Viet-Anh on Software Logo

What is: Laplacian Pyramid?

Data SourceCC BY-SA -

A Laplacian Pyramid is a linear invertible image representation consisting of a set of band-pass images spaced an octave apart, plus a low-frequency residual. Formally, let d(.)d\left(.\right) be a downsampling operation that blurs and decimates a j×jj \times j image II so that d(I)d\left(I\right) is a new image of size j2×j2\frac{j}{2} \times \frac{j}{2}. Also, let u(.)u\left(.\right) be an upsampling operator which smooths and expands II to be twice the size, so u(I)u\left(I\right) is a new image of size 2j×2j2j \times 2j. We first build a Gaussian pyramid G(I)=[I_0,I_1,,I_K]G\left(I\right) = \left[I\_{0}, I\_{1}, \dots, I\_{K}\right], where I_0=II\_{0} = I and I_kI\_{k} is kk repeated application of d(.)d\left(.\right) to II. KK is the number of levels in the pyramid selected so that the final level has a minimal spatial extent (8×8\leq 8 \times 8 pixels).

The coefficients h_kh\_{k} at each level kk of the Laplacian pyramid L(I)L\left(I\right) are constructed by taking the difference between adjacent levels in the Gaussian pyramid, upsampling the smaller one with u(.)u\left(.\right) so that the sizes are compatible:

h_k=L_k(I)=G_k(I)u(G_k+1(I))=I_ku(I_k+1)h\_{k} = \mathcal{L}\_{k}\left(I\right) = G\_{k}\left(I\right) − u\left(G\_{k+1}\left(I\right)\right) = I\_{k} − u\left(I\_{k+1}\right)

Intuitively, each level captures the image structure present at a particular scale. The final level of the Laplacian pyramid h_Kh\_{K} is not a difference image, but a low-frequency residual equal to the final Gaussian pyramid level, i.e. h_K=I_Kh\_{K} = I\_{K}. Reconstruction from a Laplacian pyramid coefficients [h_1,,h_K]\left[h\_{1}, \dots, h\_{K}\right] is performed using the backward recurrence:

I_k=u(I_k+1)+h_kI\_{k} = u\left(I\_{k+1}\right) + h\_{k}

which is started with I_K=h_KI\_{K} = h\_{K} and the reconstructed image being I=I_oI = I\_{o}. In other words, starting at the coarsest level, we repeatedly upsample and add the difference image h at the next finer level until we return to the full-resolution image. Source: LAPGAN

Image : Design of FIR Filters for Fast Multiscale Directional Filter Banks