There are several motivations for employing the directional filter banks. The directional representation implemented by the Directional Filter Bank (DFB) is useful for applications exploiting computational aspects of visual perception [5]. While the DFB can produce directional information, it lacks multi scale feature pr property in contrary to wavelet transform. One way of achieving multi scale decomposition is to combine a **Laplacian** **Pyramid** with DFB [6]. In comparison to the other compact representation techniques such as wavelet and sub band coding, the **Laplacian** **Pyramid** (LP) has the advantage that it provides greater freedom in designing the decimation and interpolation filters [7]. Thresholding plays an important role in the edge detection of images. This is justified because errors at this point are propagated throughout the detection system. Therefore, to obtain a robust thresholding measure, is a major problem in image segmentation. Thresholding is of two types namely Global and local Thresholding. In global thresholding, one threshold value is selected for the entire image, which is computed from the global information. However, global thresholding results in a poorer segmentation of the image when the illumination of the background is uneven. Therefore, there is a need for a local threshold value that changes dynamically over the image and this technique is called local Thresholding. [8].

applied to remaining sub bands i.e. LH, HL, and HH. The output of LL band is passed to **Laplacian** **Pyramid** to further remove noise. Finally the filtered bands are reconstructed using inverse of proposed method. The proposed method is equated with existing denoising methods and it is discovered that the aimed denoising method gives better PSNR and RMSE in terms of existing method. Thus the picture after removing the noise will have a good visual effect and the detail edges of the image are preserved.

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**Laplacian** **pyramid**-based modified nonlinear diffusion The proposed **Laplacian** **pyramid**-based modified nonlinear diffusion (LPMND) method is illustrated in Fig. 1. The **Laplacian** **Pyramid** is a decomposition of the original image into a hierarchy of images so that each level corresponds to a different band of image frequencies [34]. More details are reported in Additional file 1: Appendix A1. In our method, the degraded image is decomposed into 3 levels with the approximation image as the highest level.Actually noise and useful signal components of an image will be reflected in different levels after decomposition using the **Laplacian** **pyramid**. Regard to a degraded image, as noise has high frequency, it mainly exists in the lower **pyramid** level. How- ever, we need notice that although noise mainly exists in lower level, some also exists in higher level and should be discarded. Similarly, some useful information of structure may exist in the lower level and should be retained. The proposed modified nonlinear diffusion (MND) is therefore applied on the image in each level. Obviously, there are three steps as shown in Fig. 1: (1) transforming the image to be processed into its **pyramid** domain, (2) restoring all **pyramid** images by performing MND filter, and (3) recon- structing the **Laplacian** **pyramid** using the processed **pyramid** images in each level.

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The core of the approach is our novel boosting **Laplacian** **pyramid**, which is based on the structure of boosting the detail and base signal, respectively, and the boosting process is guided by the proposed exposure weight. Our approach can effectively blend the multiple exposure images for static scenes while preserving both color appearance and texture structure. Our experimental results demonstrate that the proposed approach successfully produces visually pleasing exposure fusion images with better color appearance and more texture details than the existing exposure fusion techniques and tone mapping operators.

Table 1 shows performance comparison of each method for the RGB Woman image in terms of the computation time, in which a PC with 2.0 GHz Pentium IV (1 GB RAM) is used. Note that the computation time represents the time taken to interpolate an input image by a factor of two horizon- tally and vertically. Cubic-spline interpolation cannot e ﬀ ec- tively produce high-quality images with the smallest amount of computation because of the simplicity of the algorithm. In contrast, Li and Orchard’s method produces high-quality images with the highest computational load. In the proposed algorithm, once Gaussian/**Laplacian** **pyramid** is generated (most of the time is taken in this processing), final results

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The **pyramid** has been introduced here as a data struc- ture for supporting scaled image analysis. The same structure is well suited for a variety of other image processing tasks. Applications in data compression and graphics, as well as in image analysis, will be de- scribed in the following sections. It can be shown that the **pyramid**-building procedures described here have significant advantages over other approaches to scaledanalysis in terms of both computation cost and complexity. The **pyramid** levels are obtained with fewer steps through repeated REDUCE and EXPAND operations than is possible with the standard FFT. Fur- thermore, direct convolution with large equivalent weighting functions requires 20- to 30-bit arithmetic to maintain the same accuracy as the cascade of convolu- tions with the small generating kernel using just 8-bit arithmeticcompact code.The **Laplacian** **pyramid** has been described as a data structure composed of band- pass copies of an image that is well suited for scaled- image analysis. But the **pyramid** may also be viewed as an image transformation, or code. The **pyramid** nodes are then considered code elements, and the equivalent weighting functions are sampling functions that give node values when convolved with the image. Since the original image can be exactly reconstructed from it’s **pyramid** representation, the **pyramid** code is complete. There are two reasons for transforming an image from one representation to another: the transformation may isolate critical components of the image pattern so they are more directly accessible to analysis, or the transformation may place the data in a more compact form so that they can be stored and transmitted more efficiently.

The contourlet transform can effectively overcome the disadvantages of wavelet; contourlet transform is a multi- scale and multi-direction framework of discrete image. In this transform, the multiscale analysis and the multi-direction analysis are separated in a serial way. The **Laplacian** **pyramid** (LP) [18] is first used to capture the point discontinuities, then followed by a directional filter bank (DFB) [17] to link point discontinuities into linear structures.

This is a two dimensional array of size iw by ik. Fil denotes a filter. This is a two dimensional array of floating point numbers of size fw by fk. Usually ( fk, fw ) << ( ik, iw). The output O(x, y) is an image of same size as Im. The value of O at any pixel is generated by position Fil on Im(x, y) such that the top right pixel of Fil coincides with Im(x, y) and then multiplying the values of Im and Fil for all the pixels of Im covered by Fil, and finally summing these values.Next procedure is to apply low pass filtering to the image. This includes firstly preparation of inputs and the kernel.i.e. the inputs include gray image and a Box filter(2x2). Then there are 8 levels of processing performed . Thus 8 level Gaussian pyramids are obtained using convolution (i.e.for each of the images). Here the few levels out of 8 levels of **pyramid** for the 1 st image are depicted in Figure18,Figure19,Figure20.Next is the application of bandpass filtering.This involves the generation of 7 levels of the **Laplacian** **pyramid** by subtracting the consecutive levels of gausssian pyramids. Here few levels that are visible are depicted in Figure21, Figure22, Figure23, Figure24, Figure25.Here bilinear interpolation to upsample the image of lower size is used so that the two images used during subtraction have the same size.Here 2 approaches were used for computing the Gaussian **pyramid**. First approach is by the use of Box filter where size of image gets reduced at each level. Second approach is zero crossings where the size of the image remains fixed and during convolution process results in black border around the lower and left periphery. These are seen in Figure26, Figure27, Figure28, Figure29, Figure30 and Figure31. **Laplacian** computation with equal size images are shown in Figure32,Figure33 and Figure34. Then multi-scale edge estimation(convolution,segmentation

All the methods above produce a certain fusion result with a changeless information and details. However, it is uncertain that the fixed fusion result image is just what we need because it is a difficult problem to judge what result is “best”. For example, sometimes we expect as many details as possible, but sometimes we focus more on overall bright and dark effect. Instead of evaluating the quality of HDR image based on complex methodology [12], we propose a new strategy that lets users choose the detail display level as they want freely by adjusting a simple parameter. Our method is based on multi-resolution **Laplacian** **pyramid** weighted blending as [10], but we use Local **Laplacian** Filtering to reconstruct the **Laplacian** **pyramid** coefficients, and use a remapping function to manipulate the image detail level controlled by an user parameter. Our method can generate clear and detail-preserving fused result without halos and have a greater flexibility on detail display for the needs of different users.

result. Disturbing seams appear at the border of each segmentation. To address the seam problem, we use a technique proposed by Mertens et al. in [7], which is first inspired by [1]. It is originally to fuse several images with different exposures. We use it to achieve seamless blending of our tone mapped images. First, the images are decomposed into a **Laplacian** **pyramid**, which basically contains band-pass filtered versions at different scales [1]. Blending is then carried out for each level separately. Let the l-th level in a **Laplacian** **pyramid** of an image A be defined as L{R} l , and G{B} l for a Gaussian **pyramid** of image B. Then, we blend the coefficients in a similar fashion to Eq. (7)

Meanwhile, Goodfellow et al. [9] proposed a GAN, which has become more popular and well- known in deep-learning field. Various kinds of GANs have been proposed in recent years. Under the guidance of network structure, the **Laplacian** **Pyramid** of Generative Adversarial Networks (LapGAN) produces sharp images with the **Laplacian** **pyramid** [11]. Deep Convolutional Generative Adversarial Networks (DCGAN) has shown great feature representations by using fully convolutional networks in the generator [26], instead of deterministic spatial pooling functions. Even more excellently and effectively, there are perceptual loss function including a content loss and an adversarial loss in Super-Resolution using a Generative Adversarial Network (SRGAN) [27], which achieves state-of-the-art performance. Besides, other deep-learning algorithms also have good results. Such as DRCN [28], LAPSRN [5], InfoGAN [30], CGAN [31] and CycleGAN [32]. According to related work, an advanced architecture is crucial for GAN algorithm.

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The discrete Fourier-Wavelet transform is a combination of two well known image transforms: the **Laplacian** **pyramid** [6] and the windowed Fourier transform. In some ways, it is sim- ilar to the octave band Gabor representation proposed in [7], but avoids some of the more unpleasant numerical properties of the Gabor functions. Although the **pyramid** is overcom- plete, by some 33% in 2-D, this becomes negligible in 3-D (14%) unless overlapping windows are used in the WFT. Thus in 1-D, in the continuous domain, a prototypical FWB vector has the form

Image fusion is a method to combine two relevant images. So the resulting image that is attained has to be more instructive. In computer vision, Multisensor Image fusion is the method of combining relevant information from two or more images into a single image .The resultant image will be more informative than any of the input images [1] . In remote sensing applications, the increasing availability of space borne sensors gives a motivation for different image fusion algorithms. Several situations in image processing need high spatial and high spectral resolution in a single image. Most of the available equipment is not capable of providing such data convincingly. Image fusion techniques allow the integration of dissimilar information sources. The fused image can have complementary spatial and spectral resolution characteristics. However, the standard image fusion techniques can distort the spectral information of the multispectral data however merging .In satellite imaging, two kinds of images are available. The panchromatic image acquired by satellites is transmitted with the maximum resolution available and the multispectral data are transmitted with coarser resolution [2] . This will usually be two or four times lower. At the receiver station, the panchromatic image is merged with the multispectral data to convey more information [6] .Numerous procedures exist to implement image fusion. The very basic one is the high pass filtering technique. Later techniques are based on Discrete Wavelet Transform, uniform rational filter bank, and **Laplacian** **pyramid**.

Abstract: In severe situations like accidents occur, majority of registered cases are for bone or head injury. For proper diagnosis, both CT scan and MRI scan are required to study the damage occurred for skull as well as for the internal organ injury of brain for the development of any brain tumors. If a combination of both images is present in a single image, then diagnosing the patient would be easier. Image Fusion is a method used to combine two input images to generate a combined complementary information contained image. For medical image processing, the resultant image is required to be highly reliable, low cost in terms of storage cost, uncertainty, etc. Also the information in both CT scan and MRI scan must be retained in the fused image for reliable study and assessment for diagnosis. This paper deals with pixel level fusion methods and their generic multiresolution fusion scheme. This scheme utilizes the low pass residuals and high pass residuals to segregate the information of two input images that are to be fused. The linear and nonlinear methods are used to develop the fused image. The fused image is evaluated in terms of fusion metrics such as standard deviation, entropy, fusion mutual information, etc. The methods like **laplacian** **pyramid**, ratio **pyramid**, principal component analysis, average methods prove to be better options for medical image fusion.

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This paper proposes the image fusion based on counterlet transform and discrete wavelet transform. The DWT and CT transform are used to extract the best features from different blur input images. The images are portioned based on dimensional reduction methods such as **Laplacian** **pyramid** and different coefficients from discrete wavelet transform to enhance the mean square error (MSE) and peak signal to noise ratio (PSNR) for exhibit the good appearance of output image i.e. image fusion. Hybrid DWT architecture has the advantage of lowers computational complexities and higher efficiencies. The algorithm is written in system MATLAB software. Image fusion based on contoulet transform and discrete wavelet transform gives better MSE and PSNR results as compared to existing methods.

Thickness distribution of sheet metal in an incremental forming process is numerically investigated using finite element code. Product shape is a quadrangular **pyramid**. A two-pass SPIF method is applied to form the same product obtained with single-pass SPIF method, in order to find the effect of the first method on the thickness distribution of the final parts. The effect of the first **pyramid**, in a two- pass SPIF method, is also investigated. It’s concluded that: The two-pass SPIF method makes the thickness more homogenous and a critical drop in thickness in the walls can be avoided in this case. In the fillets where tool changes direction of displacement, two-pass SPIF method brings the minimal thickness areas near the corners of the **pyramid**, and the shape becomes thinner. As the depth of the first **pyramid** in a two pass SPIF method increases, the thickness of the **pyramid** walls increases and the minima thickness values in the fillets decreases.

II. **LAPLACIAN** AND NORMALIZED **LAPLACIAN** To obtain the number of spanning trees of a graph through the evolution of the determinant of a matrix , we use Matrix-Tree Theorem. A tree is a connected graph that includes no cycles. A spanning subgraph H of a graph G is the one with V(H) = V(G) and every edge e ∈ E(H) belongs to E(G). As we can choose any subset of the edges of G to be E(H) we infer that if |E(G)| = q then the number of spanning subgraphs of G is 2 q . A spanning tree is a spanning subgraph and a tree. Number of spanning subgraphs in a graph is a pertinent graph structural parameter and is denoted by τ(G). Cauchy-Binet Theorem provides a nice generalization of a vital property concerning determinants that if A and B are n×n matrices, then |AB|=|A||B| to the case then when A and B are rectangular matrices. Interestingly it turns that |AB| is a square matrix.

To sum, the orientation is consistent with the idea that the **pyramid** pertained to Sirius, but, in contrast to Belmonte et al. (2005), I believe the southeast edge is the more accurate indicator to that effect than the eastern face. This also sup- ports the idea that the **pyramid**, more than a shrine, was designed with scientific intent. I agree with the authors that the Yebu **Pyramid** was not oriented with the intent to follow the course of the Nile. The fact that an edge or corner rather than a face was an orientation marker has precedence in both late predynastic and early dynastic Egypt when kings built large enclosures, so-called “fortresses of the gods” (Wilkinson, 2000: 18-19). These enclosed rectangular spaces proba- bly symbolized the heavens, i.e. Nut . They typically had a gate at the eastern as- pect of the southeast corner and at the northern aspect of the northeast corner. This architectural theme was preserved for at least 400 years prior to when the Yebu **Pyramid** was built (Wilkinson, 2000: 18). It is possible that these two gates opened symbolic passages to Sirius/Horus/ Sopdu and Alkaid in Ursa Major ( Mesekhtiu ), the ox thigh of the Egyptian Zodiac representing the god Seth . This physical lay-out on the ground correlates well with the stellar theme of the Py- ramid Texts and an allusion to the Ogdoad (water lilies; see discussion). Accord- ing to James P. Allen’s translation (Allen, 2005: 67) of inscriptions on Teti ’s burial chamber’s west wall gable, the first spell recited to Nut reads:

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The energy E ( G ) of a graph G is equal to the sum of the absolute values of the graph eigenvalues, namely the sum of the eigenvalues of the adjacency matrix A ( G ) of G. The ori- gin of this concept comes from the π-electron energy in the H ¨uckel molecular orbital model, but has also gained purely mathematical interest. In the past decade many kinds of energy have been introduced. In 2006, Gutman and Zhou defined the **Laplacian** energy of a graph as the sum of the absolute deviations of the eigenvalues of its **Laplacian** matrix [15]. The signless **Laplacian**, the distance, the incidence and many other versions of energy associated with a graph were defined, see [23]. In 2010, Cavers, Fallat and Kirkland first studied the Normalized **Laplacian** energy of a graph known as the Randic energy related to the Randic index [3].

119. Jennifer has designed a stone sculpture composed of a square-based right **pyramid** and a right rectangular prism. The pyramid’s base has sides 25 cm long and the **pyramid** has a slant height of 18 cm. The rectangular prism is 30 cm wide, 20 cm deep, and 40 cm high. Calculate the total exposed surface area of the sculpture.

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