SYDE 575: Digital Image Processing Using Discrete Gradients: Edge Detection & Edge Enhancement/Sharpening Exact Blur Compensation Textbook 10.2.5 (edge detection), 3.6 (sharpening), blackboard (exact blur compensation review ztransform on your own) Edge or Gradient Detection Why are edges important? Psychovisually, most important characteristic that HVS

identifies. 1) Difficult to distinguish two similar grey levels unless side-by-side with a boundary separating 2) Phase vs. magnitude: reconstruct an image with only phase and structure is retained; reconstruct using only magnitude and an unintelligible image is produced (Lab #3) First Derivatives

Implemented as magnitude of gradient in image processing f = f x

f y f f =

x 2 2 f + y

1/2 These are continuous implement in discrete fashion (Dx, Dy) for digital image processing First Difference Implement discrete derivative using first difference e.g., Dx = f(m + 1, n) f(m,n) = [ -1 1 ] (as a mask) Dy = f(m, n + 1) f(m,n) = [ -1 1 ] = Dx (as a mask)

Another method: Roberts Cross Operator | Df(m,n) | = | f(m,n) f(m+1,n-1) | + | f(m,n-1) f(m+1,n) | First Difference First difference operators are not effective Very sensitive to noise because of small spatial extent Solution: use a larger mask Gradient Masks

Source: Gonzalez and Woods Example: Defect Detection Source: Gonzalez and Woods 1-D Example Edge Detector Characteristics a) Negative weights required b) Zero DC gain c) Non-causal

d) Typically odd dimension e) Odd symmetry Canny Edge Detector (1986) Advanced method for edge detection Canny specified 3 issues an edge detector must address: 1) Error rate: edge detector must respond only to edges and detect them all 2) Localization: distance between detected edges and true edges must be minimized 3) Response: do not identify multiple edge

pixels where only a single edge exists Canny (cont.) Canny assumed step edge w/ point Gaussian noise Tried to derive single filter to optimize edge detection based on these 3 criteria for given edge model Outcome too complex to be solved analytically! But a reasonable solution is a derivative of a Gaussian So,

step * Gaussian * edge detector = ?? Canny (cont.) Step 1: convolve image with derivative of a Gaussian Step 2: non-maximum suppression Thins edge boundary to 1-pixel thick Threshold based on direction of gradient Magnitude of gradient at edge pixel should be greater than magnitude of gradients on each side of edge Step 3: hysteresis Two thresholds, Th and Tl

Any gradient bigger than Th, automatically an edge Iteratively, any pixel connect to Th with gradient > Tl automatically an edge as well Spatial Filtering: Sharpening Goal: highlight or enhance details in images Some applications:

Photo enhancement Medical image visualization Industrial defect detection Basic principle:

Averaging (blurring) is analogous to integration Therefore, logically, sharpening accomplished by differentiation Comparing First and Second Derivatives First derivative f = f ( x + 1) f ( x ) x

Second derivative 2 f = f ( x + 1) + f ( x 1) 2 f ( x) 2 x Derivatives of Digital Function

Second-order derivatives have stronger response to fine detail (e.g., thin lines and points) Second-order derivatives have a non-zero response to ramps First-order derivatives have stronger

response to step changes Second-order derivatives produce double response at step changes Example Source: Gonzalez and Woods Laplacian Second-order derivatives in 2D described

by the Laplacian 2 2 f f f = 2 + x y 2 2

Discrete Laplacian Filter 2 2 f f 2 f = 2 + 2 x y

2 f = f ( x + 1, y ) + f ( x 1, y ) 2 f ( x, y) 2 x 2 f = f ( x , y + 1) + f ( x , y 1) 2 f ( x , y ) 2 y

Laplacian Filter Source: Gonzalez and Woods 1 Laplacian in Practice How can the second derivative (Laplacian) be used? Example Source: Gonzalez and Woods

Sharpening using Unsharp Masking Process used for many years in publishing Subtract blurred version of image from the image itself to produce sharp image gmask(x,y) = f(x,y) m(x,y) g(x,y) = f(x,y) + k gmask(x,y) output

input edge map k=1 -> unsharp masking k>1 -> high boost filtering Example

Source: Gonzalez and Woods Impulse Response for Unsharp Masking Example Source: Gonzalez and Woods Edge Enhancement Filter Characteristics

a) Even symmetry b) Negative weights (surrounding) c) DC gain of 1 These aspects characterize lateral inhibition (see earlier in the course) Neighbouring inputs inhibit response of a pixel Well-known as a model of neural interaction in the HVS Exact Blur Compensation Use of complex number to solve difference

equation representing inverse system Use of z-transform for solving inverse system for general blur model Blackboard notes