Lecture notes: Histogram, convolution, smoothing

Transcription

1 Lecture notes: Hstogram, convoluton, smoothng Hstogram. A plot o the ntensty dstrbuton n an mage. requency (# occurrences) ntensty The ollowng shows an example mage and ts hstogram: I we denote a greyscale mage as I[ then the hstgram H[] can be computed as I r c H [, ] [ ] r c 0 I[ c ], The hstogram s oten used n mage restoraton or cleanng. Hstogram equalzaton. Stretch the contrast evenly through the ntensty range by manpulatng the hstogram. The dstrbuton o ntensty s remapped to come as close as possble to unorm:

2 We desre to nd a transorm T or each orgnal ntensty to a new value so that the hstogram becomes unorm. T ( ) Howeve because the uncton H[] s dscrete the output wll only be approxmately unorm. Assumng we have an mage o ROWS by COLS 8-bt pxels, the hstogram equalzaton transorm can be wrtten as T( ) x0 H[ x]* ROWS * COLS * 55 where the summaton on H[] computes how much o the mage has an ntensty less than or equal to (ths s the cumulatve hstogram), the racton /(ROWS*COLS) normalzes these percentages (ths s the normalzed cumulatve hstogram), and the value 55 scales the output to the desred range The ollowng shows the mage rom above ater hstogram equalzaton, along wth the equalzed hstogram:

3 In C code, t can be computed as ollows: unsgned char nt nt double *mage; ROWS,COLS; hst[56],x; nhst[56],chst[56]; or (x=0; x<56; x++) hst[x]=0; or (x=0; x<rows*cols; x++) hst[mage[x]]++; or (x=0; x<56; x++) /* normalzed dstrbuton */ nhst[x]=(double)hst[x]/(double)(rows*cols); chst[0]=nhst[0]; or (x=; x<56; x++) /* cumulatve dstrbuton */ chst[x]=chst[x-]+nhst[x]; or (x=0; x<rows*cols; x++) /* remap pxels accordng to chst */ mage[x]=(unsgned char)(55.0 * chst[mage[x]]); What purpose does hstogram equalzaton serve? It tends to sharpen the detals vsble n an mage, by ncreasng ther contrast. For a human vewe ths can be qute useul. For a machne vson system, t s generally useless, as no new normaton s gleaned through the process. Convoluton. Combnng local-area normaton. Image convoluton can be wrtten as W W dr W dcw O [ I[ r d c d* [ d d where the range W...+W s a wndow o local-area normaton. The uncton [] s called a lte and weghts how much each pxel n the local area contrbutes to the output. I[] s the nput mage and O[] s the output mage. Smoothng. Suppressng nose n an mage. Consder a porton o an mage X n whch a pxel X s corrupted by nose. How could we go about suppressng ths nose, and determnng a good value or the pxel? One way s to take the average o all the pxels n the local neghborhood. For example, we could convolve the mage wth W= and

4 Ths s a mean lter. Mean lterng s good when nothng s known about the type o nose aectng the mage. Oten we assume that the nose has a Gaussan dstrbuton (or no better reason that because lots o naturally occurrng thngs have a Gaussan dstrbuton). In ths case we can perorm Gaussan smoothng usng a Gaussan-shaped lter: dr dc [ d d e where s the standard devaton o the Gaussan nose, and the stu n ront o e s a normalzng constant (may need to be adjusted). Suppose the corrupted pxel X s a spke, caused by a temporary loss or saturaton o sgnal? In that case, averagng would be bad, because the spke would clearly bas the mean. Ths type o nose s oten called salt-and-pepper nose. A medan lter s good or spke nose. Each pxel X s replaced by the medan (mddle) value n ts local neghborhood. A medan lter cannot be mplemented by convoluton. When workng wth a segmentaton, another convenent smoothng lter s the mode lter. Each pxel X s replacted by the mode (most commonly occurrng) value n ts local neghborhood. A mode lter cannot be mplemented by convoluton. The ollowng shows the above mage smoothed wth a 3x3 mean, medan, and mode lter. Note the very derent results. An example o salt-and-pepper nose wll be demonstrated n class, along wth the result rom usng these derent methods to smooth t.

5 Separable lters. Convoluton can be slow as W gets large. Separatng a D lter nto two D lters can greatly speed convoluton. O [ O[ W dcw W dr W I[ c d* O [ r d * c [ d r [ dr] For example, the mean lter could be mplemented usng W= and separatng [] nto the lters c r The choce o whch lter c or r to convolve rst s arbtrary. Note the need o an ntermedary result mage O []. Sldng wndow. In the case where W s large, convoluton can also be sped up by usng the summaton rom the preceedng pxel. For example: W W How does the summaton *W der rom *W? Only by the subtracton and addton o a sngle column at each end. As W gets large, computng the summaton ths way can save a great deal o tme. The sldng wndow and separable lter trcks can be appled togethe speedng the computaton even more.