20-Bio-B4 Robotics · December 2015
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format: National Exams, December 2015 — 04-Bio-B4 Digital Image Processing. Three hours, open book (any paper notes or textbooks permitted, but no calculator or computer). Six questions of equal value (20 marks each); five constitute a complete paper and only the first five appearing in the answer book are marked. All six are solved here, because this set is a study resource rather than an examination script. Every question is essay/descriptive (definitions, algorithm/system design) except the convolution-size and complexity items in Question 2, the median-filter nonlinearity proof in Question 2(d)(ii), and the illustrative numeric examples worked into Questions 4 and 6.
Reference texts (the books a candidate should have reviewed for this subject):
Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.
Digital watermarking embeds an imperceptible (or occasionally visible) pattern into an image, typically by perturbing transform-domain coefficients (DCT/DWT mid-frequencies) or pixel LSBs so the mark survives ordinary processing (compression, cropping, mild filtering) while remaining invisible to the eye. It is used for copyright/ownership proof, source-leak tracing (a unique watermark per distributed copy), and tamper/authenticity detection (a fragile watermark that breaks under editing).
Downsampling reduces an image's spatial resolution by keeping only every $D$-th sample along each axis, which reduces storage/bandwidth and speeds subsequent processing (thumbnails, multi-resolution pyramids, coarse-to-fine search). Done naively (point decimation) it violates the sampling theorem for any content above the new, lower Nyquist rate and introduces aliasing; correct practice always low-pass filters the image first to remove those frequencies before discarding samples (worked quantitatively in Question 4(d)).
The CIE family defines colour independently of any particular sensor or display, starting from the CIE XYZ tristimulus values (derived from human colour-matching experiments) and re-parameterizing them into perceptually motivated spaces such as CIELab, whose $L^*$ (lightness) and $a^*, b^*$ (colour-opponent) axes are designed so that Euclidean distance approximates perceived colour difference ($\Delta E$). It is used wherever colour must be compared, communicated, or reproduced consistently across devices — colour management/ICC profiles, print-to-screen colour matching, and perceptually-weighted image-quality or segmentation metrics.
The 2D FFT is the separable, fast $O(MN\log(MN))$ algorithm for computing the 2D discrete Fourier transform of an $M\times N$ image (applying 1D FFTs along the rows, then along the columns of the result), decomposing the image into a sum of complex sinusoids at every spatial frequency. It underlies fast convolution/filtering (Question 2(c)), frequency-domain restoration (Wiener deconvolution), spectral analysis of periodic texture, and image compression schemes that operate in a transform domain.
SIFT and SURF are algorithms that detect distinctive keypoints (typically at blob-like local extrema found across a scale-space, e.g. a difference-of-Gaussians pyramid) and describe each with a local gradient-orientation histogram, giving a descriptor that is invariant (or near-invariant) to image scale, in-plane rotation, and moderate illumination/viewpoint change. They are used for image matching and registration, panorama stitching, object recognition, and camera-pose/structure-from-motion estimation, wherever the same physical point must be re-identified across images taken from different distances or angles.
The Gabor transform convolves an image with a set of sinusoidal gratings windowed by a Gaussian envelope, each tuned to a specific spatial frequency and orientation, giving a joint space-frequency-orientation decomposition that is a close engineering model of simple-cell receptive fields in the mammalian visual cortex. It is used for texture analysis and classification, iris recognition, and as an oriented, multi-scale edge/feature detector where both the location and the orientation of local structure matter.
Unlike the critically-sampled discrete wavelet transform (which downsamples at every level and is therefore not shift-invariant), overcomplete variants — the undecimated/à-trous transform, and complex/dual-tree wavelets with directionally-selective, approximately shift-invariant subbands — skip the downsampling step (or add extra oriented filter pairs), producing a redundant representation. They are used wherever shift-invariance or better directional selectivity matters more than compactness: denoising (avoiding the pseudo-Gibbs artifacts a critically-sampled DWT produces near edges), and texture/edge analysis needing several oriented sub-bands per scale.
