Compressing the Laplacian Pyramid

1. Compressing the Laplacian Pyramid Georgios Georgiadis Hamidreza Khazaei March 9, 2010 EE 398A – Image and Video Compression

2. Introduction • Multiresolution representation of images is widely used e.g. Laplacian Pyramid (LP) • Since the LP is an overcomplete representation of the image, we show how a critical representation can be achieved that would lead to higher compression gains • We compare the (R,D) performance of the critical LP with the original LP using arithmetic and entropy encoding. We also investigate the (R,D) performance of the critical LP by using various popular filters and different number of decomposition levels G. Georgiadis, H. Khazaei: Compressing the Laplacian Pyramid

3. The Laplacian Pyramid (LP) • LP was introduced as a technique for image encoding • Image is represented by the detail signals and the coarse signal resulting from the last level of decomposition i.e. multiresolution representation • LP is an overcomplete representation but it decorrelates the image Coarse signals Detail signals G. Georgiadis, H. Khazaei: Compressing the Laplacian Pyramid

4. Critical Representation of the LP • To achieve higher compression gain: Modify the LP to a complete representation without losing the decorrelation property. • For simplicity, assume one level of decomposition and an one dimensional input image . The coarse signal • , where H is the decimation matrix. The detail signal • , • where G is the interpolation matrix and (i.e. assuming biorthogonal filters). Note that: G. Georgiadis, H. Khazaei: Compressing the Laplacian Pyramid

5. Critical Representation of the LP • Hd = 0 means that the detail signal is orthogonal to the subspace spanned by the rows of H. • Row space of H is K-dimensional. Hence d lies in a N-K dimensional subspace. Since , we can find a basis to represent d by decomposing • using SVD getting . has N-K eigenvalues that are 1, and K eigenvalues that are 0. Hence K diagonal elements of Σ are zero. • We can separate , where corresponds to the N-K non zero entries of Σ and • corresponds to the rest. G. Georgiadis, H. Khazaei: Compressing the Laplacian Pyramid

6. Critical Representation of the LP • Therefore, the columns of span the subspace in which d lies and hence we can project d to a lower dimensional subspace by: • Thus we can represent d with N-K coefficients rather than N. 2-D case Projection of the detail signal to a lower dimensional subspace results in coefficient savings G. Georgiadis, H. Khazaei: Compressing the Laplacian Pyramid

9. Critical Representation of the LP G. Georgiadis, H. Khazaei: Compressing the Laplacian Pyramid

10. Critical Representation of the LP G. Georgiadis, H. Khazaei: Compressing the Laplacian Pyramid

11. Conclusions • By representing the Laplacian Pyramid in its critical representation we achieve higher compression gains • The critical LP is a complete representation of an image that achieves decorrelation of the image pixels • We compared the (R,D) performance of the critical LP and the original LP by entropy and arithmetic encoding them. • We investigated the (R,D) performance of three filters • We compared the (R,D) performance of various levels of decomposition • Possible future work may include evaluating (R,D) performance of other encoders and using filters that do not satisfy the biorthogonality condition G. Georgiadis, H. Khazaei: Compressing the Laplacian Pyramid