This research consists of two applications of image processing, namely, image compression and image denoising. Image compression aims to reduce the size of an image without losing too many features. This is often used to store a large number of images such as fingerprints. Denoising is a technique for removing noise from an image while preserving as many of the edges and other detailed features as possible. This research studies the use of different discrete wavelets and the Dual Tree Complex Wavelets in the image compression and denoising process. The wavelet transform decomposes the original image into approximation and detail coefficients, where the approximation coefficients are calculated by averaging and the detail coefficients by taking differences. The wavelet transform is also invertible so that the image can be reconstructed again using the approximation and detail coefficients. The compression and denoising method consists of three steps: decomposition, thresholding and reconstruction. The difference between compression and denoising lies in the threshold part. For image compression, a percentage of the detail coefficient is chosen as the threshold. The wavelets db6, sym5, coif3, bior4.4 and rbio1.5 are chosen to compress the images. The images are tested with compression rates ranging from 5:1 to 43:1. Based on the Structural Similarity Index Measurement (SSIM), the bior4.4 wavelet performs best. For denoising, the threshold is optimised to obtain a denoised image. The discrete wavelets used are db4, coif3, bior2.8. Of these, the bior2.8 wavelet performs the best on images used in this research. Therefore, the bior2.8 wavelet is compared with the Dual Tree Complex Wavelet (DTCW). Based on the Peak Signal-toNoise-Ratio (PSNR), the denoised image using the DTCW performs better than the bior2.8 wavelet. Overal, wavelets are a powerful tool in image processing. The different wavelets each have their own characteristics. The choice of the optimal wavelet depends on the application and cannot be generalised.