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Sxi xj 19 faculty of engineering robotics technology

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Unformatted text preview: ogy MECH 4041 B. Eng (Hons.) Mechatronics S. Venkannah Mechanical and Production Engineering Department Substituting this relation into the expression for the cross-ratio gives In summary, the cross-ratio is an invariant of any sets of four collinear points in projective correspondence. It is unaffected by the relative position of the line or the position of the optical centre, as shown in Figure 7. Figure below: The cross-ratio of every set of four collinear points shown in this figure has the same value. Recognition using invariants There are two stages to model-based recognition using invariants: 1. Model acquisition. Models of objects to be recognized are acquired directly from images. For planar objects, this involves computing their plane projective invariants. Their outline is also stored for the verification process. 2. Recognition. Invariants are computed for geometric invariants in the target image. If the invariant value corresponds to one in the model library, a recognition hypothesis is generated. This hypothesis is either confirmed or denied by verification: The model outline from the acquisition image is projected onto the target image. If the projected edges overlap image edges sufficiently then the hypothesis is verified. 20 Faculty of Engineering Robotics Technology MECH 4041 B. Eng (Hons.) Mechatronics S. Venkannah Mechanical and Production Engineering Department Invariants: They provide a simple means of comparison They can provide position and orientation information (i.e. Moments etc..) Too simplistic for some applications Do not give a unique means of identification Neural nets: Neural nets have seen an explosion of interest since their re-discovery as a pattern recognition paradigm in the early 1980s. the value of some of the applications for which they are used may be arguable, but there is no doubt that they represent a tool of great value in various areas generally regarded as “difficult”, particularly speech and visual pattern recognition. Most neural approaches are based on combinations of elementary processors (neurons), each of which takes a number of inputs and generates a single output. Associated with each input is a weight, and the output (in most cases) is then a function of the weighted sum of inputs; this function may be discrete or continuous, depending on the variety of network in use. A simple neuron is shown in the model below. The inputs are denoted by ν1,, ν2,…. And the weights by w1, w2, …: the total input to the neuron is then the sum of the weighted inputs. ν1 w1 ν2 f (ν,w) Output y Σ νn Wn Where θ is a threshold associated with this neuron. Also associated with the neuron is a transfer function f(x) which provides the output; common examples are f(x) = 0 if x ≤ 0 1 if x > 0 f(x) = 1 /(1 + e-x) this model saw a lot of enthusiastic use during an early phase, culminating in Rosenblatt’s perceptron. 21 Faculty of Engineering Robotics Technology MECH 4041 B. Eng (Hons.) Mechatronics S. Venkannah Mechanical and Production Engineering Department The general idea of collections of these neurons is that they are interconnected (so that the output of one becomes the input of another, or others)- this idea mimics the high level of interconnection of elementary neurons found in brains, which is thought to explain the damage resistance and recall capabilities of humans. Such an interconnection may then take some number of external inputs and deliver up some number of external outputs. What lies between then specifies the network: This may mean a large number of heavily interconnected neurons, or some highly structured (e.g., layered) interconnection, or, pathologically nothing. Typical uses of such a structure Classification: if the output vector (m –dimensional) is binary and contains only a single one, the position of the one classifies the input pattern into one of m categories. Auto-association: Some uses of neural networks cause then to re-generate the input pattern at the outputs (so m=n and vi = yi); the purpose of this may be to derive a more compact vector representation from within the network intervals. General association: At their most interesting, the vectors v and y represent patterns in different domains, and the network is forming a correspondence between them. Feed forward networks: The standard approach to use such networks is to obtain a training set of data – a set of vectors for which the ‘answer’ is already known. This used to teach a network with some training algorithm, such that the network can perform the association accurately. Then, in classification mode, unknown patterns are fed into the net and it produces answers based on generalizing what it has learned. Back propagation proceeds by comparing the output of the network to that expected, and computing an error measure based on sum of square differences. Back propagation trains strictly layered networks in which it is assumed that at least one layer exists between input and output. Back propagation algorithm (Fr...
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