  The method of canonical differential equations is an important tool in the
calculation of Feynman integrals in quantum field theories. It has been
realized that the canonical bases are closely related to $d$-dimensional
$d\log$-form integrands. In this work, we introduce the concept of generalized
loop-by-loop Baikov representation, and clarify its relation and difference
with Feynman integrals using the language of intersection theory. We then
utilize the generalized Baikov representation to construct $d$-dimensional
$d\log$-form integrands, and discuss how to convert them to Feynman integrals.
We describe the technical details of our method, in particular how to deal with
the difficulties encountered in the construction procedure. Our method provides
a constructive approach to the problem of finding canonical bases of Feynman
integrals, and we demonstrate its applicability to complicated scattering
amplitudes involving multiple physical scales.
