= Solution
Let $\Pi$ be a Poisson point process on $\mathbb R^d$ with intensity measure
$$
\mu(A)=\int_A\lambda(x)\,dx,
$$
where $\lambda$ is nonnegative and locally integrable. Let $f:\mathbb R^d\to\mathbb R^s$ be measurable and suppose its pushforward measure
$$
\nu(B)=\mu(f^{-1}(B))
=\int_{f^{-1}(B)}\lambda(x)\,dx
$$
is locally finite. The <mapping theorem for Poisson point processes> says that the image counting measure
$$
f(\Pi)=\sum_{x\in\Pi}\delta_{f(x)}
$$
is a Poisson random measure with intensity $\nu$.
For $f(\Pi)$ to be a spatial Poisson process in the usual simple sense, one also assumes that $\nu$ is diffuse:
$$
\nu(\{y\})=0\qquad(y\in\mathbb R^s).
$$
This prevents collisions with positive probability. If $\nu$ is absolutely continuous, its Radon--Nikodym derivative $\widetilde\lambda$ is the image intensity function, characterized by
$$
\boxed{\int_B\widetilde\lambda(y)\,dy
=\int_{f^{-1}(B)}\lambda(x)\,dx.}
$$
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