This is a rewrite of the OP's question to emphasize what I think are the research level issues here.

Let $\mathscr{R}$ be a bounded convex body in $\mathbb{R}^n$ and let $H : \mathbb{R}^n \to \mathbb{R}^r$ be a surjective linear map for $r<n$. How can we compute the volume of $H(\mathscr{R})$? Of course, the answer to this question will depend on how $\mathscr{R}$ is given. I don't know what the OP intended, but here are some options I can see:

$\mathscr{R}$ is a convex polytope, given as a list of vertices.

$\mathscr{R}$ is a convex polytope, given as a list of facet inequalities

$\mathscr{R}$ is a $\{ f(x_1, \ldots, x_n) \leq c \}$, for $f$ some convex polynomial. We could generalize this to $\{ f_1 \leq c_1,\ f_2 \leq c_2,\ \cdots,\ f_N \leq c_N \}$ for some list of convex polynomials $f_j$.

There is some polynomial function $\phi$ sending $\mathbb{R}^n$ to symmetric $k \times k$ matrices, and $\mathcal{R}$ is the set of $\vec{x}$ so that $\phi(\vec{x})$ has at least $\ell$ nonnegative eigenvalues. (This sort of formulation is very common in semidefinite programming.)

There will probably also be different answers depending on whether we are considering $r$ and $n$ bounded, $r$ bounded with $n \to \infty$, or both $r$ and $n$ going to $\infty$.

The original question is below.

Consider a convex body $\mathscr{R}\subset \mathbb{R}^n$ and a rank-$r$ matrix $\mathbf{H}=[\mathbf{h}_1,\cdots,\mathbf{h}_n]\in \mathbb{R}^{r\times n}$. Assume that the $r$-dimensional volume of $\mathbf{H}\mathscr{R}=\{\mathbf{Hr}:\mathbf{r}\in\mathscr{R}\}$ is finite and nonzero.

How to compute it?

This problem is extended by the previous one (The $r$-dimensional volume of the Minkowski sum of $n$ ($n\geq r$) line sets).