely correct. The Optical Flow Constraint Equation (OFCE), expressed as $I_{x}u+I_{y}v+I_t=0$, provides only one equation at each pixel $(x,y)$, while there are two unknowns: the horizontal optical flow component, $u$, and the vertical optical flow component, $v$.

Here’s a breakdown:

• Variables:

  • $I_x$: The partial derivative of image intensity with respect to $x$. (Spatial Gradient in x direction)
  • $I_y$: The partial derivative of image intensity with respect to $y$. (Spatial Gradient in y direction)
  • $I_t$: The partial derivative of image intensity with respect to time $t$. (Temporal Gradient)
  • $u$: The horizontal component of the optical flow vector at the pixel.
  • $v$: The vertical component of the optical flow vector at the pixel.

• Underconstrained Nature: Because we have two unknowns ($u$ and $v$) and only one equation, we cannot uniquely determine the values of $u$ and $v$ at a single pixel based solely on the OFCE. There are infinitely many combinations of $u$ and $v$ that could satisfy the equation. This is explicitly mentioned on slides 6 and 7. This situation is referred to as an underconstrained or ill-posed problem.

• Geometric Interpretation (Slide 7): The OFCE defines a line in the $u-v$ plane. Any point ($u$, $v$) on this line satisfies the brightness constancy constraint at that pixel. The slope of this line is determined by the image gradients ($I_x$ and $I_y$), and its position is influenced by the temporal gradient ($I_t$). The equation of that line can be written as:

v = -\frac{I_x}{I_y}u - \frac{I_t}{I_y} $$ provided $I_y \neq 0$. * **Aperture Problem:** This underconstrained nature is directly related to the aperture problem (Slide 8). When observing motion through a small aperture (or locally within an image), it's impossible to determine the true motion direction unambiguously. Only the component of motion *perpendicular* to the edge (i.e., along the image gradient) can be estimated reliably. * **Solutions:** To overcome this underconstrained problem, additional constraints or assumptions are necessary. The Horn-Shunck and Lucas-Kanade methods, described in the slides, introduce such constraints: * **Horn-Shunck (Slides 9-27):** Imposes a *global smoothness constraint*, assuming that the optical flow field varies smoothly across the image. * **Lucas-Kanade (Slides 31-45):** Assumes that the optical flow is *locally constant* within a small neighborhood around each pixel. In summary, your statement is correct: the OFCE by itself is underconstrained, requiring additional constraints to solve for the optical flow.