Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

Optimal Latching Control of a Wave Energy Device in Regular and Irregular Waves

https://hal.archives-ouvertes.fr/hal-00463034/document

Comparison of latching control strategies for a heaving wave energy device in random sea

Control of wave energy converters (WECs) has been very often limited to hydrodynamic control to absorb the maximum energy possible from ocean waves. This generally ignores or significantly simplifies the performance of real power take-off (PTO) systems. However, including all the required dynamics and constraints in the control problem may considerably vary the control strategy and the power output. Therefore, this paper considers the incorporation into the model of all the conversion stages from ocean waves to the electricity network, referred to as wave-to-wire (W2W) models, and identifies the necessary components and their dynamics and constraints, including grid constraints. In addition, the paper identifies different control inputs for the different components of the PTO system and how these inputs are articulated to the dynamics of the system. Examples of pneumatic, hydraulic, mechanical or magnetic transmission systems driving a rotary electrical generator, and linear electric generators are provided.

https://www.mdpi.com/1996-1073/9/7/506/pdf

A Review of Wave-to-Wire Models for Wave Energy Converters

Towards simulating floating offshore oscillating water column converters with Smoothed Particle Hydrodynamics

The aim of this paper is to provide the key theoretical and numerical aspects of the NEMOH software. NEMOH is a numerical solver for computations of first order hydrodynamic coefficients in frequency domain. It is based on linear free surface potential flow theory. It has been developed in Ecole Centrale de Nantes for 30 years. It has been released in open source in January 2014. 
This paper starts with recalling the assumptions and the resulting boundary value problem for the free surface flow around a body with an arbitrary condition for the normal velocity on the body surface. Then, it is recalled how the 3D flow problem can be transformed into the 2D problem of a source distribution on the body surface using Green's second identity and the appropriate Green function. Discretization of the mathematical problem using the constant panel method leads to a linear matrix problem whose coefficients are the influence coefficients. Depending on the distance between the source and field point, the influence coefficients are calculated analytically or approximately. Irregular frequencies are removed by adding constraints on the interior flow problem. Hydrodynamic coefficients for usual diffraction and radiation problems can be obtained by specifying to the solver the appropriate body conditions. The flexibility in specifying the body conditions allows dealing with complex structures, e.g deformable wave energy converters. Eventually, it is explained how far field coefficients and free surface elevation can be obtained from the solver outputs.

Theoretical and numerical aspects of the open source BEM solver NEMOH

Simple analytical relations for the bow wave generated by a ship in steady motion are given. Specifically, simple expressions that define the height of a ship bow wave, the distance between the ship stem and the crest of the bow wave, the rise of water at the stem, and the bow wave profile, explicitly and without calculations, in terms of the ship speed, draught, and waterline entrance angle, are given. Another result is a simple criterion that predicts, also directly and without calculations, when a ship in steady motion cannot generate a steady bow wave. This unsteady-flow criterion predicts that a ship with a sufficiently fine waterline, specifically with waterline entrance angle 2αE smaller than approximately 25°, may generate a steady bow wave at any speed. However, a ship with a fuller waterline (25°<2αE) can only generate a steady bow wave if the ship speed is higher than a critical speed, defined in terms of αE by a simple relation. No alternative criterion for predicting when a ship in steady motion does not generate a steady bow wave appears to exist. A simple expression for the height of an unsteady ship bow wave is also given. In spite of their remarkable simplicity, the relations for ship bow waves obtained in the study (using only rudimentary physical and mathematical considerations) are consistent with experimental measurements for a number of hull forms having non-bulbous wedge-shaped bows with small flare angle, and with the authors' measurements and observations for a rectangular flat plate towed at a yaw angle.

Simple analytical relations for ship bow waves

The recent progress of computers and numerical algorithms enables us today to use the translating and pulsating Green function in panel methods for seakeeping calculations. Two different methods of calculations of this function, its derivatives and their integrations over panels or segments, are briefly presented and have been introduced into the seakeeping codes Aquaplus developed at Ecole Centrale de Nantes and Poseidon at Laboratoire d'Etudes Aerodynamiques (LEA), Centre d'Etudes Aerodynamiques et Thermiques (CEAT). Both codes interchange the Fourier and boundary integrals on panels or waterline segments, the last part being performed analytically. These methods have been used to compute flows around Wigley or Series 60 model ships. To check the numerical results, an experimental setup has been developed at the CEAT that measures forces and moments on a model in forced harmonic oscillations of pitch or heave. Tests have been performed in the recirculating water channel of Ecole Centrale de Nantes on two L = 1.2 m Series 60 models of C B = 0.6 and 0.8 block coefficients. Unsteady wave patterns have been recorded using a resistive wave probe. The experimental results are compared with the numerical ones.

Comparison between numerical computations and experiments for seakeeping on ship models with forward speed

Measurements of the bow waves generated by a rectangular flat plate, immersed at a draught D = 0.2 m, towed at constant speed U = 1.75 m s−1 in calm water and held at a heel angle 10° and a series of nine yaw angles α = 10°, 15°, 20°, 25°, 30°, 45°, 60°, 75° and 90° are reported. The measurements show that bow wave unsteadiness is significantly larger for the flat plate towed at yaw angles 30° ≤ α ≤ 90° than at 10° ≤ α ≤ 20°, which are associated with the unsteady and overturning bow wave regimes, respectively, separated by the boundary with g ≡ acceleration of gravity. These measurements of bow wave unsteadiness provide preliminary experimental validation of the foregoing simple theoretical relation for the boundary between the unsteady and overturning bow wave regimes for non-bulbous wedge-shaped ship bows with insignificant rake and flare. Extension of this relation to more complicated ship bows, notably bows with rake and flare, is also considered.

Boundary between unsteady and overturning ship bow wave regimes

The basic computational task of the thin-ship theory of free-surface potential flow about a ship that advances at constant speed along a straight path in calm water, of large depth and lateral extent, is considered. Specifically, a straightforward method for evaluating the pressure and the wave profile at a ship hull (the wave drag, hydrodynamic lift and pitch moment, and sinkage and trim are also considered) in accordance with Michell’s thin-ship theory is given. A main ingredient of this method is a simple analytical approximation to the local-flow component in the expression for the Green function (associated with the classical Michell–Kelvin linearized free-surface boundary condition) of thin-ship theory. This practical Green function is used to evaluate and analyze steady flow about a four-parameter family of ship bows with rake and flare. In particular, the variations of the bow-wave height and location with respect to the draft-based Froude number, the entrance angles at the top and bottom waterlines, and the rake angle are explored via a systematic parametric study. This parametric study provides estimates—immediately useful for design—of the influence of rake and flare on the height and the location of a ship bow wave, and shows that rake and flare effects can be significant, especially at low Froude numbers.

Gérard Delhommeau

Papers

Theoretical and numerical aspects of the open source BEM solver NEMOH

Simple analytical relations for ship bow waves

Comparison between numerical computations and experiments for seakeeping on ship models with forward speed

Boundary between unsteady and overturning ship bow wave regimes

Thin-ship theory and influence of rake and flare