web app: sparse spike convolution

Sparse spike convolution model

Let $a : T\rightarrow H$ be a continuous function from a compact set $T\subset \mathbb{R}^d$ to the Hilbert space $H$, e.g. $L^2(\Omega)$. An observation vector $b\in H$ is said to be a sparse spike convolution of the discrete measure $x = \sum_{i=1}^n \beta_i\delta_{t_i}$ if
$$b = \sum_{i=1}^n \beta_i a(t_i) = \int_{t\in T} a(t) \text{d} x(t)=: Ax$$
Here $\delta_t$ denotes a spike (a.k.a. Dirac mass) located at $t\in T$. We may think of $x$ as a light sources with true location $t_i$ and amplitude $\beta_i$, and $b=Ax$ as an acquisition image via a convolution kernel $a$.

The following web application provides a visualization of above sparse spike convolution model. Click on the symbol > on the top left corner to modify the parameters. Here we consider $T=[0,1]^2\subset \mathbb{R^2}$ and $H=L^2(T)$ and $a(t)(z) = 2^{-||z-t||^2/w^2}$ for all $z, t\in T$, i.e. the kernel $a$ is a shifted Gaussian-like function.

• positions of spikes: $t_1, ..., t_n$ in $T$
• amplitudes of spikes: $\beta_1,..., \beta_n$ in $\mathbb R$
• half_width: $w$

Links: blog post, github, streamlit web app