An improved least squares Laplacian pyramid for image compression

Abstract

Abstract. This paper describes two ways of improving Burt and Adelson's Laplacian pyramid, a technique developed for image compression. The Laplacian pyramid is a multi-resolution image representation that captures the loss of information occurring through repeated reduction of the spatial resolution. The generation of this data structure involves the use of two complementary functions: EXPAND, which increases the size of an image by a factor of 2, and REDUCE, which performs the reverse operation. The first modification is the adjunction of a pre-filter to the initial EXPAND function in order to guarantee an image extrapolation that is an exact interpolation of the coarser resolution level. The second refinement is a REDUCE operation modified to minimize information loss. The corresponding least squares Laplacian pyramid (LSLP) is generated by adding a post-filter to the initial REDUCE function. These new functions have an efficient implementation using recursive algorithms. Preliminary experiments indicate improved performance: for a Gaussian-like kernel (a = 3), the new EXPAND function exhibits a 2 to 2.5 dB attenuation of the first level of the Laplacian pyramid, while the complete scheme (LSLP) leads to a 4.7 to 8.5 dB improvement in the two images used to test the procedure. For comparable compression ratios, the subjective image quality for the LSLP appears to be significantly better. A theoretical relationship between the present approach and the family of quadrature mirror filter image pyramids is also derived.Zusammenfassung. Diese Arbeit beschreibt zwei Methoden, die Laplacepyramide yon Burr und Adelson zu verbessern, welche zur Bildkompression dient. Die Laplacepyramide ist eine Bilddarstellung dutch Mehrfachaufl6sung, welche den Informationsverlust erfasst, der durch die wiederholte Reduktion der rS.umlichen Aufl6sung entsteht. Fiir die Erzeugung dieser Datenstruktur werden zwei komplement/ire Funktionen gebraucht : EXPAND, welche das Bild um einen Faktor zwei vergr6s-sert, und REDUCE, welche die inverse Operation durchfiihrt. Die erste Modifikation besteht im Hinzufiigen eines Vorfilters zur EXPAND Funktion, um eine Bildextrapolation zu erreichen, welche eine pr/izise Interpolation des gr6beren Aufl6sungsni-veaus ist~ Die zweite Verbesserung betrifft die REDUCE Funktion, welche modifiziert wird, um einen minimalen Informationsverlust zu erreichen. Die entsprechende Laplacepyramide kleinster Quadrate (LSLP) wird durch Hinzufiigen eines Nachfilters zur REDUCE Funktion erzeugt. Diese zwei neuen Funktionen erlauben eine effiziente Realisierung mithilfe rekursiver Algorithmen. Erste Versuche deuten auf eine deutliche Verbesserung hin: fiir einen Gauss-artigen Kern (a = 3) erreicht die neue EXPAND Funktion eine D/impfung von 2 bis 2.5 dB im ersten Niveau der Laplacepyramide, w/ihrend die vollst/indige Methode (LSLP) eine Verbesserung yon 4.7 bis 8.5 dB fiir zwei Testbilder erreicht. Fiir vergleichbare Bildkompressionsfaktoren ist die subjektive Bildqualit/it von LSLP wesentlich besser. Ferner wird ein the...

Centered pyramids

“…Let be a discrete function and a Riesz basis of a discrete space The orthogonal projection of into is (21) where the coefficients are given by (22) Proof: The prove follows the one given in Appendix A. The -orthogonality of the error implies that (23) (24) which gives (22).…”

“…One of the main concerns there is data compression and the multiresolution decomposition is usually optimized for maximum energy compaction [21], [22], [26]. While the wavelet theory has resulted in the construction of improved image pyramids, most multiscale algorithms that assign image labels still operate with the oldest and most primitive form: the nearest neighbor or Haar pyramid.…”

The Pairing of a Wavelet Basis With a Mildly Redundant Analysis via Subband Regression

“…A standard argument (cf. [22]) then yields (9) where is the -dual of (10) Note that this result can also be obtained as a special case of the multichannel solution which is derived in the next section. The solution is equivalent to digital filtering of the auxiliary signal followed by down-sampling by a factor two.…”

“…Subsection III.B) which is simpler to formulate because there is a single wavelet involved. In that setting, the original analysis wavelet space at resolution is (22) where is the biorthogonal analysis wavelet associated with the given wavelet basis. The analysis wavelet is usually expressed in terms of the analysis scaling function (23) where is biorthogonal to and where corresponds to the wavelet analysis filter in the perfect reconstruct filterbank (cf.…”

Steerable Pyramids and Tight Wavelet Frames in $L_{2}({\BBR}^{d})$

“…The limitation is that the approach does not generalize well to dimension greater than 2 [22]. 2) Laplacian-like pyramid decompositions in the spirit of Burt and Adelson [23]: Such pyramids can be designed to have good isotropy and energy compaction properties [24], [25]; they can also be specified using orthogonal scaling functions, which automatically yields a tight frame [26].…”