text: The discrete wavelet transform can be found at the heart of many image-processing algorithms. Until now, the transform on general-purpose processors (CPUs) was mostly computed using a separable lifting scheme. As the lifting scheme consists of a small number of operations, it is preferred for processing using single-core CPUs. However, considering a parallel processing using multi-core processors, this scheme is inappropriate due to a large number of steps. On such architectures, the number of steps corresponds to the number of points that represent the exchange of data. Consequently, these points often form a performance bottleneck. Our approach appropriately rearranges calculations inside the transform, and thereby reduces the number of steps. In other words, we propose a new scheme that is friendly to parallel environments. When evaluating on multi-core CPUs, we consistently overcome the original lifting scheme. The evaluation was performed on 61-core Intel Xeon Phi and 8-core Intel Xeon processors.
  @cite_0: We construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity. The order of regularity increases linearly with the support width. We start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction. The construction then follows from a synthesis of these different approaches.
  @cite_4: Multiresolution representations are effective for analyzing the information content of images. The properties of the operator which approximates a signal at a given resolution were studied. It is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j+1 and 2 sup j (where j is an integer) can be extracted by decomposing this signal on a wavelet orthonormal basis of L sup 2 (R sup n ), the vector space of measurable, square-integrable n-dimensional functions. In L sup 2 (R), a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi (x). This decomposition defines an orthogonal multiresolution representation called a wavelet representation. It is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters. Wavelet representation lies between the spatial and Fourier domains. For images, the wavelet representation differentiates several spatial orientations. The application of this representation to data compression in image coding, texture discrimination and fractal analysis is discussed. >
summary: Usually, the 2-D transform @cite_4 is defined as the tensor product of 1-D transforms. Unlike the 1-D case, the 2-D transform splits the input signal into a quadruple of wavelet coefficients (LL, HL, LH, and HH). To describe 2-D matrices, the predict and update operators must be extended into two dimensions. Considering the separable lifting scheme, the predict and update lifting steps can be applied in both directions sequentially. It should be noted that the horizontal and vertical steps can be arbitrarily interleaved. The 2-D lifting then follow from a sequence | . | . | . | . Note the synchronization @math before the matrices. As the sequence can be hard to imagine, the individual matrices are illustrated in Figure for the CDF 5 3 wavelet @cite_0 . For multiple lifting pairs, the scheme is separately applied to each such pair. Recall that the separable lifting scheme has the smallest possible number of arithmetic operations and the highest number of steps.