# 2d convolution in analytic form

I have an image of a 2D-sinusoidal pattern $$f(x, y)$$ with wavelength $$\lambda$$ which I would like to convolve with a 2D circular pill-box function $$h(x,y)$$ of radius $$r$$.

The image is given by $$f(x,y) = \dfrac{1}{2}\bigg[1 + \sin\bigg(\dfrac{2\pi x}{\lambda}\bigg)\bigg].$$ Similarly the circular pill-box function is given by $$h(x,y)= \dfrac{1}{\pi r^2} \begin{cases} 1& \text{if } x^2 + y^2 \leq r^2\\ 0 & \text{otherwise} \end{cases}$$ $$\qquad$$ where the scaling constant $$\dfrac{1}{\pi r^2}$$ ensures that the area of the filter is one.

The 2D convolution integral may be expressed as $$g(x,y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(\tau_u, \tau_v)\, h(x - \tau_u, y-\tau_v) \, \mathrm{d}\tau_u \, \mathrm{d}\tau_v$$

$$\qquad$$ where $$\tau$$ is a dummy variable to represent the shift of one function with respect to the other.

How do I evaluate the convolution integral so that I can express the convolved image $$g(x,y)$$ in analytic form.

• What do you mean by simplifying in to a special form; could you please be more explict : ) If i plug in the equations of $f(x,y)$ and $h(x,y)$ in the convolution integral, shouldn't I get an expression for the convolved image? Sep 27, 2021 at 11:04
• If I plug in the expressions for $f(x,y)$ and $g(x,y)$ and evaluate the integral, shouldn't I obtain an expression for $g(x,y)$. For eg $g(x,y) = \mathrm{sinc\bigg(\dfrac{A}{B}\bigg)} \dfrac{1}{2}\bigg[1 + \sin\bigg(\dfrac{2\pi x}{\lambda}\bigg)\bigg]$, where $A$ and $B$ are some constants. Sep 27, 2021 at 11:14