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Convolution
Published in Aditi Majumder, M. Gopi, Introduction to Visual Computing, 2018
From the above discussions, we are now capable of visualizing or generating 2D filters like a 2D box filter or a 2D high pass filter (Figure 3.19).. Now, when considering 2D filters, there is another important property to be aware of. This is called separability. Let us consider a p × q2D filter given by h[i][j] where 1 ≤ i ≤ p and 1 ≤ j ≤ q. If h can be separated into two 1D filters, a and b of size p and q respectively, such that h[i][j] = a[i] × b[j], then h is a separable filter. As an example, let us consider a 3 × 3 box filter where p = q = 3. We know that h is a constant function where h[i][j]=19 $ h[i][j] = \frac{1}{9} $ . Now, consider two filters a and b, each of size 3 such that a[i]=13, $ a[i] = \frac{1}{3}, $ 1 ≤ i ≤ p and b[j]=13, $ b[j] = \frac{1}{3}, $ 1 ≤ j ≤ q. You can think of a and b as two 1D box filters, one in horizontal direction and the other in vertical direction. Note that in this case, h[i][j] is indeed equal to a[i]b[j], ∀(i,j) $ \forall (i, j) $ . Therefore, a 2D box filter is separable.
Classification of benign and malignant masses using bandelet and orthogonal ripplet type II transforms
Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2018
G. Prathibha, B. Chandra Mohan
The value of for curvelet, contourlet and ridgelet transform is more compared to wavelet transform since it can resolve 1D singularities along horizontal and vertical directions only. Ridgelet transform provides information about orientation of linear edges in images since it is based on Radon transform which is capable of extracting lines of arbitrary orientation that can resolve 1D singularities along an arbitrary direction. But ridgelet transform cannot resolve 2D singularities which can be overcome by curvelet proposed by Candes et al. (2006) that provides optimally sparse representation of objects with edges, optimal image reconstruction in severely ill-posed problems and optimal sparse representation of wave propagators better than wavelet transform. The curvelet transform is developed in continuous domain and then discretised for sampled data which can be overcome by constructing a discrete-domain multiresolution and multidirection expansion using non-separable filter banks which results in a flexible multiresolution, local and directional image expansion using contour segments known as contourlet transform proposed by Do and Vetterli (2005) that has two key features that improves over the separable two dimensional wavelet transform, namely directionality and anisotropy. The contourlet filter bank can provide a flexible multiscale and directional decomposition for images. The curvelet transform provides optimally sparse representations of objects which display curve-punctuated smoothness. In the proposed work, the entire mammogram is not considered for classification, instead an ROI is considered. Hence, a curvelet cannot capture the local variations in ROI because a curve can be approximated as ridgelet. Hence, curvelet is unsucessful compared to ridgelet.
Medical image interpolation based on 3D Lanczos filtering
Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2020
Thiago Moraes, Paulo Amorim, Jorge Vicente Da Silva, Helio Pedrini
Lanczos resampling is a separable filter, that is, it is possible to first apply it in the horizontal direction, then in the vertical direction and, finally, in the depth direction to be used in the 3D interpolation. Since the kernel may have negative values, the range of values of the output image may be wider than the input image. Thus, the output image must be rescaled after the interpolation process to generate the same value interval as the input values.
Deep learning-based automated COVID-19 classification from computed tomography images
Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2023
The Gaussian filtering was chosen over other filters because it is less computationally expensive to implement thanks to its filter separability property. The 2D Gaussian filter is separable into two 1D filters and can be expressed as the outer product of the two, which in turn means that the filter can be split into two passes, horizontal and vertical (Talbi et al. 2015). With a square image x[k,l] of size N×N and a square filter kernel h[n,m] such as the Laplacian filter of size M×M, the raw 2D convolution to produce the cropped output image of size N×N requires about N2×M2 Multiply – Accumulates (MACs). The raw 2D convolution between x[k,l] and the filter h[n,m] is implemented by using two ‘for loops’ to range through each output pixel and two additional for loops to perform the 2D convolution at that pixel location. Hence a total of four nested for-loops are required resulting in a complexity of O(N2×M2). With a separable filter h[n,m], such as the Gaussian filter, we can have , where f[n] and g[m] are the one-dimensional filters. In this case, the convolution between the image x[k,l] and the filter h[n,m] can be performed without a raw 2D convolution sum, by the following approach: First, perform a 1D convolution between columns of x[k,:] and the 1D filter f[n], which requires about N×M MACs to complete. This operation should be performed for each column of x[k,l] by proceeding along its horizontal, N many, columns. Hence a total of N×M×N MACs will be required to complete the first step to produce the intermediate image w[o,p].Then, apply a similar algorithm, with the filter g[m] and rows of the intermediate image w[:,p], which will require a similar number of MACs as N×M×N.