  The Karhunen-Lo\`eve transform (KLT) is often used for data decorrelation and
dimensionality reduction. The KLT is able to optimally retain the signal energy
in only few transform components, being mathematically suitable for image and
video compression. However, in practice, because of its high computational cost
and dependence on the input signal, its application in real-time scenarios is
precluded. This work proposes low-computational cost approximations for the
KLT. We focus on the blocklengths $N \in \{4, 8, 16, 32 \}$ because they are
widely employed in image and video coding standards such as JPEG and high
efficiency video coding (HEVC). Extensive computational experiments demonstrate
the suitability of the proposed low-complexity transforms for image and video
compression.
