2 2 f f + = 0 2 2 x y Laplacian operator is - - PowerPoint PPT Presentation

Laplacian Operator

• The Laplacian operator is based on the

Laplace equation given by

• Laplacian operator is discretized version of the

above equation and is based on second derivatives along x and y directions

2 2 2 2

f f x y ∂ ∂ + = ∂ ∂

IIT Bombay Slide 75 GNR607 Lecture 13-16 B. Krishna Mohan

Laplacian Operator

• Filter coefficients
• The discrete version of the second

derivative operator:

• [1 -2 1] and [1 -2 1]T in the horizontal

and vertical directions

• Superimposing the two,

we get the discrete Laplace

• perator

0 -1 0

• 1 4 -1

0 -1 0

IIT Bombay Slide 76 GNR607 Lecture 13-16 B. Krishna Mohan

Properties of Laplace Operator

• Isotropic operator – cannot give
• rientation information
• Any noise in image gets amplified
• Faster since only one filter mask

involved

• Smoothing the image first prior to

Laplace operator is often needed for reliable edges

IIT Bombay Slide 77 GNR607 Lecture 13-16 B. Krishna Mohan

Zero-Crossing Edge Detectors

• First derivative maximum: exactly where

second derivative zero crossing

• In order to detect edges, we look at pixels

where the intensity gradient is high, or the first derivative magnitude is maximum

• First derivative maximum implies a zero when

the second derivative is computed

• Edges are located at those positions where

there is a positive value on one side and a negative value on the other side, in other words a zero-crossing

IIT Bombay Slide 78 GNR607 Lecture 13-16 B. Krishna Mohan

A step edge, whose first derivative is an impulse, and whose second derivative shows a transition from a positive to a negative Edge location corresponds to the point where a sign change occurs from positive to negative (or vice versa) IIT Bombay Slide 79 GNR607 Lecture 13-16 B. Krishna Mohan

Zero-Crossing Edge Detectors

• Laplacian of a function I(r,c)

2 2 2 2 2 2 2 2 2

) ( c I r I I c r I ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ = ∇

Two commonly used masks for Laplacian operator IIT Bombay Slide 80 GNR607 Lecture 13-16 B. Krishna Mohan

Zero Crossing Edge Detector

• Direct operation on the image using the

Laplacian operator results in a very noisy result

• Derivative operator amplifies the high

frequency noise

• Preprocess the input image by a

smoothing operator prior to application of the Laplacian

IIT Bombay Slide 81 GNR607 Lecture 13-16 B. Krishna Mohan

Zero Crossing Edge Detector

• The Gaussian shaped smoothing operator

is found to be ideal as a preprocessing

• perator
• Therefore the Laplacian operator is

applied on Gaussian smoothed input image

• ZC(image) = Laplacian [gaussian(image)]

IIT Bombay Slide 82 GNR607 Lecture 13-16 B. Krishna Mohan

LOG operator

• Both Laplacian operator and Gaussian
• perator are linear, and hence can be

combined into one Laplacian of Gaussian (LoG) operator

• Laplacian[Gaussian(image)] =

[Laplacian(Gaussian)](image)

IIT Bombay Slide 83 GNR607 Lecture 13-16 B. Krishna Mohan

LOG operator

• Laplacian[Gaussian(image)] =

[Laplacian(Gaussian)](image)

2 2 2 2 2 2 2 2 2

1 2 ( ) 2 4 2 1 2 ( ) 2 4 2 1 2 2 ( ) 2 4 2

1 ( , ) (1 ) 2 1 [ (1 )] 2 1 (2 ) 2

r c r c r c

r LOG r c e c e r c e

σ σ σ

πσ σ πσ σ πσ σ

+ − + − + −

= − − + − − + = − − Verify!

IIT Bombay Slide 84 GNR607 Lecture 13-16 B. Krishna Mohan

LOG operator

• LoG operator is a sampled version of the function
• For a given value of σ, the size of the Gaussian

filter is -3σ to +3σ

• Computationally more expensive due to

convolution with large filter masks

2 2 2

1 2 2 ( ) 2 4 2

1 ( , ) (2 ) 2

r c

r c LOG r c e

σ

πσ σ

+ −

+ = − −

IIT Bombay Slide 85 GNR607 Lecture 13-16 B. Krishna Mohan

Zero-Crossing Edge Detectors

Properties

• Edges depend on the value of σ
• For small value of σ all edges are detected
• For large value of σ only major edges are detected
• Any minor difference in intensity between neighbors

can be captured using LoG filter

• Significant zero crossings can be identified using

suitable threshold

IIT Bombay Slide 86 GNR607 Lecture 13-16 B. Krishna Mohan

Zero-Crossing Edge Detectors

• A pixel at (m,n) is declared to have a zero

crossing if f’’(m,n) > T and f’’(m+δm, n+δn) < -T OR f’’(m,n) < -T and f’’(m+δm, n+δn) > T

IIT Bombay Slide 87 GNR607 Lecture 13-16 B. Krishna Mohan

Edge Detection in Multispectral Images

• Simple approaches:

– Compute gradient by taking Euclidean distance between multispectral vectors of data at adjacent pixels instead

• f differences in gray levels

– Find independent gradients for different bands, edges and combine edges – Find independent gradients, combine gradients, and find edge from multiband gradient

IIT Bombay Slide 88 GNR607 Lecture 13-16 B. Krishna Mohan

Edge Detection in Multispectral Images

IIT Bombay Slide 89 GNR607 Lecture 13-16 B. Krishna Mohan

Edge Detection in Multispectral Images

IIT Bombay Slide 90 GNR607 Lecture 13-16 B. Krishna Mohan

Image Sharpening

For example,

• Sharpened image =

Original image + k. gradient magnitude

• Scale factor k can determine whether

gradient magnitude is added as it is or a fraction of it. The sum may be rescaled to 0-255 to display like an image

IIT Bombay Slide 92 GNR607 Lecture 13-16 B. Krishna Mohan

IIT Bombay Slide 93 GNR607 Lecture 13-16 B. Krishna Mohan

Original image (left), Sharpened Image (right)

Unsharp Masking

• Sample convolution mask

0 0 0 0 0 0 1 1 1 0 1 0 + 0 1 0 - (1/9) 1 1 1 0 0 0 0 0 0 1 1 1 G = F + | (F – Fmean) |

IIT Bombay Slide 94 NR607 Lecture 13-16 B. Krishna Mohan

IIT Bombay Slide 95 NR607 Lecture 13-16 B. Krishna Mohan

Line Enhancement

Difference between a line and an edge Line is a physical entity Edge is a perceptual entity

IIT Bombay Slide 96 GNR607 Lecture 13-16 B. Krishna Mohan

IIT Bombay Slide 97 NR607 Lecture 28 B. Krishna Mohan

Lines

Line Enhancement

Detection of a physical line involves

High to low transition  Low to high transition OR Low to high transition  High to low transition

IIT Bombay Slide 98 GNR607 Lecture 13-16 B. Krishna Mohan

Line Enhancement Masks

• These masks look for positive to negative

and negative to positive transitions in vertical/horizontal/diagonal directions

IIT Bombay Slide 99 GNR607 Lecture 13-16 B. Krishna Mohan

Summary of Gradient Operators

• Edges or boundaries convey very important

information for image understanding

• Gradient operators emphasize the local intensity
• r other property differences thereby making

visible object boundaries

• Gradient operations in normal course are only the

first step in reliable edge extraction

IIT Bombay Slide 100 GNR607 Lecture 13-16 B. Krishna Mohan

Summary of Neighborhood Operators

• Image processing operations involving

neighborhoods of pixels are important in many tasks

• Smoothing filters are composed of non-negative

coefficients which add up to 1

• Gradient filters are composed of both positive and

negative coefficients which must add up to 0 so that in images where there is no edge, the output is zero.

IIT Bombay Slide 101 GNR607 Lecture 13-16 B. Krishna Mohan