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Review of Linear algebra Fourier transform Image Analysis - Lecture 2 Fredrik Kahl 4 September 2009 Fredrik Kahl Image Analysis - Lecture 2 Review of Linear algebra Fourier transform Lecture 2 Contents ◮ Linear Algebra ◮ Linear spaces, bases ◮ Scalar product, orthogonality, ON-bases ◮ Sub-spaces, projections ◮ The Fourier transform ◮ Deﬁnition, examples, 1D and 2D ◮ Discrete Fourier Transform, Fast Fourier Transform ◮ Application to Images ◮ Master thesis proposal of the day Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Linear algebra: Linear spaces The following linear spaces are well-known: x1 n : all n × 1-matrices, x = . , ◮ R . . xi ∈ R xn x1 n : all n × 1-matrices, x = . , ◮ C . . xi ∈ C xn Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Basis Deﬁnition e1 , . . . en ∈ Rn is a basis in Rn if ◮ they are linearly independent ◮ they span Rn . Example (3D space) e1 , e2 , e3 ∈ R3 is a basis in R3 if they are not located in the same plane. Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Canonical Basis Example (canonical basis) 1 0 0 0 1 0 e1 = . , e2 = . , ... en = . . . . . . . 0 0 1 is called the canonical basis in Rn . x1 . x = . = x1 e1 . . . + xn en . . xn Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Coordinates Let e1 , e2 , . . . , en be a basis. Then for every x there is a unique set of scalars ξi such that n x= ξi e i . i=1 These scalars are called the coordinates for x in the basis e1 , e2 , . . . , en . Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Image matrix An M × N image, f , is described by the matrix f (0, 0) ... f (0, N − 1) . .. . f = . . . . . , f (i, j) ∈ C f (M − 1, 0) . . . f (M − 1, N − 1) f (i, ·)=i:th row, f (·, j)=j:th column. Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Row-stacking Row-stacking: f T (0, ·) f T (1, ·) ˜= f . ∈ RMN (CMN ) . . f T (M − 1, ·) Properties: f + g = ˜+ g f ˜ λf = λ˜, f λ∈R A linear space! Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Canonical basis 0 ... 0 ... 0 . . 1 . . χ(i, j) = . . , 0 ... 0 ... 0 with the 1 at position (i, j). Using this canonical basis we can write f = f (i, j)χ(i, j) . i,j Idea for image transform: Choose another basis that is more suitable in some sense. Image matrices can thus be seen as vectors in a linear space. Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Scalar product Deﬁnition Let A be a (complex) matrix. Introduce ¯ A∗ = (A)T . Deﬁnition Let x and y be two vectors in Rn (Cn ). The scalar product of x and y is deﬁned as x ·y = ¯ xi yi = x ∗ y . Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Orthogonality Deﬁnition The scalar product of two matrices (images) is deﬁned as M−1 N−1 f ·g = ¯(i, j)g(i, j) . f i=0 j=0 x, y ∈ R(C) are orthogonal if x · y = 0. This is often written x⊥y ⇔ x ·y =0 . The length or the norm of x is deﬁned as ||x|| = |xi |2 = (x ∗ x)1/2 . Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Coordinates: ξ2 x e2 e1 ξ1 Norm: x2 x = (x1 , x2 ) ||x|| x1 Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Orthogonal basis (ON-basis) Deﬁnition {e1 , . . . , en } is an orthonormal (ON-) basis in Rn (Cn ) if 0 i=j ei · ej = 1 i=j Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Coordinates in ON-basis Theorem Assume that {e1 , . . . , en } is orthonormal (ON) basis and n x= ξi e i . i=1 Then n ξi = ei · x = ei∗ x, ||x||2 = |ξi |2 i=1 Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Illustration e2 · x x e2 e1 e1 · x Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Orthogonal projection Deﬁnition Let {a1 , . . . , ak } ∈ Rn , k ≤ n, span a linear subspace, π, in Rn , i.e.: k π = {w |w = xi ai , xi ∈ R} . i=1 The orthogonal projection of u ∈ Rn on π is a linear mapping P, such that uπ = Pu and deﬁned by min ||u − w || = ||u − uπ || . w ∈π Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Orthogonal projection (ctd.) The orthogonal projection is characterized by 1. uπ ∈ π 2. u − uπ ⊥w for every w ∈ π u u − uπ π uπ Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Example: Matrix-basis What is the orthogonal projection of f 1 2 3 f = 4 5 6 7 7 7 onto the space spanned by (e1 , e2 , e3 ) 1 1 1 1 1 1 1 0 −1 1 1 1 e1 = 1 1 1 , e2 = √ 0 0 0 , e3 = √ 1 0 −1 3 6 −1 −1 −1 6 1 0 −1 1 1 1 Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Example: Matrix-basis (ctd.) Since (e1 , e2 , e3 ) is orthonormal the coordinates are √ √ x1 = f · e1 = 14, x2 = f · e2 = −15/ 6, x3 = f · e3 = −4/ 6. The orthogonal√ is projection√ then ˆ = 14e1 − 15/ 6e2 − 4/ 6e3 f 1 5 1 2 3 1.5 2 6 26 f = 4 5 6 , ˆ = 4 4 3 f 2 53 , 1 7 7 7 1 5 6.5 7 6 76 Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Example: ’Face’-basis What is the orthogonal projection of f onto the space spanned by (e1 , e2 , e3 ) Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Example: ’Face’-basis (ctd.) Since (e1 , e2 , e3 ) is orthonormal, the coordinates are x1 = f · e1 = −2457, x2 = f · e2 = 303, x3 = f · e3 = −603. The orthogonal projection is then ˆ = −2457e1 + 303e2 − 603e3 f Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Uniqueness of the projection Let a ∈ π and b ∈ π be two solutions to the minimisation problem. Set 2 f (t) = u − ta − (1 − t)b = ... 2 = u−b + t2 a − b 2 − 2t(a − b) · (u − b), t ∈ R. This is a second degree polynomial with minimum in t = 0 and t = 1 ⇒ f (t) is a constant function and thus ⇒ a = b. Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Characterization of the projection Let f (t) = u − uπ + ta 2 , where a ∈ π. It follows that f ′ (0) = 2(u − uπ ) · a = 0, i.e. (u − uπ ) ⊥ a. Conversely: Assume w ∈ π. The property that (u − uπ ) ⊥ a, for every a ∈ π gives that 2 2 u −w = u − uπ + uπ − w = 2 2 2 u − uπ + uπ − w ≥ u − uπ , i.e. uπ solves the minimization problem. Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection An important result Let A = [a1 . . . ak ] be a n × k matrix and π = {w |w = Ax, xi ∈ Rn } Lemma If {a1 , . . . , ak } are linearly independent Rn then A∗ A is invertible. Proof: Do it on your own. (Use SVD if you are familiar with it.) Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Projection onto the subspace spanned by A Theorem if the columns of A are linearly independent, then the projection of u on π is given by uπ = x1 a1 + . . . + xk ak , x = (A∗ A)−1 A∗ u . Proof: Use the characterization of the projection (above). a∗ (u − uπ ) = 0 i ⇒ A∗ (u − Ax) = 0 ⇒ A∗ u = A∗ Ax ⇒ x = (A∗ A)−1 A∗ u Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection The pseudo-inverse Deﬁnition A+ = (A∗ A)−1 A∗ is called the pseudo-inverse of A. Observe that if A is quadratic and invertible then A+ = A−1 . Theorem If {a1 , . . . , ak } are orthonormal, then the projection of u on π is given by uπ = y1 a1 + . . . + yk ak , yi = a∗ u . i Proof: This follows from A∗ A = I. Fredrik Kahl Image Analysis - Lecture 2 Linear space, basis Review of Linear algebra Scalar product Fourier transform Orthogonal projection Illustration u y2 a2 uπ y1 a1 π Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Fourier transform Deﬁnition Let f be a function from R to R. The Fourier transformen of f is deﬁned as +∞ (Ff )(u) = F (u) = e−i2πxu f (x)dx . −∞ Theorem Under the right assumptions on f , the following inversion formula +∞ f (x) = ei2πux F (u)du −∞ holds. Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Examples Example δ(x) → 1(u) sin(2πu) rect(x) → 2 = 2 sinc(2πu) 2πu Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Illustrations F 1 x u F 1 2 −1 1 x u Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Properties c1 f1 (x) + c2 f2 (x) → c1 F1 (u) + c2 F2 (u) (linearity) 1 u f (λx) → F( ) (scaling) |λ| λ f (x − a) → e−i2πua F (u) (translation) e−i2πxa f (x) → F (u + a) (modulation) f (x) → F (−u) (conjugation) df → 2πiuF (u) (differentiation I) dx dF −2πixf (x) → (differentiation II) du Example: δ(x − 1) → e−i2πu Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Review of Distributions ∞ Test functions: ϕ ∈ C0 . A distribution u is a functional (with certain constraints) acting on the space of test functions. This action is written (u, ϕ). E.g. (δ, ϕ) = ϕ(0), Dirac delta function. Had u been a ordinary function, then (u, ϕ) = u(x)ϕ(x)dx, i.e. the scalar product. Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Properties of Distributions Properties ◮ Differentiation: (u ′ , ϕ) = −(u, ϕ′ ). ◮ Convolution: u ∗ ϕ(x) = (u, ϕ(x − ·)). E.g. δ ∗ ϕ(x) = (u, ϕ(x − ·)) = ϕ(x). ◮ ˆ ˆ Fourier transform: (u , ϕ) = (u, ϕ). E.g. ˆ ˆ ˆ (δ, ϕ) = (δ, ϕ) = ϕ(0) = ϕ(x)dx = (1, ϕ) ˆ i.e. δ = 1. Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform The discrete Fourier transform (DFT) Sample f (x) and F (u). x u 0 N −1 0 N −1 ∆x ∆u N−1 F (n∆u ) ∼ e−i2πk ∆x n∆u f (k∆x )∆x , k =0 N−1 f (k∆x ) ∼ ei2πk ∆x n∆u F (n∆u )∆u . n=0 Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform The discrete Fourier transform (DFT) (ctd.) 1 This works particularly well if ∆x ∆u = N: N−1 1 F (n∆u ) ∼ e−i2πkn/N f (k∆x ), ∆x k =0 N−1 1 1 f (k∆x ) ∼ ei2πkn/N F (n∆u ). N ∆x n=0 F and f are extended to periodic functions with periods N∆u and N∆x respectively. Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Deﬁnition of the discrete Fourier transform Let the vector (f (0), f (1), . . . , f (N − 1)) . represent the discretized version of f (x). Deﬁnition The discrete Fourier Transformen (DFT) of f is N−1 ku F (u) = f (k)ωN , u = 0, . . . , N − 1 , k =0 where ωN = e−i2π/N . Represent the sequence F (u) with the vector (F (0), F (1), . . . , F (N − 1)) . Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Important assumption All sequences are assumed to be period with period N. 0 N −1 21 3 Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Properties of DFT There are similar formulas for the discrete Fourier Transform (as compared to that of the continuous Fourier Transform), e.g. (f (−k0 ), f (1 − k0 ), . . . , f (N − 1 − k0 )) → N−1 → f (k − k0 )ω ku = [l = k − k0 ] k =0 N−1−k0 = f (l)ω (l+k0 )u = l=−k0 N−1−k0 N−1 k0 u lu k0 u =ω f (l)ω = [f periodic] = ω f (l)ω lu = l=−k0 l=0 k0 u −i2πk0 /N =ω F (u) = e F (u) Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform DFT in matrix form Let f (0) F (0) . . . . f = , . F = . . f (N − 1) F (N − 1) Deﬁnition The Fourier Matrix FN is given by 1 1 1 ... 1 1 ω ωN2 ... ωNN−1 N 2 4 2(N−1) FN = 1 ωN . ωN ... ωN . . . . . . .. . . . . . . . N−1 2(N−1) (N−1)(N−1) 1 ωN ωN . . . ωN Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Computation of DFT Theorem F = FN f Proof: Use the deﬁnition of DFT and of the Fourier Matrix Consequence: DFT can be computed using matrix multiplication. Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Properties of the Fourier matrix Lemma −1 1 F N FN = NI ⇐⇒ FN = FN N Proof: Multiply F N with FN and use N−1 Np p 1 − ωN ωN ωN = 1, (ωN )j = =0 . 1 − ωN j=0 Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform The inversion formula This lemma gives us the following inversion formula Theorem N−1 1 1 −ku f = F F ⇐⇒ f (k) = F (u)ωN , k = 0, . . . , N − 1 N N u=0 Proof: 1 1 F = Ff ⇒ F F = F Ff = If = f . N N Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Example Example N = 2, ω = −1: 1 1 F2 = . 1 −1 1 4 f = ⇒F = . 3 −2 Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Example Example N = 4, ω = −i: 1 1 1 1 1 −i −1 i F4 = 1 −1 1 −1 . 1 i −1 −i Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Factorization of the Fourier matrix N = 22 = 4, ω = ω4 , τ = ω 2 1 1 1 1 1 ω ω 2 ω3 F4 = 1 ω 2 ω 4 = ω6 1 ω3 ω6 ω9 1 1 1 1 1 0 0 0 1 ω 2 ω 3 ω 0 0 1 0 =1 ω 4 ω 2 = ω 6 0 1 0 0 1 ω6 ω3 ω9 0 0 0 1 1 1 1 1 1 τ ω ωτ = P 1 1 −1 −1 4 1 τ −ω −ωτ Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Factorization of the Fourier matrix (ctd.) P4 denotes a 4 × 4 permutation matrix (a matrix with zeros and ones, where each row and each column only contains one one). 1 0 F2 0 ω F2 F2 0 F4 = P4 = I D2 P4 1 0 I −D2 0 F2 F2 − F2 0 ω where D2 = diag(1, ω) . Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform The Fast Fourier Transform (FFT) Theorem The Fourier Matrix can be factorized as I DN FN 0 F2N = P2N , I −DN 0 FN where 2 N−1 DN = diag(1, ω2N , ω2N , . . . , ω2N ) . and P2N is a permutation matrix of order 2N × 2N that maps (x(0), x(1), . . . , x(2N − 1)) −→ (x(0), x(2), . . . , x(2N − 2), x(1), x(3), . . . , x(2N − 1)) . Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform The Fast Fourier Transform (FFT) (ctd.) Corollary Calculation of FN f thus involves two calculations of FN/2 f , which involves 4 calculations of FN/4 f , etc. This algorithm is called the Fast Fourier Transform. Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Calculational complexity Let µn be the number of multiplications needed for calculating DFT of order 2n . Factorization gives µn = 2µn−1 + 2n−1 . A solution to this recursion formula is n2n N log2 N µn = = om N = 2n . 2 2 Example N = 1024 = 210 ◮ FFT gives µ ∼ 104 multiplications. ◮ DFT gives N 2 ∼ 106 multiplications. Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Two-dimensional Fourier Transform Deﬁnition Let f (x, y) be a function from R2 to R. The Fourier transform of f is deﬁned as +∞ Ff (u, v) = F (u, v) = e−i2π(ux+vy ) f (x, y)dxdy . −∞ This can be written (using u = (u, v), x = (x, y)): +∞ F (u) = e−i2πu·xf (x)d x . −∞ Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform The inversion formula in 2D Theorem Under certain conditions on f , the following inversion formula +∞ f (x, y) = ei2π(ux+vy ) F (u, v)dudv −∞ holds. Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Properties of the 2D Fourier transformation Properties: (in addition to those for the 1-D Fourier Transform) f1 (x)f2 (y) → F1 (u)F2 (v) (separability) f (Qx) → F (Qu) (rotation) where Q denotes an orthogonal matrix. Example rect(x) rect(y) → 4 sinc(2πu) sinc(2πv) δ(x)1(y) → 1(u)δ(v) δ(x) rect(y) → 1(u)2 sinc(2πv) f (x − 1) + f (x + 1) → (e−i2πu + ei2πu )F (u) = = 2 cos(2πu)F (u) Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform A useful fact If f real (usual case for images): ◮ even f → real F ◮ odd f → imaginary F ◮ F (u) = F (−u) Observe: F (u, v) is in general complex valued. It is common to illustrate the transform with |F (u, v)|. Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform DFT and FFT in two dimensions The discrete Fourier Transform (DFT) of f is deﬁned as M−1 N−1 ux vy F (u, v) = f (x, y)e−i2π( M + N ) = x=0 y =0 M−1 N−1 ux vy = f (x, y)ωM ωN = x=0 y =0 1 ωNv u 2u (M−1)u 2v ωN = 1 ωM ωM . . . ωM f , . . . (N−1)v ωN x = 0, . . . , M − 1, y = 0, . . . N − 1 . Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform DFT in Matrix form Let the matrix F represent the Fourier transform of the image f (x, y): F = FM f FN or F = FM (FN f T )T . i.e. the DFT in two dimensions can be calculated by repeated use of the one-dimensional DFT, ﬁrst for the rows, then for the columns. Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform f DFT DFT F Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform FFT on images Let the M × N-matrix f represent an image f (x, y). Extend the image periodically Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform FFT on images (ctd.) FFT gives a double periodic function fftshift It is common to move the origin to the middle of the image for illustration purposes. Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Interpretation of the Fourier Transform ◮ Usually, the gray-levels of the Fourier Transform images are scaled using c log(1 + |F (u, v)|). ◮ The middle of the Fourier image (after fftshift) corresponds to low frequencies. ◮ Outside the middle high components in F corresponds to higher frequencies and the direction corresponds to "edges"in the images with opposite orientation. Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Example What does the original image look like if this is the Fourier transform? Left: Magnitude, Right: Phase Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Answer Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Masters thesis suggestion of the day: Crossword reader/solver Construct a system for automatic crossword scanning and solution. Idea: Take an image, ﬁnd squares with text and without, interpret the text. Is it possible to solve crosswords automatically? Fredrik Kahl Image Analysis - Lecture 2 Continuous Fourier transform Review of Linear algebra Discrete Fourier Transform (DFT,FFT) Fourier transform Two-dimensional Fourier Transform Review - Lecture 2 ◮ Linear Algebra ◮ Subspaces ◮ Projections, Pseudo-inverse ◮ Image matrix ◮ Fourier Transform in 1 and 2 dimensions ◮ Discrete Fourier Transform in 1 and 2 dimensions ◮ Fast Fourier Transform (FFT) Fredrik Kahl Image Analysis - Lecture 2

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