Orthogonal Projection

A polyhedral object in 3-dimensional space is often well represented by a set of points and line segments that act asits features. By a nice viewpoint of an object we mean a projective view in which all (or most) of the features of theobject, relevant for some task, are clearly visible. Such a view is often called a non-degenerate view or projection. Inthis paper we are concerned with computing non-degenerate orthogonal and perspective projections of sets of pointsand line segments (objects) in 3-dimensional space. We outline the areas in which such problems arize, discuss recentresearch on the computational complexity of these problems, illustrate the fundamental ideas used in the design ofalgorithms for computing non-degenerate projections, and provide pointers to the literature where the results can befound.Keywords: Data visualization, nice viewpoints, knot theory, computer vision, graph drawing, computer graphics,degeneracies, general position, orthogonal projections, perspective projections, robustness of algorithms, computa-tional geometry.

Orthogonal subspace projection (OSP) has been successfully applied to hyperspectral image processing. In order for OSP to be effective, the number of bands must be no less than that of signatures to be classified so that there are sufficient dimensions to accommodate individual signatures to discriminate one another via orthogonal projection. This intrinsic constraint is not an issue for hyperspectral images since they generally have hundreds of bands which are more than the number of signatures resident within images. It, however, may not be true for multispectral images where the number of signatures to be classified is greater than the number of bands such as 3-band SPOT images. This paper presents a generalization of OSP, called generalized OSP (GOSP) to relax this constraint in such a fashion that OSP can be extended to multispectral image processing in an unsupervised fashion. The idea of GOSP is to create new additional band images nonlinearly from original multispectral images so as to achieve sufficient dimensionality prior to OSP classification. It is then followed by an unsupervised OSP classifier, called automatic target detection and classification algorithm (ATDCA) for classification. The effectiveness of the proposed GOSP is evaluated by a 3-band SPOT and a 4-band Landsat MSS images. The experimental results has shown that GOSP significantly improves the classification performance of OSP.

The simplex growing algorithm (SGA) developed by Chang et al. (A growing method for simplex-based endmember extraction algorithms. IEEE Transactions on Geoscience and Remote Sensing 44(10): 2804–2819, 2006b) has been used for finding endmembers and was studied in Chap. 10 of Chang (Real time progressive hyperspectral image processing: endmember finding and anomaly detection, Springer, New York, 2016). It can be considered a sequential version of the well-known N-finder endmember finding algorithm (N-FINDR) developed by Winter (Proceedings of 13th international conference on applied geologic remote sensing, Vancouver, BC, Canada, pp. 337–344, 1999a; Image spectrometry V, Proceedings of SPIE 3753, pp. 266–277, 1999b) to find endmembers one after another by growing simplexes one vertex at a time. However, one of the major hurdles for N-FINDR and SGA is the calculation of a simplex volume (SV), as discussed in Chap. 2, which poses a great challenge in designing any algorithms using a SV to find endmembers. This chapter develops an orthogonal projection (OP)-based approach to SGA, called OPSGA, which essentially resolves this computational issue. The idea is based on a geometric SV (GSV) from structures of simplexes. If we consider a j-vertex simplex S j specified by j previously found endmembers as a base and the next endmember mj+1 to be found as a new vertex to be added to S j to form a new (j + 1)-vertex simplex, Sj+1, then calculating the GSV of this new (j + 1)-vertex simplex, Sj+1, is equivalent to multiplying the GSV of S j , which is considered a base with the OP on the base S j from mj+1, which is considered its height. As a result, finding mj+1 to yield a Sj+1 with the maximal GSV is equivalent to finding mj+1 with the maximal OP on S j . On this interpretation, OPSGA converts the issue of calculating a determinant-based SV (DSV) to finding OP without actually computing matrix determinants. Accordingly, OPSGA can be considered a technique for calculating a GSV by OP (GSV-OP). To further reduce the computational complexity, OPSGA is also extended to a recursive Kalman filtering-like recursive hyperspectral sample processing of OPSGA (RHSP-OPSGA), which has several advantages and benefits in terms of computational savings and hardware implementation over N-FINDR and SGA.

Revisitation of the product of two orthogonal projectors

To compare the advantages and disadvantages of different techniques for fitting Zernike polynomials to surfaces. Two different methods, Orthogonal Projection and Gram-Schmidt orthogonalization, are compared in terms of speed and performance at fitting a complex object. Orthogonal Projection provides an extremely rapid fitting of a surface, but leaves residual high frequency noise. The Gram-Schmidt technique provides a more accurate fit of the original object, but consumes much more computing time. Orthogonal Projection, and its associated noise, may be tolerated in classifying corneal topography. Orthogonal Projection has a distinct advantage in calculation time over other methods for fitting surfaces. This advantage may be exploited in cases where accurate surface fitting is not necessary, but only general features need to be extracted for classification. If fit accuracy is needed, then slower fitting techniques, such as Gram-Schmidt, should be used.

Effective methods often rely on simple mathematical operators. Among these operators, orthogonal projections have been widely used because of their simplicity in compensating for detrimental factors. This efficiency depends largely on the way these tools are prepared. This article links the mathematical basics of orthogonal projections to the notion of vectoral subspaces, highlighting which information should be removed in the process and the important practical properties concerned with optimizing this technique. This review covers several methods involving orthogonal projections and focuses specifically on their practical use. This concerns the identification of detrimental information and its removal together with adjusting the dimension of the projection. The methodology discussed in this review will enable the reader to optimize orthogonal projections for any given situation.

The structure of theC*-algebra with identity generated by two orthogonal projections is well understood. All irreducible representations of this algebra are either two-dimensional or one-dimensional. The situation becomes unpredictable in the case of theC*-algebra generated by three orthogonal projections. Even in the more specific case when two of the projections commute, the algebra under consideration may have infinite dimensional irreducible representations.

With the progress of hardware technology, the development of real‐time adaptive algorithms is becoming more and more important. The essential questions concerning adaptive algorithms are: 1) increasing the speed of the algorithm; and 2) speeding up the convergence. It seems that the block signal processing and the orthogonal projection on the subspace of state‐vectors are effective methods to solve the forementioned questions 1) and 2), respectively.This paper proposes a block orthogonal projection algorithm (BOPA) which combines the block processing with the orthogonal projection operation. Next, an analysis of this algorithm is presented and the fundamental properties of the relation between the block length and the convergence characteristics are shown. Also, the efficiency of the proposed algorithm is verified by computer simulation. This algorithm is based on a block‐by‐block adaptation where the operation of orthogonal projection on the state‐vector space is performed repetitively for arbitrary blocks.

We report the use of dual-domain regression models, which were built utilizing a wavelet prism decomposition and paired with transfer by orthogonal projection, for the calibration transfer of near-infrared (NIR) spectra. The new method is based on obtaining specific frequency components for a spectrum via wavelet analysis, projecting the frequency components of the primary instrument onto the subspace orthogonal to the mean instrumental difference between spectra from the primary and the secondary instrument, and weighting each frequency component model according to the cross-validation error of the frequency components of the projected primary instrument's spectra to generate a stacked ensemble model robust to contributions to the spectra from instrumental variations. The method, which does not require property values from the secondary data set, is tested on three NIR data sets, and is compared with orthogonal projection in the wavelength domain, orthogonal signal correction, and with model updating approaches. For the data sets we examined, we show that the prediction performance of the new method is competitive with orthogonal projections in the wavelength domain, as well as orthogonal signal correction and model updating approaches, both of which require property values for spectra from the secondary instrument. Examination of the spectral data reconstructed from the projected frequency components indicates that aspects of the data that may be attributable to instrumental or physical phenomena (i.e., instrumental baseline shifts or discretized intensity changes which may be attributed to scatter) are suppressed, but those associated with the chemical phenomena are retained. The benefits of orthogonal projection on each individual frequency component are further corroborated by the fact that the models based on frequency component projection generalize better to unseen instruments compared with the other methods.

The equalization using the traditional blind estimation is based on the channel outputs and knowledge of the probabilistic property of input signal. But it is difficult for conventional implementation with on-line processing because the traditional algorithms need high-order momentum. We present a new method using the orthogonal projection, in order to enable the on-line processing of blind equalization. First, we estimate the characteristics of the channel using both the skewness and the kurtosis of the output of the channel. Secondly, we will design an equalizer using the orthogonal projection onto the received signal space. The proposed method is based on designing an equalizer parameter so that the matrix P/sub N,N/=W/sub N,N/H/sub N,N/ may be the orthogonal projection matrix onto the received signal space, where W/sub N,N/ and H/sub N,N/ demote the impulse response matrix of an equalizer to be designed and that of the channel respectively. The impulse response matrix of an equalizer is basically expressed with Moore-Penrose inverse matrix. The proposed blind equalization algorithm can be implemented with on-line processing, it is expected that the convergence characteristics of the proposed equaliser are better than that of the traditional device.

A property of orthogonal projectors

Sum and intersection of linear subspaces in a vector space over a field are fundamental operations in linear algebra. The purpose of this survey paper is to give a comprehensive approach to the sums and intersections of two linear subspaces and their orthogonal complements in the finite-dimensional complex vector space. We shall establish a variety of closed-form formulas for representing the direct sum decompositions of them-dimensional complex column vector space 𝔺mwith respect to a pair of given linear subspaces 𝒨 and 𝒩 and their operations, and use them to derive a huge amount of decomposition identities for matrix expressions composed by a pair of orthogonal projectors onto the linear subspaces. As applications, we give matrix representation for the orthogonal projectors onto the intersections of a pair of linear subspaces using various matrix decomposition identities and Moore–Penrose inverses; necessary and su˚cient conditions for two linear subspaces to be in generic position; characterization of the commutativity of a pair of orthogonal projectors; necessary and su˚cient conditions for equalities and inequalities for a pair of subspaces to hold; equalities and inequalities for norms of a pair of orthogonal projectors and their operations; as well as a collection of characterizations of EP-matrix.

A new stable blind multiuser detector based on signal subspace orthogonal projection is presented. By carrying out a pseudo-eigendecomposition (PED) analysis, a technique proposed for the first time, the signal of the interested user can be approximated by the orthogonal projection of a received multiuser signal onto the related subspace of the interested user. Simulations show that compared with the previous subspace-based MMSE blind multiuser detector, the new detector offers better performance without leading to an increase in the computational complexity.

We study the convergence of the iterations in a Hilbert space $V,x_{k+1}=W(P)x_k, W(P)z=w=T(Pw+(I-P)z)$, where $T$ maps $V$ into itself and $P$ is a linear projection operator. The iterations converge to the unique fixed point of $T$, if the operator $W(P)$ is continuous and the Lipschitz constant $\left\|(I-P)W(P)\right\|<1$. If an operator $W(P_1)$ satisfies these assumptions and $P_2$ is an orthogonal projection such that $P_1P_2=P_2P_1=P_1$, then the operator $W(P_2)$ is defined and continuous in $V$ and satisfies $\left\|(I-P_2)W(P_2)\right\|\leq \left\|(I-P_1)W(P_1)\right\|$.

In the design and manufacturing of shell structures it is frequently necessary to construct trimming curves on surfaces. The novel method introduced in this paper was formulated to be coordinate independent and computationally efficient for a very general class of surfaces. Generality of the formulation is attained by solving a tensorial differential equation that is formulated in terms of local differential properties of the surface. In the method proposed here, a space curve is mapped onto the surface by tracing a surface curve whose points are connected to the space curve via surface normals. This surface curve is called to be an orthogonal projection of the space curve onto the surface. Tracing of the orthogonal projection is achieved by solving the aforementionned tensorial differential equation. For an implicitely represented surface, the differential equation is solved in three-space. For a parametric surface the tensorial differential equation is solved in the parametric space associated with the surface representation. This method has been tested on a broad class of examples including polynomials, splines, transcendental parametric and implicit surface representations. Orthogonal projection of a curve onto a surface was also developed in the context of surface blending. The orthogonal projection of a curve onto two surfaces to be blended provides not only a trimming curve design tool, but it was also used to construct smooth natural maps between trimming curves on different surfaces. This provides a coordinate and representation independent tool for constructing blend surfaces.