  The Quaternion Fourier transform (QFT) is one of the key tools in studying
color image processing. Indeed, a deep understanding of the QFT has created the
color images to be transformed as whole, rather than as color separated
component. In addition, understanding the QFT paves the way for understanding
other integral transform, such as the Quaternion Fractional Fourier transform
(QFRFT), Quaternion linear canonical transform (QLCT) and Quaternion
Wigner-Ville distribution. The aim of this paper is twofold: first to provide
some of the theoretical background regarding the Quaternion bound variation
function. We then apply it to derive the Quaternion Fourier and linear
canonical inversion formulas. Secondly, to provide some in tuition for how the
Quaternion Fourier and linear canonical inversion theorems work on the
absolutely integrable function space.
