Total Variation Theory and Its Applications
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1 Total Variation Theory and Its Applications 2nd UCC Annual Research Conference, Kingston, Jamaica Peter Ndajah University of the Commonwealth Caribbean, Kingston, Jamaica September 27, 2018 Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
2 Outline 1 Introduction Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
3 Outline 1 Introduction 2 Derivations of The Total Variation Functional Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
4 Outline 1 Introduction 2 Derivations of The Total Variation Functional 3 Total Variation Diminishing Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
5 Outline 1 Introduction 2 Derivations of The Total Variation Functional 3 Total Variation Diminishing 4 Image Denoising Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
6 Outline 1 Introduction 2 Derivations of The Total Variation Functional 3 Total Variation Diminishing 4 Image Denoising 5 Edge Detection Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
7 Outline 1 Introduction 2 Derivations of The Total Variation Functional 3 Total Variation Diminishing 4 Image Denoising 5 Edge Detection 6 TV Image Filter Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
8 Outline 1 Introduction 2 Derivations of The Total Variation Functional 3 Total Variation Diminishing 4 Image Denoising 5 Edge Detection 6 TV Image Filter 7 Comparing Marr-Hildreth and Total Variation Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
9 Outline 1 Introduction 2 Derivations of The Total Variation Functional 3 Total Variation Diminishing 4 Image Denoising 5 Edge Detection 6 TV Image Filter 7 Comparing Marr-Hildreth and Total Variation 8 Object Recognition Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
10 Outline 1 Introduction 2 Derivations of The Total Variation Functional 3 Total Variation Diminishing 4 Image Denoising 5 Edge Detection 6 TV Image Filter 7 Comparing Marr-Hildreth and Total Variation 8 Object Recognition 9 Book Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
11 Introduction Introduction The total variation of a real-valued function f defined on an interval [a,b] is a measure of the 1 D arc length from a to b. Mathematically we can define the total variation of f as V b a (f) = sup p n f(x i ) f(x i 1 (1) i=1 where the supremum runs over the set of all partitions P = {a = x 0,x 1,...,x n = b}. (1) is useful if we are measuring the arc length of a one-dimensional function f. to extend the notion to more general curves, we may begin by observing that equation (1) can be multiplied and divided by x to obtain V b a (f) = sup p n i=1 f(x i ) f(x i 1 x x (2) Equation (2) can be transformed into the continuous case by taking limits as x 0 and n. Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
12 Derivations of The Total Variation Functional Derivations of The Total Variation Functional: Method I This method may be called the method of first principles. A function f defined on an interval [a,b] is said to be of bounded variation if there is a constant C > 0 such that n f(x k ) f(x k 1 ) C for every partition k=1 a = x 0 < x 1 <... < x n = b of [a,b] by points of subdivision x 0,x 1,...,x n. If f is of bounded variation, then by the total variation of f is meant the quantity n Va b (f) = sup f(x k ) f(x k 1 ) k=1 where the least upper bound is taken over all partitions of the interval [a, b]. The above definitions work quite well for Riemann integrable functions. Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
13 Derivations of The Total Variation Functional Derivations of The Total Variation Functional: Method II This method uses measure theory to describe the total variation functional. It is useful only when describing the total variation qualitatively. Let λ be a charge on X and let P,N be a Hahn decomposition for λ. The positive and negative variations of λ are the finite measuresλ +,λ defined for E in X by λ + (E) = λ(e P) λ (E) = λ(e N) The total variation of λ is the measure λ defined for E in X by λ = λ + (E)+λ (E) If f is Lebesgue-integrable and f belongs to L(X,X,µ) with respect to a measure µ on X, and if λ is defined for E in X by λ(e) = fdµ, Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) E Its Applications September 27, / 47
14 Derivations of The Total Variation Functional Derivations of The Total Variation Functional: Method II If f is Lebesgue-integrable and f belongs to L(X,X,µ) with respect to a measure µ on X, and if λ is defined for E in X by λ(e) = fdµ, then λ is a charge and so that λ (E) = λ + (E) = λ (E) = E f dµ = E E E E f + dµ f dµ f + dµ+ f dµ (4) E Here, f is any Lebesgue-integrable function. The sum of the positive and negative variations of the function f gives its total variation. Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
15 Derivations of The Total Variation Functional Derivations of The Total Variation Functional: Method III A function µ L 1 (Ω) whose partial derivatives in the sense of distributions are measures with finite total variation in Ω is called a function of bounded variation. This class of functions is usually denoted by BV(Ω). Thus, u BV(Ω) if and only if there are signed measures µ 1,µ 2,...,µ n defined in Ω such that for i = 1,2,...,n, and Du (Ω) < ud i ϕdx = ϕdµ i (5) for all ϕ C0 (Ω). The gradient of µ will therefore be a vector valued measure with finite total variation: Du = sup{ udivvdx : v = (v 1,...,v n ) C 0 (Ω;R n ), Ω Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
16 Derivations of The Total Variation Functional Derivations of The Total Variation Functional: Method III The divergence of a vector field is denoted by divv and is defined by divv = n D i v i = i=1 n i=1 v i x i If u BV(Ω), the total variation Du may be regarded as a measure, for if f is a non-negative real-valued continuous function with compact support in Ω, we may define Du (f) = sup{ udivvdx : v = (v 1,...,v n ) C0 (Ω;Rn ), Ω v(x) f(x) x Ω} (8) Du is additive, continuous under monotone convergence and a non-negative Radon measure on Ω. Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
17 Derivations of The Total Variation Functional Derivations of The Total Variation Functional: Method III The space of absolutely continuous u with u L 1 (R 1 ) is contained in BV(R 1 ). In the same manner in R n, a Sobolev function is also BV. That is, W 1,1 (Ω) BV(Ω). If u W 1,1 (Ω) then, Ω udivvdx = Ω n D i uvdx (9) and the gradient of u has finite total variation with Du (Ω) = Du dx. (10) Ω i=1 Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
18 Derivations of The Total Variation Functional Derivations of The Total Variation Functional: Method III This measures both the positive and negative variations implicitly rather than explicitly. If u C 1 (Ω), then Du dx = u(x) dx (11) Let Ω R 2 be an image surface. Then we have Ω Ω u(x) dx = Ω ( ) u 2 + x ( ) u 2 dxdy (12) y which is the total variation of an image u with two independent variables x,y. Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
19 Derivations of The Total Variation Functional Ndajah and Kikuchi Derivation: Method IV We derive the total variation of an image using the vector gradients. Images are two dimensional spatial functions. In general, to find the total variation of an n-dimensional mathematical object, we consider the directional derivative of a scalar function f( x) = f(x 1,x 2,...,x n ) along a unit vector u = (u 1,...,u n ). The directional derivative is defined to be the limit f( x +h u) f( x) u f( x) = lim (13) h 0 + h This means that the directional derivatives exist along any unit vector u, and one has u f( x) = f( x) u (14) Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
20 Derivations of The Total Variation Functional Ndajah and Kikuchi Derivation: Method IV For an image we represent the directional derivatives by i, j so that { x f( x) = f( x) i = f x y f( x) = f( x) j = f y The components x f( x) and y f( x) are orthogonal so that and the inner product is f( x) = i f x +j f y (15) (16) f( x), f( x) = f 2 = ( ) f 2 ( ) f 2 + dxdy (17) x y Therefore, total variation in two dimensions can be written as f dxdy (18) Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
21 Total Variation Diminishing Total Variation Diminishing In numerical methods, total variation diminishing (TVD) is a property of certain discretization schemes used to solve hyperbolic partial differential equations. It is mostly applied in computational fluid dynamics. This concept was first introduced by Ami Harten. In systems described partial differential equations, such as u t +a u x = 0 (19) u TV = x dx (20) A numerical method is said to be total variation diminishing (TVD) if, TV(u n+1 ) TV(u n ) Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
22 Total Variation Diminishing Total Variation Diminishing In Computational Fluid Dynamics, TVD scheme is used to capture sharper shock predictions without leaving any misleading oscillations when variation of field variable is discontinuous. On the other hand, to capture the variation, fine grids ( x = very small) are needed and the computations can become heavy and uneconomic. The use of coarse grids with central difference scheme, upwind scheme, hybrid difference scheme, and power law scheme gives false shock predictions. The TVD scheme enables sharper shock predictions on coarse grids saving computation time and as the scheme preserves monotonicity there are no spurious oscillations in the solution. Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
23 Total Variation Diminishing Total Variation Diminishing Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
24 Total Variation Diminishing Total Variation Diminishing Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
25 Total Variation Diminishing Total Variation Diminishing Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
26 Image Denoising Image Denoising Rudin, Osher and Fatemi (1990) developed finite difference schemes were developed there which were used to enhance mildly blurred images significantly while preserving the variation of the original image. They concluded that the space of BV functions is the proper class for many basic image processing tasks. Thus, the constrained minimization problem is min Ω ux 2 +u2 ydxdy (21) subject to the constraints involving u 0. They took the two constraints as above: Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
27 Image Denoising Image Denoising This constraint signifies the fact that the white noise η(x,y) is of zero mean and 1 2 (u u 0) 2 = σ 2 Ω where σ > 0 is given. The second constraint uses a priori information that the standard deviation of the noise η(x,y) is σ. Thus, there is a linear and nonlinear constraint. They arrive at the Euler-Lagrange equations 0 = x u x ux 2 +uy 2 + y u y u 2 x +u 2 y in Ω, with u n = 0 on the boundary of Ω = Ω λ 1 λ 2 (u u 0 ) (22) Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
28 Image Denoising Image Denoising The solution uses a parabolic equation with time as an evolution parameter, or equivalently, the gradient descent method. This means that we solve u t = x u x u 2 x +u2 y + y u y u 2 x +u2 y λ(u u 0 ) (23) for t > 0, x,y Ω, u(x,y,0) given u n = 0 on Ω As t increases, we approach a denoised version of the image. We must compute λ(t). If steady state has been reached, the left hand side of equation (23) vanishes. We then have λ = 1 2σ 2 ux 2 +uy 2 (u 0) x u x + (u 0) y u y dxdy (24) ux 2 +u2 y ux 2 +u2 y Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
29 Image Denoising Image Denoising This gives a dynamic value for λ(t), which appears to converge as t. This is merely the gradient projection method of Rosen in his 1961 paper titled The Gradient Projection Method for Nonlinear Programming part II, nonlinear constraints published by SIAM. The numerical method in two dimensions is: x i = ih, y j = jh i,j = 0,1,...,N with Nh = 1 t n = n t, n = 0,1,... u n ij = u(x i,y i,t n ), u 0 ij = u(ih,jh)+σϕ(ih,ih) Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
30 Image Denoising Image Denoising So the numerical approximation to equations (23) and (24) is u n+1 ij = u n ij+ t h [ x + y ( x + un ij ( x + un ij )2 +(m( x + un ij, y un ij )2 ) 1 2 ( ) y + un ij y + un ij +(m( x + un ij, x un ij )2 ) 1 2 for i,j = 1,...,N with boundary conditions u n 0j = un 1j, un Nj = un N 1,j, un i0 = un in = un i,n 1 Here, x u ij = (u i 1,j u ij ) tλ n (u n ij u 0 (ih,ih)), (25) Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
31 Image Denoising Image Denoising and similarly for y u ij. λ n is defined discretely as λ n = h 2σ 2 [ ( ( x + un ij )2 +( y + un ij )2 ( x + u0 ij )( x + un ij ) ( x + un ij )2 +( y + un ij )2 ( y + u0 ij )( y + kun ij ) ( x + un ij )2 +( y + un ij )2 (26) A step size restriction is imposed for stability: t h 2 c Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
32 Image Denoising Image Denoising Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
33 Image Denoising Image Denoising Reference Image: MSE = 255, SSIM = 1 (a) Denoised Image: λ = 60,τ = 0.01, MSE = 255, SSIM = (b) Denoised Image: λ = 12, τ = 0.01, MSE = 255, SSIM = (c) Denoised Image: λ = 2, τ = 0.01, MSE = 255, SSIM = (d) Denoised Image: λ = 1, τ = 0.01, MSE = 255, SSIM = (e) Denoised Image: λ = 0.5, τ = 0.01, MSE = 255, SSIM = (f) Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
34 Edge Detection Image Edge Detection We show edge detection using total variation and compare the result with the Marr-Hildreth method. The scaling factor σ enables us to tune the edge detection process. G(x,y) = e x2 +y 2 2σ 2 (27) to smooth the image u(x,y). This is accomplished by a convolution of both functions i.e. G(x,y) u(x,y). σ is the standard deviation of G(x,y) and it acts as a scaling factor, blurring out noise and structures with scales below σ. Therefore, we get no absolute definition of edges. We only talk about edges at a certain scale. Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
35 Edge Detection Image Edge Detection The convolution D(x,y) = G(x,y) u(x,y) (28) yields a smoothed image at a scale σ to which the Laplacian operator is applied i.e. 2 (G(x,y) u(x,y)). This operation is commutative and gives the same result as ( 2 G(x,y) ) u(x,y) = 2 D(x,y) (29) The expression 2 G is called the Laplacian of a Gaussian (LoG) and is expressed as [ x 2 2 +y 2 2σ 2 G(x,y) = ]e x2 +y 2 2σ 2 (30) σ 4 Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
36 Edge Detection Image Edge Detection Likewise, for the total variation approach, we first smooth the image by convolving u(x,y) and the Gaussian function G(x,y) just as in the case of the Laplacian approach. We then apply the total variation operator to the convolved image D(x,y) i.e. 2 D(x,y) D. This operation is not commutative as in the case of the LoG. Below we show results of edge detection based on the LoG images and total variation filtered images. Edge detection for both kinds of images do not follow exactly the same procedure. The traditional method of detecting edges in LoG images is by zero crossings. Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
37 Edge Detection Image Filters Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
38 Edge Detection Image Filters Figure: Laplacian Filtered Image Showing Edges Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
39 TV Image Filter TV Image Filter We obtained a minimization of the total variation functional by means of the Euler-Lagrange minimization method. We use the resulting nonlinear steady state partial differential equation given by J min u = 2 u u with the Dirichlet boundary condition u n = 0 to filter images in place of the Laplacian. The changes in the image texture are well captured in the filtered image. Also, the Laplacian forms double edges during image filteration process. The total variation filter overcomes this disadvantage by producing only single a edge. (31) Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
40 TV Image Filter TV-Filtered Image TV filter Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
41 Comparing Marr-Hildreth and Total Variation Figure: Gaussian Smoothed Image with σ = 2 Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47 Comparing Marr-Hildreth and Total Variation
42 Comparing Marr-Hildreth and Total Variation Figure: Marr-Hildreth Edge Detection with σ = 2 Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47 Comparing Marr-Hildreth and Total Variation
43 Comparing Marr-Hildreth and Total Variation Figure: Total Variation Edge Detection with σ = 2 Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47 Comparing Marr-Hildreth and Total Variation
44 Comparing Marr-Hildreth and Total Variation Figure: Gaussian Smoothed Image with σ = 3 Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47 Comparing Marr-Hildreth and Total Variation
45 Comparing Marr-Hildreth and Total Variation Figure: Marr-Hildreth Edge Detection with σ = 3 Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47 Comparing Marr-Hildreth and Total Variation
46 Comparing Marr-Hildreth and Total Variation Figure: Total Variation Edge Detection at σ = 3 Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47 Comparing Marr-Hildreth and Total Variation
47 Comparing Marr-Hildreth and Total Variation Figure: Gaussian Smoothed Image with σ = 4 Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47 Comparing Marr-Hildreth and Total Variation
48 Comparing Marr-Hildreth and Total Variation Figure: Marr-Hildreth Edge Detection with σ = 4 Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47 Comparing Marr-Hildreth and Total Variation
49 Comparing Marr-Hildreth and Total Variation Figure: Total Variation Edge Detection at σ = 4 Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47 Comparing Marr-Hildreth and Total Variation
50 Comparing Marr-Hildreth and Total Variation Figure: Gaussian Smoothed Image with σ = 5 Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47 Comparing Marr-Hildreth and Total Variation
51 Comparing Marr-Hildreth and Total Variation Figure: Marr-Hildreth Edge Detection with σ = 5 Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47 Comparing Marr-Hildreth and Total Variation
52 Comparing Marr-Hildreth and Total Variation Figure: Total Variation Edge Detection at σ = 5 Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47 Comparing Marr-Hildreth and Total Variation
53 Object Recognition Object Recognition Edge detection algorithms are at the heart of object detection and recognition in computer vision systems. Processes such as image segmentation and pattern recognition depend on image edge detection. These primary techniques combined with deep learning is what seems to give machines the ability to recognize objects and act accordingly in different situations. An example is self-navigating unmanned ground vehicles sometimes called driverless cars. For the car to successfully travel on a road, it must be able to recognize lane marks. The algorithm central to this ability is image edge detection algorithm. Earlier methods used for this purpose include the Canny and Marr-Hildreth algorithms. As we have seen in previous sections, these algorithms will fail after scale 4 (i.e. σ = 4). This means that if the road marks are not bright enough, the vehicle will not be able to recognize them. Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
54 Object Recognition Figure: Driverless Car Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47 Lane Recognition
55 Book Total Variation Book Peter Ndajah (University of the CommonwealthTotal Caribbean, Variation Kingston, Theory and Jamaica) Its Applications September 27, / 47
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