Given an alphabet with n characters. let the characters be numbered from 1 to n, and let the frequency of character i be fi = for i=1, – 1, with the last character n having frequency fn = ns. for example, if n=6, then the frequencies of the first five characters are 1/2, 1/4, 1/8, 1/16, 1/32, respectively, and the frequency of the last character is 1/32. answer the following questions: a) (5 points) show that the frequencies of all characters sum to 1 (as they should) for any n. b) (5 points) show what the huffman encoding is for each character. in building the encoding tree, the larger frequency child should be on the left, and if there are two groups of characters with the same frequency, the one with the smaller index should be on the left. c) (5 points) what is the expected number of bits per character? let the encoding length of character i be denoted as li, then the expected or average number of bits for the encoding is computed as e-li · fi. use this formula to derive an closed form expression (for any n).

Work with a monitor that’s calibrated and profiled. for critical image editing, calibration and profiling are essential. otherwise, the image you see on your monitor looks different on other monitors or when printed.plan to use adjustment layers to adjust the tonal range and color balance of your image. adjustment layers let you go back and make successive tonal adjustments without discarding or permanently modifying data from the image layer. keep in mind that using adjustment layers adds to the file size of the image and demands more ram from your computer. accessing the color and tonal commands in the adjustments panel automatically creates adjustment layers.if you don’t want to use adjustment layers, you can apply adjustments directly to an image layer. remember that some image information is discarded, when making a color or tonal adjustment directly to an image layer.for critical work and maximum preservation of image data, it’s best if the image you work with is 16 bits per channel (16‑bit image) rather than 8 bits per channel (8‑bit image). data is discarded when you make tonal and color adjustments. the loss of image information is more critical in an 8‑bit image than a 16‑bit image. generally, 16‑bit images have a larger file size than 8‑bit images.duplicate or make a copy of the image file. working on a copy of your image preserves the original in the event you want to use the image in its original state.remove any flaws such as dust spots, blemishes, and scratches from the image before making color and tonal adjustments.open the info or histogram panel in expanded view. as you evaluate and correct the image, both panels display invaluable feedback on your adjustments.you can make a selection or use a mask to confine your color and tonal adjustments to part of an image. another way to apply color and tonal adjustments selectively is to set up your document with image components on different layers. color and tonal adjustments are applied to only one layer at a time. only the image components on the targeted layer are affected.