2. 1 CS6825: Simple Geometric Operations

3. 2Scaling This is what happens when you resize an image. If you transform the image from an MxN to a PxQ image where P>M and Q>N, then we call this magnification. This is what happens when you resize an image. If you transform the image from an MxN to a PxQ image where P>M and Q>N, then we call this magnification. Example: Suppose we want to shrink our image from MxN to M/2xN/2. The following algorithm will do this, notice that we simply average 2x2 blocks of pixels to get a value for each pixel in the new smaller image. /*SHRINKING BY FACTOR of 2*/ S= 2; /*the scale factor*/ S= 2; /*the scale factor*/ /*Visit each pixel of the new smaller*/ /*Visit each pixel of the new smaller*/ /* image and calculate its value*/ /* image and calculate its value*/ for(r=0; r<M/S; r++) for(r=0; r<M/S; r++) for(c=0; c<N/S; c++) for(c=0; c<N/S; c++) { Pnew[r,c] = (P[r*S, c*S] + P[r*S+1, c*S] + P[r*S, c*S+1] + P[r*S+1, c*S+1])/4; } { Pnew[r,c] = (P[r*S, c*S] + P[r*S+1, c*S] + P[r*S, c*S+1] + P[r*S+1, c*S+1])/4; } The above algorithm, averages or interpolates the values in a 2x2 block to get a single value. This kind of interpolation is called linear interpolation The above algorithm, averages or interpolates the values in a 2x2 block to get a single value. This kind of interpolation is called linear interpolation What if the scale is not 2? What if the scale is not 2? What if the scale factor is not the same in the row and column directions? What will happen to the image? What if the scale factor is not the same in the row and column directions? What will happen to the image? What would be the algorithm if you were magnifying the image? What would be the algorithm if you were magnifying the image? Instead, a faster, but, arguably not as good in quality, would be to simply choose one of the pixels in the 2x2 block of the original image as the pixel value for the corresponding pixel in the new smaller image. Instead, a faster, but, arguably not as good in quality, would be to simply choose one of the pixels in the 2x2 block of the original image as the pixel value for the corresponding pixel in the new smaller image.

4. 3Rotation Suppose that you wanted to rotate and image. What does it mean to say you want to rotate an image by 45 degrees? You must apply the laws of trigonometry to rotate an image around its center point by the angle desired. The following equations govern the relationship between the old position (r,c) and the new position (r',c'): Suppose that you wanted to rotate and image. What does it mean to say you want to rotate an image by 45 degrees? You must apply the laws of trigonometry to rotate an image around its center point by the angle desired. The following equations govern the relationship between the old position (r,c) and the new position (r',c'): r' = r*cos(angle) + c*sin(angle) c' = c*cos(angle) - r*sin(angle) c' = c*cos(angle) - r*sin(angle) Consider the following: What will you do if the r',c' calculated are not integer values? What will you do if the r',c' calculated are not integer values? How would you implement this algorithm in a discrete MxN image? How would you implement this algorithm in a discrete MxN image? What happens to the corners of an image when you rotate it 45 degrees? What happens to the corners of an image when you rotate it 45 degrees?

5. 4 Translation When performing translation, usually you are moving parts of images to different locations within the same image. To specify a translation, you specify the portion of the image and the destination in row, column values of this portion. This destination is usually signified as the new location of the upper left corner the rectangular image portion specified. Given this is the case, the following algorithm will implement the translation operation. Note that really a copying of the portion is what is achieved. You may choose to set the original portion of the image not overlapping with the coppied version to black as is done by programs such as Photoshop. When performing translation, usually you are moving parts of images to different locations within the same image. To specify a translation, you specify the portion of the image and the destination in row, column values of this portion. This destination is usually signified as the new location of the upper left corner the rectangular image portion specified. Given this is the case, the following algorithm will implement the translation operation. Note that really a copying of the portion is what is achieved. You may choose to set the original portion of the image not overlapping with the coppied version to black as is done by programs such as Photoshop. Rd = Row value of destination; Cd = Column value of destination; Rs,Cs = Upper left hand corner of image portion want to translate; Re,Ce = Lower right hand corner of image portion. for(r=Rs; r<Re; r++) for(c=CS; c<Ce; c++) Rd = Row value of destination; Cd = Column value of destination; Rs,Cs = Upper left hand corner of image portion want to translate; Re,Ce = Lower right hand corner of image portion. for(r=Rs; r<Re; r++) for(c=CS; c<Ce; c++) { Pnew[Rd + r, Cd + c] = P[r,c]; } { Pnew[Rd + r, Cd + c] = P[r,c]; } Consider the following: What would you do if the designated destination region was either partly or wholey outside of the image boundaries? What would you do if the designated destination region was either partly or wholey outside of the image boundaries?

6. 5 Mirroring The process of mirroring is that of "reflecting" the image values around a horizontal or vertical line going through the center of the image. Below is the algorithm for doing a vertical mirroring. The process of mirroring is that of "reflecting" the image values around a horizontal or vertical line going through the center of the image. Below is the algorithm for doing a vertical mirroring. /*Vertical Mirroring*/ /*Vertical Mirroring*/ for(r=0; r<M; r++) for(r=0; r<M; r++) for(c=0; c<N; c++) for(c=0; c<N; c++) { Pnew[r,c] = P[r, N-c]; } { Pnew[r,c] = P[r, N-c]; } Consider the following: How would you write a program to mirror the image around an arbitrary line specified by two endpoints? How would you write a program to mirror the image around an arbitrary line specified by two endpoints?