Refining the position of nanoparticles on an image

<p>After <a class="wikilink" href="/locating_particles_for_nanoparticle_tracking/">locating particles for nanoparticle tracking</a> using gray dilation, we will end up with a lot of false positives (for example, single-pixel noise). In <a class="litnote tooltip" href="/literature/@crocker1996">@crocker1996<span class="tooltiptext">Methods of Digital Video Microscopy for Colloidal Studies</span></a> they propose calculating the centroid around one of those pixels by performing the following computation:</p> <p>$$\begin{pmatrix}\epsilon_x \ \ \epsilon_y \end{pmatrix}=\frac{1}{m_0}\sum_{i^2+j^2\le w^2}\begin{pmatrix}i\ j\end{pmatrix}A(x+i, y+j)$$</p> <p>Where</p> <p>$$m_0 = \sum_{i^2+j^2\le w^2} A(x+i, y+j)$$</p> <p>The equations above are the common centroid calculation in a region. That is why it is important to subtract the background appropriately as done in <a class="wikilink" href="/image_restoration_for_nanoparticle_tracking/">image restoration for nanoparticle tracking</a> or there would be a bias towards the brighter regions of the image. </p> <p>Bear in mind that $$\epsilon$$ represents the variation from the previously identified pixel as the likely center of the particle. In principle, if $$|\epsilon|&gt;0.5$$ then we should move the likely candidate to a new region and recalculate the position. </p> <p>It is important to note that <a class="litnote tooltip" href="/literature/@crocker1996">@crocker1996<span class="tooltiptext">Methods of Digital Video Microscopy for Colloidal Studies</span></a> was written before the onset of single-molecule localization microscopy, and therefore the approach may be slightly extemporaneous to today’s approach, but nonetheless valid.</p>