June 30 th, 2008 Lesson notes taken from professor Hongmei Zhu class.

Transcription

1 P. 1 June 30 th, 008 Lesson notes taken from professor Hongmei Zhu class. Sharpening Spatial Filters. 4.1 Introduction Smoothing or blurring is accomplished in the spatial domain by pixel averaging in a neighborhood (or spatial integration). What averaging filter does is replacing the value of every pixel in an image by the average of the intensity levels in the neighborhood defined by the filter mask. The most obvious application of smoothing is noise reduction. But, averaging filters have the undesirable effect that they blur edges. To undo this effect we can use sharpening techniques. Often one prefers to have sharpened images to reflect the real details like the figures below. To be able to read something from a moving object (license plate of a moving car), or to be able to read a bar code (one dimensional image), sometimes it is necessary to apply sharpening first. The principal objective of sharpening an image is to highlight transitions in intensity. Therefore, sharpening can be done by spatial differentiation. Different types of Sharpening GIMP has various sharpening options in enhance and edge detection, such as sharpen, unsharp mask, the differences of Gaussian, Edge, Laplace, Sobel, etc. Outline Motivation Basics Laplacian filter Unsharp masking, highboost filtering 4.. Basics - Foundation of Sharpening Derivatives and Differences The derivatives of a digital image are usually defined in terms of differences. This is how we calculate the first derivative f (x). From calculus we know that therefore f ( x+δ x) = f( x) + f '( x) Δx Or f ( x+ 1) = f( x) + f '( x) which we can derive that f '( x) = f( x+ 1) f( x) (1) To calculate the second derivative at x, f (x), use formula (1) two times. f '( x) f ( x+ Δx) f( x), Δx

2 f ''( x) = f '( x+ 1) f '( x) = [ f ( x+ ) f( x+ 1)] [ f( x+ 1) f( x )] f( x ) f( x 1) f( x) Or f (x-1) = f(x + 1) -f(x) + f(x 1) f = ( x 1) f ( x) Illustration of derivatives in a 1-D digital function (p159) P. We approximate the 1st order derivatives by f = + x and the nd order derivatives by f = f ( x+ 1) f ( x) + f ( x 1) x Properties of derivatives The 1st derivatives satisfy three properties which are a) zero in areas of constant b) nonzero at the onset of an intensity step or ramp c) nonzero along ramps The nd derivatives satisfy three properties which are a) zero in areas of constant b) nonzero at the onset of an intensity step or ramp c) zero along ramps if constant slopes Edges in digital images often are ramp-like transitions in intensity, in which case the first derivative of 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 one pixel thick, separated by zeros. Therefore, second derivatives are much easier to implement, and enhance fine details much better than the first derivatives.

3 4.3. Sharpening using Laplacian In Gimp, Filter Edge detect - Laplace Laplacian filters (p161) are isotropic The Laplacian for a function f ( x,y) is defined as filter. Isotropic filters are rotation invariant, in the sense that rotating the f f + image and then applying the filter gives x y the same result as applying the filter to where the image first and then rotating the f result. = f ( x + 1,y) + f ( x 1,y) f ( x,y) x f = f ( x,y+ 1) + f ( x,y 1) f ( x,y ). y The discrete Laplacian is f f + = f ( x + 1,y) + f ( x 1,y) + f ( x,y+ 1) + f ( x,y 1) 4f ( x,y) x y In (a) and (c) derivatives are calculated along the x and y axis, vertical and horizontal directions. In (b) and (d) derivatives are calculated along four directions including two diagonal. P. 3 Laplacian filters (p163) (a) shows a slightly blurred image of the North Pole of the moon. (b) shows the result of filtering this image with the Laplacian filter that calculates derivatives only in two directions. ( c) is the scaled image. g( x, y) = f f max f f min min After scaling, the dominant features are edges and sharp intensity discontinuities. The background, previously black, is now gray. The grayish appearance is typical of Laplacian images that have been scaled properly.

4 P. 4 Laplacian filters (p16) Because the Laplacian is a derivative operator, it highlights the intensity discontinuous in an image and de-emphasizes the regions with slowing varying intensity levels. It produces grayish edge lines and discontinuities, superimposed on a dark, featureless background. Thus, we can get a better sharpened ( ) ( ) image by ( ) ( ) f x,y f x,y g x,y = f x,y + c + x y where c = -1 if the center value in the filter is negative; otherwise, c = 1; (d) shows the result obtained using equation above with c = -1. The detail in this image is clearer and shaper than in the original image. (e) shows the result of repeating the procedure with the filter in fig 3.37/b, which calculates the derivatives in four directions. There is a significant improvement in sharpness. Results such as those in fig (c ) and (d) have made Laplacian a tool of choice for sharpening digital images Unsharp Masking, Highboost Filtering (p16) Unsharpen Masking (k=1) and Highboost Filtering (k>1) A process is often used for the printing and publishing industry to sharpen images, involving subtracting an unsharp (smoothed) version of an image from its original: f x,y Blur the original image ( ) Subtract the blurred image from the original (unsharp mask) g x,y = f x,y f x,y mask ( ) ( ) ( ) Add the mask to the original (add a weighted portion of the mask back to the original image: g x,y = f x,y + k g x,y k 0 ( ) ( ) ( ) mask When k = 1, we have unsharp masking, as defined above g( x, y) = f ( x, y) f ( x, y), when k > 1, the process is referred to as highboost filtering. Choosing k < 1 de-emphasizes the contribution of the unsharp mask.

5 P. 5 f ( x, y) blurred image g( x, y) = f ( x, y) f ( x, y) unsharp masking g( x, y) = f ( x, y) + k[ f ( x, y) f ( x, y)] highboost masking Mechanics of Unsharpen Masking (p164) (a) is the original signal (b) shows the result of imaging ( c) is the unsharp mask, obtained by subtracting the blurred signal from the original. (d) is the final sharpened result, obtained by adding the mask to the original signal Example (p165)

6 GIMP tutorial: smart sharpening taken from Original image After Sharpening P. 6 Step 1 Here is the original image loaded into GIMP. It is noticeably soft, probably due to bad out of focus. It really shows up in prints. Note: be sure to do any other editing that you want to on the image first, especially down sampling or upsampling (resizing). Sharpening should always be the last step that you do. Step : Note: be sure to do step that you do. Duplicate the image by pressing Ctrl+D or right-click and select <Image> Image -> Duplicate. It might be wise to minimize the original image window now. You can open it later to compare to the result. Right-click on the duplicate image and select <Image> Image -> Mode -> Decompose. In the decompose dialog box, select LAB (or HSV; see Note 1). Check the "As layers" checkbox (see Note ) and click OK. This will create a new window with the image decomposed into the Luminosity ("brightness"), A and B channels as 3 layers. This will allow us to isolate and sharpen the luninosity channel which is a) the most important, and b) should be the cleanest in terms of noise At this point I usually turn off visibility of the A and B layers in the Layers dialog by clicking on the "eyes".

7 P. 7 Step 3 Go to the duplicate image. Right-click and select <Image> Filters -> Edge Detect -> Edge. In the Edge Detect dialog box, select a parameter value of the appropriate size for the edges in your image. You may need to experiment with this. For the example image I chose 6 Now convert the edges image to grayscale by right clicking on it and selecting <Image> Image -> Mode -> Grayscale. This image is going to become our sharpening mask. Step 4 The next step is to adjust the tonality a little so that areas that need sharpening are really white and anything that doesn't need sharpening at all is really black. Open the Levels dialog (<Image> Image -> Colors - > Levels). Bring the black point up and the white point down to filter out any insignificant edges. You may need to play around with the exact positions of the sliders. Click OK when the image outlines the significant edges in white and most everything else is black. How much you'll need to tweak this really depends on the image. There wasn't a lot of spurious edges shown here, so a small adjustment was all that was necessary. Step 5 Now apply a gaussian blur (<Image> Filters -> Blur -> Gaussian Blur) to smooth out the edges a little. A radius of between 3 and 10 pixels (horizontal and vertical) should be enough. In the example I used 7

8 P. 8 Step 6 If you want you can use Levels again to adjust the white and black points of the mask. I find this is usually a good idea after the blur. In this example I pulled the white point slider down a bit. Step 7 Open the Layers dialog. Select the LAB image (if HSV, value) in the drop down box at the top and then click on the Channels tab. Click on the new channel button ( "Sharpening Mask. ) at the bottom of the dialog to create a new channel. Name it Step 8 Right-click in the blurred/edges image window and <Image> Select -> All. Right-click again and select <Image> Edit -> Copy. (Alternatively you can use keyboard shortcuts Ctrl+A and Ctrl+C in succession). In the Layers dialog, make sure the Sharpening Mask channel is selected. Go back to the LAB (value) window and paste (<Image> Edit -> Paste or Ctrl+V). Select the Layers tab in the Layers dialog and click on the anchor button ( the floating image. ) to anchor

9 P. 9 Step 9 Click on the Channels tab in the Layers dialog. You should see a tiny version of the sharpening mask in the channel icon, indicating that you properly pasted the sharpening mask into the new channel. At this point you may wish to click on the eye icon to make the sharpening mask invisible. Select the Sharpening Mask channel and click the "Channel to Selection" button ( ). You should see "marching ants. Step 10 Important: in the Layers dialog, click on the Layers tab and make sure the luminosity layer (if HSV, Background layer of Value window) is selected, and is the only one selected. We only want to sharpen the luminosity channel Apply an unsharp mask (<Image> Filters -> Enhance -> Unsharp Mask). You may want to experiment with the settings here. You will find that you can use much higher amounts of sharpening (than if the entire image were selected) without significantly degrading the image. In the example here I used Radius=1, Amount=.0 (00%?), Threshold=0. Now get rid of the selection (right-click and <Image> Select -> None) and zoom in (= key) to examine the results. If you're not happy with the results, undo and try the sharpen step again with different parameters.

10 P. 10 Step 11 It's time to reconstruct our original image from the LAB (HSV) components. Right-click in the luminosity (value) window and select (<Image> Image -> Mode -> Compose). In the Compose Options dialog, make sure that you have selected LAB (HSV) on the left and on the right that the Luminosity, A and B (Hue, Saturation and Value) components are matched up with their respective images. Click OK; this should create a new composite image Final Step Examine the resulting image, zooming in to examine edge detail.