*... axis Along Y-axis Examples of Linear Transformations 2D reflection Along X-axis Along Y-axis Properties of Linear Transformations Any linear transformation ...*

*Proof: We find a basis: Define a linear transformation VW: We show the basis: Spans: T:VW. We show Independence Suppose Example: V=Fm W=Fn.*

*Linear Transformation Affine Transformation Simple 3D Rotation Full 3D Rotation 3D Rotation + Translation Just like 2D case Linear Algebra and SVD (Some slides adapted ...*

*... Implicit representations Explicit representations Transforming geometry Translation Linear transformations Geometry of 2D linear trans. Linear transformation ...*

*From previous lectures … Transformation groups Affine transformation Not a linear transformation! Can be made linear in HOMOGENEOUS COORDINATES Affine group (contd ...*

*Pointwise: .^ Euclidean transformations Stretching = tilting and projecting (with weak perspective) Linear Transformation Affine Transformation Files Functions Format ...*

*... Psym = subset of Sym that are positive definite Orth = set of all orthogonal tensors Orth + = set of all rotations Second rank tensors provide a linear transformation of ...*

*... ey+f)= (u(x,y),v(x,y)) Affine transformations of the plane An affine function has the form: (x,y) (ax+by+c, dx+ey+f) An affine function is a linear transformation ...*

*... the information of a single bitplane Piecewise Linear Transformations Piecewise Linear Transformations BP 7 BP 5 BP 0 Exercise: How does the transformation ...*

*Shifting and rescaling: linear transformations Original data x1, x2, . . . xn Linear transformation: x* = a + bx, (intercept a, slope b) x x* 0 a Shifts data by a ...*

*... Linear system of equations/column space Mathematical Background Mathematical Background Example – Null Space Mathematical Background Linear Transformation ...*

*... falls at the top of the weak range, indicating a stronger relationship that the one represented by the linear fit line. This result hints that a squared transformation ...*

*Finite Elements: Basis functions 1-D elements coordinate transformation 1-D elements linear basis functions quadratic basis functions cubic basis functions*

*Linear Transformation. Transformations consist of making the scale larger, so that negative ...*

*... equations to get X0 The same operation to obtain Y0 Physical interpretation: 6 parameters: Cx, Cy, , , Dx0, Dy0, and in linear form: Polynomial Transformation ...*

*... combinations of points and lines also work 3 collinear points in either image are a degenerate configuration preventing a unique solution Direct Linear Transformation ...*

*... is useful only if the ratio is large For a small range the log function is almost linear Square Root Transformation: For a ...*

*Homogeneous Matrices N-dimensional affine transformation is n+1 dimensional linear transformation A 3D homogeneous transformation matrix is a 4x4 matrix Treat 3D ...*

*V1 V2 Percentage of explained variance Number of non-zero loadings * * * SVD on microarray data is a linear transformation from genes x arrays space to eigen-genes x ...*

*... transformation that minimises the sum of squared differences between images: Step 1: Linear transformation (12-parameter affine) Step 2: Non-linear transformation ...*

*... Point representation We can use a column vector (a 2x1 matrix) to represent a 2D point x y A general form of linear transformation ...*

*Power law transformation Display distortion correction Power law – contrast manipulation Piece-wise Linear Transformation Contrast Stretching Gray level Slicing Bit ...*

*You just completed the transformation of Given the following function, How will the graph look? Given the following function, How will the graph look?*

*... JC Science, 290: 2319–2323, 2000 Nonlinear Dimensionality Reduction by Locally Linear Embedding Roweis and Saul Science, 2323-2326, 2000 Linear transformation ...*

*Kernel Functions Instead of trying to fit a non-linear model, we can map the problem to a new space through a non-linear transformation and use a linear model in the new ...*