Representation as fiter bank. Assumption for coding: Certain viewing distance, playback size Certain viewing angle for the eye.

Transcription

1 Representation as fiter bank Assumption for coding: Certain viewing distance, playback size Certain viewing angle for the eye. - Also applies to the original to a certain extent, since the eye must be far enough away so that it can no longer distinguish pixels. Example: 3 megapixel image printed on DIN-A0 poster size: looks like a mosaic, not like a continuous image. Question: what if the edge length of the image cannot be divided by 8? A: Fill with zeros, or better: mirror the ends (for better frequency response), or simply cut of the additional pixels. Representation of the DCT decomposition as a bank of flters (flter bank): Unterabtastung um Faktor 8: Faltung (*) mit Impulsant Wort Kleinere Bilder der Teilbänd Original Filter Lässt bestimmte Ortsfrequenzen durch (z.b. 0,0 im Orts-Frequenz Bereich) Lässt nur jeden 8 Wert durch, entspr. Kürzere Kantenläng

2 Application of the DCT to a row of pixles (for example): Efect can be seen as a set of 8 flters. Each of these 8 flters has an impulse response corresponding to the 8 mirrored columns of the transform matrix (i.e. the Cos(...) terms). These 8 flters flter our signal (convolution with the signal). After fltering, a downsampling is performed by a factor of 8 (i.e. only every 8th value of the fltered signal is left). -> In this way, the total number of values remains constant. (See pictures of the subband signals in lecture 8). Important point: The shown transformation (DCT) and the representation by flter banks are equivalent, only other mathematical representations of the same facts. The representation as transformation is usually more favorable for the implementation, the representation as flter bank more favorable for the design of e.g. image processing or coding algorithms. -> one flter -> one subband image This means that an appropriate flter is needed for each of the 8x8 images of the subbands. (64 flters). According to the 64 pictures of the subbands shown above. - We have applied DCT separably horizontally and vertically to the image:

3 Vertikale Teilbänder (für jede der 8 horizontale Teilbänder) Ein DCT horizontale Frequenzen Filter 7 Filter 7 Original Down-sampler Filter 1 Filter 1 Filter 0 Analyse (Coder) Filter 0 8x8 Teilbänder, wie im Bild oben 8 horizontale Teilbänder Image from subband images (8x8=64 images)

4 Filter 0,...,7: the corresponding DCT flters (reversed columns 0 to 7) - Cascade the fltering for horizontal / vertical frequencies Rectangular/square images result as sub-band images. - Each flter flters certain properties from the image. Decoder: An image is reconstructed from sub-bands, frst using up-sampling to resize the subband images to the original size, and the fltering (suitable interpolation) (Note: Bandpass Nyquist!) Bild wird vergrößert durch Einfügen von 7 Nullen nach jedem Pixel Signal in Original-Größe Teilbänder 8 Filter s 7 Addition der TeilbandßBilder Up-Sampler (7 Nullen nach jedem Wert) + Bild im Originalbereich 8 Filter s 0 Können als Interpolations- Filter angesehen werfen Synthesis -> Perfect reconstruction desired

5 Specifcally for the DCT: we have the advantage that it is a square matrix invertible I.e. decoder can be constructed by the inverse matrix. Coder: Teilband- Koeff. Decoder: y = T x DCT x = T -1 y Bild Inverse DCT-Matrix N 1 x(n)= 2 N k=0 y(k) cos( π N k(n+0.5)) A k A k =. 1 für k=0 2 1 sonst (Faktor wird für Orthogonalität gebraucht) Decoder: Synthesis (reconstruction from subbands) Encoder: Analysis (decomposition into subbands)

6 Complete chain: Original - Encoder/Analysis- Synthesis/Decoder- Rektionstructed image Modifcation of the subbands in this chain: By splitting into subbands in this chain, we also have the possibility of modifying subbands in diferent ways. Exampies: - We only keep subband (0.0), i.e. DC, the remaining subbands are set to zero. -> The result is a low-pass fltered image with less detail. Image, reconstructed with only sub-band (0,0) DC:

7 For comparison: original: Observe: The reconstructed image now consists of pixel blocks of DCT size 8x8 pixels! (This can be seen after zooming in) - Observe: the picture does not look so lowpass like (with gradual transitions), but it contains abrupt transitions between the 8x8 blocks! -> "Blocking Artifacts of the DCT -> important drawback of this transform: the

8 artifacts look unnatural. - Next example: We keep the 4 lowest subbands. The result should look less blurry:

9 The used subbands: DC (0,0) 4 tiefste Bänder - Using the 4 lowestt subbands: the image indeed looks sharper, but still has some blocking artifacts (again it can be seen after zooming in).

10 - Next example: We keep all subbands, except DC: All subbands except DC - Observe: Only the edges in the image remain, i.e. the high spatial frequencies This could be a tool for extracting edges.

11 - Next example: All subbands except DC are amplifed by a factor of 2 This enhances the edges in the image by this factor. Edge enhancement by factor 2: