Abstract
In this and the following chapters, we demonstrate why multiple subspaces can be a very useful class of models for image processing and how the subspace clustering techniques may facilitate many important image processing tasks, such as image representation, compression, image segmentation, and video segmentation.
Everything should be made as simple as possible, but not simpler.
—Albert Einstein
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Notes
- 1.
Which involves further quantization and entropy coding of the so-obtained sparse signals α.
- 2.
A topic we will study in more detail from the compression perspective in the next section.
- 3.
Here, “optimality” means that the transformation achieves the optimal asymptotic for approximating the class of functions considered (DeVore 1998).
- 4.
Here, in contrast to the case of pre-fixed transformations, “optimality” means the representation obtained is the optimal one within the class of models considered, in the sense that it minimizes certain discrepancies between the model and the data.
- 5.
In fact, in the VQ model, the coefficients are assumed to be binary.
- 6.
Be aware that compared to methods in the first category, representations in this category typically need additional memory to store the information about the resulting model itself, e.g., the basis of the subspace in PCA, the cluster means in VQ.
- 7.
Therefore, b needs to be a common divisor of W and H.
- 8.
Here by default, the peak value of the imagery data is normalized to 1.
- 9.
Notice that to represent a d-dimensional subspace in a D-dimensional space, we need only specify a basis of d linearly independent vectors for the subspace. We may stack these vectors as rows of a \(d \times D\) matrix. Any nonsingular linear transformation of these vectors span the same subspace. Thus, without loss of generality, we may assume that the matrix is of the normal form \([I_{d\times d},G]\), where G is a \(d \times (D - d)\) matrix consisting of the so-called Grassmannian coordinates.
- 10.
Notice that if one uses a preselected basis, such as discrete Fourier transform, discrete cosine transform (JPEG), or wavelets (JPEG-2000), there is no such overhead.
- 11.
We also need a very small number of binary bits to store the membership of the vectors. But those extra bits are insignificant compared to \(\Omega \) and often can be ignored.
- 12.
In fact, the minimal \(\Omega \) can also be associated with the Kolmogorov entropy or with the minimum description length (MDL) of the imagery data.
- 13.
For instance, if we take all the \(b \times b\) blocks and scramble them arbitrarily, the scrambled image would be fit equally well by the same hybrid linear model for the original image.
- 14.
This is not to be confused with the subscript i used to indicate different segments \(\mathcal{I}_{i}\) of an image.
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Vidal, R., Ma, Y., Sastry, S.S. (2016). Image Representation. In: Generalized Principal Component Analysis. Interdisciplinary Applied Mathematics, vol 40. Springer, New York, NY. https://doi.org/10.1007/978-0-387-87811-9_9
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