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I'm trying to read the paper "Particle flow for nonlinear filters with log-homotopy" by Daum and Huang. ( http://144.206.159.178/ft/CONF/16415230/16415269.pdf )

As far as I understand, they reduce their problem to solving the differential equation

$$ \frac{\partial\log(f_\lambda)}{\partial x} \frac{dx}{d\lambda} + \frac{\partial\log(f_\lambda)}{\partial \lambda} = 0$$

However, they then write: For $d=1$, we can solve the equation exactly by writing

$$ \frac{dx}{d\lambda} = -\frac{\partial\log(f_\lambda)}{\partial \lambda} / \frac{\partial\log(f_\lambda)}{\partial x} $$

for non-zero gradient and $\frac{dx}{d\lambda} = 0$ otherweise, but for $d>1$, we cannot simply solve the equation by division as above.

I have so far only seen $d$ as the symbol for total derivative, so I have no idea what their explanation is supposed to mean?

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  • $\begingroup$ $d$ wouldn't stand for the dimension, would it? $\endgroup$ Commented Sep 13, 2012 at 6:38
  • $\begingroup$ Ouch! Of course, you're right. Thank you! $\endgroup$
    – Benno
    Commented Sep 13, 2012 at 12:43

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