EDGES AND CONTOURS(1)

Transcription

1 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 1 EDGES AND CONTOURS1) KOM31 Image Processing in Industrial Sstems Some o the contents are adopted rom R. C. Gonzalez, R. E. Woods, Digital Image Processing, nd edition, Prentice Hall, 008 Digital Image Processing: An Algorithmic Introduction using Java, W Burger, Mark J. Burge, Springer Verlag, 008

2 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek Toda s Topics Sharpening Filters The need or Sharpening Filters Numerical dierentiation or images First and second derivatives Laplacian Filters Gradient Filters

3 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 3 Smoothing ilters revisited What kind o noise? Additive Multiplicative Impulsive Image smoothing suppresses the noise b using the redundanc in the image data

4 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 4 Smoothing ilters revisited

5 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 5 The need or Sharpening Filters Objective o Sharpening Filter: The principal objective o sharpening is to highlight transitions in intensit. Use o image sharpening var and include applications rom electronic printing and medical imaging to industrial inspection and autonomous guidance in militar sstems. We accomplished smoothing in the spatial domain b averaging Thus, there is an analog between smoothing and integration We can conclude spatial dierentiation should be used or sharpening we are tring to ind the intensit transitions-dierences) Fundamentall the strength o the response o the derivative operator is proportional to the degree o intensit discontinuit o the image. Thus image dierentiation enhances edges and other discontinuities such as noise) and de-emphasizes areas with slowl varing intensities.

6 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 6 The need or Sharpening Filters Edge? sharp change in brightness discontinuities) Where do edges occur? Actual edges: Boundaries between objects Sharp change in brightness can also occur within object Relectance changes Change in surace orientation Illumination changes. E.g. Cast shadow boundar

7 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 7 Numerical Dierentiation on a Data

8 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 8 Numerical Dierentiation on a Data The derivatives o a digital unction are deined in terms o dierences. First Derivative: Must be zero in areas o constant intensit Must be non zero at the onset o an intensit step or ramp Must be non zero along ramps Second Derivative: Must be zero in constant areas Must be non zero at the on set and end o an intensit step or ramp Must be zero along ramps o constant slope.

9 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 9 Numerical Dierentiation on a Data Edges in digital images oten are ramp-like transitions in intensit, in which case the irst derivative o the image would result in thick edges because the derivative is nonzero along a ramp. On the other hand, the second derivative would produce a double edge on piel thick, separated b zeros. From this, we conclude that the second derivative enhances ine details much better than the irst derivative, a propert that is ideall suited or sharpening images.

10 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 10 First Derivatives Forward dierence ormula Preerred) i i+1 i i+1 i = i+1 i i+1 i Backward Dierence Formula i i i 1 i i 1 = i i 1 i i 1 Central Dierence Formula i i+1 i 1 i+1 i 1 = i+1 i 1 i+1 i 1

11 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 11 First Derivatives Data Points 0, 0)) = 1,) 1, 1)) =,4), )) = 3,8) 3, 3)) = 4,16) 4, 4)) = 5,3) Forward dierence ormula Backward Dierence Formula Central Dierence Formula

12 Second Derivatives ) ) ) = 1 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 1 For images + 1) = 1 The data is isotropic: equivalent spaces

13 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 13 Second Derivatives Using Second Derivative or Image Sharpening-The Laplacian We are interested in isotropic ilters, whose response is independent o the direction o the discontinuities in the image to which the ilter is applied. In other words isotropic ilters are rotation invariant, in the sense that rotating the image and then appling the ilter gives the same result as appling the ilter to the image irst and then rotating the result. It can be shown Roseneld and Kak [198]) that the simplest isotropic derivative operator is the Laplacian, which, or a unctionimage),) o two variables, is deined as

14 Laplacian Filter 14 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek ), 1), 1), ), ) 1, ) 1, Thereore, it ollows rom the preceeding three equations that the discrete Laplacian o two variables is ), 4 1), 1), ) 1, ) 1, ), We can now orm our sharpening ilters

15 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 15 Laplacian Filter

16 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 16 Laplacian Filter Because the Laplacian is a derivative operator, its use highlights intensit discontinuities in an image and deemphasizes regions with slowl varing intensit levels. This will tend to produce images that have graish edge lines and other discontinuities, all superimposed on a dark, eatureless background. Shapening eect is obtained b adding the Laplacian image to the original. g, ), ) c[, )] Where, ) and g, ) are the input and sharpened images, respectivel. The constant c = 1 i the Laplacian ilter in a) or b) o previous igure are used, and c = 1 i either o the other two ilters is used.

17 Equivalent Filters KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 17 Laplacian Filter To recover the image:,, g, Laplacian Operator ,, g, The edges are highlighted

18 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 18 Laplacian Filter

19 Derivative ilters, ú ú ú ú û ù ê ê ê ê ë é = Ñ 1/ ú ú û ù ê ê ë é ø ö ç è æ + ø ö ç è æ = Ñ +» Ñ KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 19 Gradient o an image The magnitude o the gradient Gradient: Vector whose direction is in direction o maimum rate o change o and whose magnitude is maimum rate o change o

20 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 0 Derivative ilters Gradient in horizontal direction Gradient in vertical direction

21 KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 1 Derivative ilters Original Image Gradient in horizontal direction Gradient in vertical direction Magnitude o gradient