# Solving for a convolved function inside an integral

I have an imaging system that produces a signal in a pixel, in part, by responding to incident radiance $$L(\lambda)$$ that passes through a filter with transmission $$T(\lambda)$$. Lumping all the other parameters under a generic constant I can write the signal: $$S = k\cdot\int L(\lambda)\ast T(\lambda)d\lambda\,$$. I'm trying to find a way to determine, or even estimate $$L(\lambda)$$. The system is such that I know $$S, k$$, and $$T(\lambda)$$. Is there a way to recover the incident radiance? Note, I am not necessarily looking for a closed form solution, approximation is fine.

• Are S and L functions of time? Your $\ast$ isn't making sense and you didn't name the source equation. Aug 12 '20 at 13:33
• S and L are not functions of time. L, the radiance, is a function of wavelength. It's convolved with the transmission T of a color filter and summed over all wavelengths to produce the signal S (pixel value), expressed in DN in the final digital image. Not sure what you mean by source equation. Aug 12 '20 at 14:20
• This might help: dsp.stackexchange.com/questions/65977/… Aug 12 '20 at 14:50
• So $L(\lambda) * T(\lambda) = \int_{-\infty}^\infty L(\lambda - l) T(\lambda) dl$? Aug 12 '20 at 18:54
• No, perhaps I'm using convolution incorrectly here. The functions are multiplied together. For instance, at 500nm let's say the radiance is 5 $(W/m^2Sr)$, and the filter transmission at that wavelength is 0.5, so the signal recorded is $2.5\cdot k$ Aug 12 '20 at 20:27