From c436d75a7ba208e3bb62b7908ae857ff382989e3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Adam=20=C5=A0t=C4=9Bp=C3=A1nek?= Date: Sun, 7 Jan 2024 01:40:57 +0100 Subject: [PATCH] Move more stuff from cafour.cz --- pa213/index.ad | 463 +++++++++++++++++++++++++++++++++++++++ pv131/01_intro.ad | 46 ++++ pv131/02_convolution.ad | 30 +++ pv131/03_fourier.ad | 58 +++++ pv131/04_more_fourier.ad | 36 +++ pv131/05_filtering.ad | 50 +++++ pv131/06_pyramids.ad | 67 ++++++ pv131/07_nonlinear.ad | 51 +++++ pv131/08_segmentation.ad | 61 ++++++ pv131/09_hough.ad | 86 ++++++++ pv131/10_description.ad | 136 ++++++++++++ pv131/11_techniques.ad | 94 ++++++++ pv131/img/07.png | Bin 0 -> 35586 bytes pv131/img/07solution.png | Bin 0 -> 14333 bytes pv131/img/2a.png | Bin 0 -> 2099 bytes pv131/img/cnn.png | Bin 0 -> 275785 bytes pv131/img/ftsobel.png | Bin 0 -> 97257 bytes pv131/img/gaussianfd.jpg | Bin 0 -> 24038 bytes pv131/img/gaussiansd.jpg | Bin 0 -> 30115 bytes pv131/img/stripefd.jpg | Bin 0 -> 15416 bytes pv131/img/stripesd.jpg | Bin 0 -> 20012 bytes pv131/index.html | 15 ++ pv131/pv131zk.ad | 334 ++++++++++++++++++++++++++++ 23 files changed, 1527 insertions(+) create mode 100644 pa213/index.ad create mode 100644 pv131/01_intro.ad create mode 100644 pv131/02_convolution.ad create mode 100644 pv131/03_fourier.ad create mode 100644 pv131/04_more_fourier.ad create mode 100644 pv131/05_filtering.ad create mode 100644 pv131/06_pyramids.ad create mode 100644 pv131/07_nonlinear.ad create mode 100644 pv131/08_segmentation.ad create mode 100644 pv131/09_hough.ad create mode 100644 pv131/10_description.ad create mode 100644 pv131/11_techniques.ad create mode 100644 pv131/img/07.png create mode 100644 pv131/img/07solution.png create mode 100644 pv131/img/2a.png create mode 100644 pv131/img/cnn.png create mode 100644 pv131/img/ftsobel.png create mode 100644 pv131/img/gaussianfd.jpg create mode 100644 pv131/img/gaussiansd.jpg create mode 100644 pv131/img/stripefd.jpg create mode 100644 pv131/img/stripesd.jpg create mode 100644 pv131/index.html create mode 100644 pv131/pv131zk.ad diff --git a/pa213/index.ad b/pa213/index.ad new file mode 100644 index 0000000..9667b88 --- /dev/null +++ b/pa213/index.ad @@ -0,0 +1,463 @@ += PA213 Advanced Computer Graphics +:page-layout: layouts/essay.tsx +:page-math: true +:page-liquid: true + +.Lectures +1. Introduction +** link:https://is.muni.cz/auth/el/fi/jaro2022/PA213/um/slides/Intro.pdf[Slides] + +2. Fluid Simulation and Rendering +** link:https://is.muni.cz/auth/el/fi/jaro2022/PA213/um/slides/FluidSimulationAndRendering.pdf[Slides] + +3. Bounding Volume Hierarchies for Ray Tracing +** link:https://is.muni.cz/auth/el/fi/jaro2022/PA213/um/slides/BoundingVolumeHierarchiesforRayTracing.pdf[Slides] +** link:https://is.muni.cz/auth/el/fi/jaro2022/PA213/um/slides/BoundingVolumeHierarchiesforRayTracing.mp4[Video] + +4. Fractals, L-Systems +** link:https://is.muni.cz/auth/el/fi/jaro2022/PA213/um/slides/LSystemsFractals.pdf[Slides] +** link:https://is.muni.cz/auth/el/fi/jaro2022/PA213/um/slides/LSystemsFractals.mp4[Video] + +5. GPU Acceleration for General Processing +** link:https://is.muni.cz/auth/el/fi/jaro2022/PA213/um/slides/Cuda.pdf[Slides] +** link:https://is.muni.cz/auth/el/fi/jaro2022/PA213/um/slides/Cuda.mp4[Video] + +6. Computational Photography +** link:https://is.muni.cz/auth/el/fi/jaro2022/PA213/um/slides/ComputationalPhotograpy.mp4[Video] + +7. Terrains +** link:https://is.muni.cz/auth/el/fi/jaro2022/PA213/um/slides/Terrains.pdf[Slides] +** link:https://is.muni.cz/auth/el/fi/jaro2022/PA213/um/slides/Terrains.mp4[Video] + +8. Participating Media in Computer Graphics +** link:https://is.muni.cz/auth/el/fi/jaro2022/PA213/um/slides/ParticipatingMedia.pdf[Slides] +** link:https://is.muni.cz/auth/el/fi/jaro2022/PA213/um/slides/ParticipatingMedia.mp4[Video] + +9. From 2D Sketch to 3D Animated Character via Quadratic Programming +** link:https://is.muni.cz/auth/el/fi/jaro2022/PA213/um/slides/2DSketchTo3DCharacter.pdf[Slides] +** link:https://is.muni.cz/auth/el/fi/jaro2022/PA213/um/slides/2DSketchTo3DCharacter.mp4[Video] + +."Ideas" +* types of BVH creation, +* Terrain map optimizations through the similar to MIN MAP method (closer with more detial further with less based on multiple different level of detail maps), +* 2 types of parallelism, +* ray-tracing / slicing in volume rendering, +* Hilbert curve. + +== Fluid Simulation and Rendering + +Smoothed Particle Hydrodynamic:: ++ +* simulate fluid using a set of particles (Lagrange) +* compute forces using Euler +* smooth properties of particles into continuous fields +* on the fields we use momentum and incompressibility +** stem:[\frac{\partial u}{\partial t} = - (u \cdot \nabla) u - \frac{1}{p} \nabla p + v \nabla^2 u + g] +** stem:[\nabla \cdot u = 0] (isn't needed because we simulate particles) +** particles move with the fluid -> stem:[-(u \cdot \nabla) u] is not needed +** we only need to solve stem:[\frac{\partial u}{\partial t} = - \frac{1}{p} \nabla p + v \nabla^2 u + g] +** we recall Newton's 2nd law of motion -> stem:[\frac{f_i}{m_i} = - \frac{1}{p(x_i)} \nabla p(x_i) + v \nabla^2 u(x_i) + g] + +=== Euler approach + +-- +- vector field +- each fixed point on a grid stores fluid velocity +- for each point in grid we store: +-- stem:[u_{i,j}] -- fluid velocity +-- stem:[p_{i,j}] -- pressure +-- ... +-- + +Incompressible constraint:: +volume of any subregion of the fluid is constant over time + +Homogeneous constraint:: +fluid density doesn't change + +Navier-Stokes equations:: ++ +* velocity -> vector field stem:[u(x, t)] +* pressure -> scalar field stem:[p(x, t)] +* partial differential equations define changes in stem:[u] over time +* stem:[\frac{\partial u}{\partial t} = - (u \cdot \nabla) u - \frac{1}{\rho} \nabla p + v \nabla^2 u + g] +* incompressibility: stem:[\nabla \cdot u = 0] +** stem:[p(x, t)] -- pressure scalar field +** stem:[\rho] -- density of fluid +** stem:[v] -- viscosity -- resistance to deformation +** stem:[g(x, t)] -- acceleration vector field, e.g. gravity +** stem:[\nabla p = \left( \frac{\partial p}{\partial x}, \frac{\partial p}{\partial y}, \frac{\partial p}{\partial z} \right)^ T] -- gradient operator +** stem:[\nabla \cdot u = ... = \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial u}{\partial z}] -- divergence operator +** stem:[u \cdot \nabla] -- directional derivative +** stem:[\nabla^2 u = \frac{\partial^2 u}{\partial^2 x} + \frac{\partial^2 u}{\partial^2 y} + \frac{\partial^2 u}{\partial^2 z}] -- Laplacian +** custom equations (i.e. temperature, smoke density) can be added + +Rendering:: ++ +* ray marching -- sample 3d texture along the ray and sum up +* first render box back faces -- into alpha +* second render box front faces -- into z +* extract start points and ray length +* special case: camera intersects volume -> use near plane instead +* "banding" artifact: caused by integral number of samples -> weigh values of samples by their distance from the camera + + +=== Lagrange approach + +-- +- fluid is a set of particles +- each particle stem:[P_i] stores: +-- stem:[m_i] -- mass +-- stem:[p_i] -- position vector +-- stem:[u_i] -- velocity vector +-- stem:[f_i] -- total external force +-- + +Newton's equations of motion:: ++ +* stem:[\frac{\partial p_i}{\partial t} = u_i] +* stem:[\frac{\partial u_i}{\partial t} = \frac{f_i}{m_i}] + +Ordinary Differential Equation (ODE):: +a differential equation of one variable (it may contain functions of that variables and its derivatives) + +Moving particles:: ++ +* using an ODE: stem:[\frac{\partial y}{\partial t} = f(y, t)] +* solving using Euler: stem:[y(t_0 + \Delta t) = y_0 + \Delta t f(y_0, t_0)] +* solving using midpoint: stem:[y(t_0 + \Delta t) = y_0 + \Delta t f(y_0 + \frac{\Delta t}{2} f(y_0, t_0), t_0 + \frac{\Delta t}{2})] + +External forces:: ++ +* gravity: stem:[m_i \cdot g], where typically stem:[g = (0, 0, -10)^T] +* Lennard-Jones force between particles + +=== Height-field surface approximation + +- model of fluid surface stem:[h(x, y, t)] +- change of stem:[h] in time is given by stem:[\frac{\partial^2 h}{\partial t^2} = v^2 \nabla^2 h] +- stem:[v] is the speed of waves +- we need an auxilary function to solve discretely + +== Bounding Volume Hierarchies (BVH) for Ray Tracing + +-- +- BVH reduce ray/scene traversal complexity from stem:[\mathcal{O}(n)] to stem:[\mathcal{O}(\log n)]. +-- + +BVH construction methods:: ++ +- top-down +- bottom-up + +Efficient BVH:: ++ +- fast insertion query +- fast construction / updates + +Surface Area Heuristic (SAH):: ++ +- Snažíme se minimalizovat povrchy bounding boxů vnitřních uzlů BVH. ++ +[stem] +++++ +C(T) = \frac{1}{SA(T)} \left[ c_T \cdot \sum_{N \in \text{inner nodes}} SA(N) ++ c_I \sum_{N \in \text{leaves}} SA(N) \cdot t_N \right] +++++ ++ +Vysvětlivka: +- stem:[SA(N)] -- surface area of node stem:[N] +- stem:[T] -- the root of the BVH +- stem:[C(T)] -- global BVH cost +- stem:[c_T] -- traversal cost +- stem:[c_I] -- intersection cost +- stem:[t_N] -- triangles per leaf + +Progressive Hierarchical Refinement (PHR):: ++ +- progressivelly refined cut of existing auxilary BVH (using LBVH) + +Parallel Locally Ordered Clustering (PLOC):: ++ +- sort triangles based on Morton codes +- process the sorted sequence bottom-up +- merge closest clusters, repeat until only one cluster remains (in parallel) + +Extended Morton Codes (EMS):: ++ +- 1st idea: encode object size +- 2nd idea: variable lengths based on size of scene bbox + +Parallel Insertion Based Optimization:: ++ +- Search for new positions of all nodes. +- Resolve conflicts by picking higher cost reduction. +- Reinsert not conflicting nodes. + +T-SAH:: +modified SAH for animations + +Ray Classification for BVH Traversal:: ++ +- ray space subdivision + precomputed compact candidate lists +- algorithm for optimized candidate list computation + +== Fractals, L-Systems + +=== Fractals + +-- +- from latin "fractus" -- broken, fractured +- in nature +- self-similar +- Koch line & snowflake +-- + +Space filling curves (SFC):: ++ +- fills a unit square / cube / hypercube / ... +- iterative construction allows arbitrary resolution +- Z-order curve / _Morton space filling curve_ / Morton code +-- **maps multidimensional data do one dimension while preserving locality of data points** +- _Hilbert curve_ +-- hranatá +-- sometimes more effective than Morton CFC +- useful in CG data structures (e.g. quadtrees, octrees, BVHs, texture mapping in GPU memory) +- _Moore curve_ +-- like Hilbert, but loopier +- _Peano curve_ +-- probably the oldest one +- _Dragon curve_ +- _Gosper curve / island_ + +Topological dimension:: ++ +- the number of parameters needed to describe a point "inside" the object +- point -> 0 +- line -> 1 (stem:[t]) +- plane -> 2 +- stem:[\R^3] -> 3 +- ... + +Fractal dimension:: ++ +- if we divide into N pieces, we need to scale by N to see one piece -> top scale: stem:[s = \frac{1}{N}], t.j. stem:[Ns^1 = 1] +- in case of squares: stem:[s = \frac{1}{\sqrt{N}}], t.j. stem:[Ns^2 = 1] +- fractal dimension stem:[D] is stem:[Ns^D = 1] -> stem:[D = \frac{\log N}{\log \frac{1}{S}}] +- measurement -> box counting method -> how many boxes cover the object? + +Lindenmayer system (L-systems):: ++ +- parallel rewriting systems +- typically model growth of plants +- Prusinkiewicz, Hanan +- D0L-system +- bracketed, open, parametric, context-sensitive, stochastic, combined + +Iterated function systems:: ++ +* fractal is made by smaller copies of itself +* contractive -- bring points closer together +* Sierpinski triangle +* Menger sponge +* dvojice stem:[(T, P)], kde stem:[T] je množina transformací a stem:[P] je množina pravděpodobností +** součet pravděpodobností = 1 +** scale transformations < 1 (contractive) + +Fractals in complex plane:: ++ +- Mandelbrot set +- Julia sets +- ??? + +Terrain generation by midpoint displacement:: ++ +- pseudo-fractal +- enhanced midpoint displacement +- ??? + +== GPU Acceleration for General Processing + +Task parallelism:: +- the problem is decomposed into parallel tasks +- tasks are complex but different +- they are difficult to synchronize +- best for a low number of high-performance processors/cores (CPUs) + +Data parallelism:: +- on the level of data structures +- same operation on multiple elements of a data structure +- best for a high number of lower-performance processors/cores (GPUs) + +== Computational Photography + +-- +- photography + computation = better pictures +- combining multiple exposures into one -- elimination of over- and under-exposures +- relighting using synthetized renders from terrain data +- dehazing +- changing depth of field +- image forensics +-- + +Obtaining position and orientation of a camera:: ++ +- GeoPose3K - a bunch of images of mountains with known location and orientation +- for each image they render depth, normals, semantic segmentation + +Cross-locate:: ++ +* for each image it produces a descriptor +* it uses these descriptors to find the closest match +* NN for creating the descriptors based on a database -- silhouette edges, semantic segmentation +** mostly T-junctions + +== Terrains + +Tile-based:: +squares and hexagones + +Mesh-based:: +regular and iregular (TIN) + +Height map:: ++ +* 2d texture storing elevation +* cannot store caves and arches +* given 4 points, the triangulation is ambiguous +* rendered by indexing and triangle strips + +Digital countours:: +control points of spline curves + +Voxels:: ++ +* 3d grids +* render cubes, marching cubes, raytracing, raymarching, machine learning + +Transitions:: ++ +* manual or automatic +* texture splatting -- combines multiple textures by applying an alpha map +** with tiles -- 16 edge transitions, 16 corner transitions + +Real Terrain Data:: ++ +* Digital Elevation Models (DEM) +* ČR: ČUZK +* World: USGS + +Larger outdoor environments:: +require a lot of bandwidth, this is solved by: ++ +* quadtrees (quads), ROAM (triangles) -- pre-defined multiresolution terrain representation +** distance from camera and geometry -> required fidelity +** frustum culling -- nodes outside of camera are not rendered at all +** require processing by the CPU +* clipmaps +** almost all happens on GPU +** terrain as a 2D elevation image -> mipmap pyramid +** caches a square window of n-by-n samples in each level (centered in the viewer, finer-level -> smaller space) +** boundaries are too visible -> smooth morphing between levels +** rendering: break rings into smaller 2D footprint pieces -> reuse in and across levels +** optimization: viewer height -> coarser levels +* tesselation: modern alternative/addition to ROAM/Clipmaps +** used with displacement maps + +=== Artifical Terrain Map (Fractal geometry) + +Mid-point displacement:: ++ +1. Split each line into two segments +2. Move the new points by random offsets (smaller each iteration) +3. Repeat (until desired resolution). ++ +Works in 2D by splitting a quad into four segments. + +Random faults:: ++ +1. Split the plane with a line. +2. Move one side up, other down. +3. Repeat (until desired smoothness). + +2D Perlin noise:: +by combining multiple noise functions fractally, we get a pretty good terrain + +Fractional Brownian Motion:: ++ +* position of a given object changes in random increments +* integral of white noise +* fractional -> memory -> increments are not dependent +** Hurst exponent, Gain factor + +Normals:: ++ +* central differences +* derivates of noise in lattice grid -> normals + +== Participating Media in Computer Graphics + +light interacts between and below surfaces + +Volume rendering:: ++ +* 2D pixel -> 3D voxel +* useful for rendering of atmospheric effects, fluids, translucent solids, biological tissues, medical data, ... + +Radiative transfer in volumes:: ++ +* can be solved analyticall for homonogenous volumes +* has to be solved by numerical approximation otherwise +* back-to-front / front-to-back compositing +** front-to-back can be terminated early when alpha ~ 1 + +Raycasting / slicing:: ++ +* Raycasting: sampling happens on equidistant points along each ray +** concentric spherical shells +** easier to implement, early termination is trivial, more flexible than slicing +* Slicing: shells are approximated using view-aligned planes +** better for memory access + +Volume rendering pipeline:: ++ +1. reconstruction -- turn sets of discrete samples into 3D functions using interpolation / filters +2. classification -- mapping of data attributes to optical properties (e.g. transparency, color) +** can happen before or after reconstruction +3. shading -- make structures more realistic by applying an illumination model +** gradient vector instead of normals +4. compositing +** accumulation -- optical model of radiative transfer (emission + absorption) +** first hit +** averaging +** maximum intensity + +Global volume illumination:: ++ +* volumetric photon mapping +** photons are shot from light sources, once a photon interacts with a medium, it is stored in a photon map +** ray casting and estimation using photon maps +* interactive approximations + +== From 2D Sketch to 3D Animated Character via Quadratic Programming + +link:monstermash.zone[MonsterMash] + +Non-linear programming:: +solving an optimization problem where some of the constraints are not linear + +Quadratic programming:: +non-linear programming with quadratic functions + +Sketch-based modeling:: ++ +1. draw a sketch +2. order parts according to depth +3. part openings are stitched onto the body +4. ARAP-L: deformation + inflation + +ARAP-L(ayered) deformation:: ++ +- can have positional constraints +- joint inflation + deformation +- produces better results than inflation, then deformation diff --git a/pv131/01_intro.ad b/pv131/01_intro.ad new file mode 100644 index 0000000..c9aabf0 --- /dev/null +++ b/pv131/01_intro.ad @@ -0,0 +1,46 @@ += 1. Introduction, Images and Their Properties + +[.question] +Jaký je rozdíl mezi pixelem a voxelem? + +Pixel:: +Picture element + +Voxel:: +Volume element + +[.question] +Nadefinujte digitální barevný obraz. + +Funkce + +[.question] +Vysvětlete rozdíl mezi izotropním a anizotropním obrazem. + +Izotropní obraz:: +Má stejnou vzorkovací frekvenci ve všech dimenzích. + +Anizotropní obraz:: +Není izotropní. + +[.question] +Odvoďte vztah, kolik úrovní intenzit má N-bitový šedotónový obraz. + +stem:[2^N], protože tolik hodnot narveme do N-bitového čísla bez znaménka. + +[.question] +Vyjmenujte 3 základní druhy šumu, které jsou přítomny v obraze pořízeném běžnou kamerou. + +Aditivní šum:: +Převážně Gaussián. Způsoben amplifikací v elektronice. + +Impulsní šum:: +Náhodné očernění nebo obělení pixelu. Sůl a pepř. + +Poissonůn šum:: +Fotony nebo co. + +[.question] +Jaký je rozsah vlnových délek, které vnímá lidské oko? + +[380, 740] nm diff --git a/pv131/02_convolution.ad b/pv131/02_convolution.ad new file mode 100644 index 0000000..7a71d34 --- /dev/null +++ b/pv131/02_convolution.ad @@ -0,0 +1,30 @@ += 2. Convolution and Linear Filters + +[.question] +Vysvětlete, co je to konvoluční jádro. + +Matice, která se postupně posouvá přes obraz a jejíž hodnoty jsou použity jako váhy přiv váženém součtu příslušných hodnot. + +[.question] +Odvoďte, jakou časovou složitost má konvoluce obrazu I o velikosti K x L s obecným konvolučním jádrem velikosti M x N. + +stem:[O(KLMN)] + +[.question] +Uveďte příklad 2D vyhlazovacího filtru, který zachovává průměrnou intenzitu obrazu. + +Gaussián. + +[.question] +Co to znamená, že filtr je LSI? Nepřekládejte, ale vysvětlete význam. + +LSI := Linear shift invariant + +stem:[O(T_(a,b) (f(x, y)))) = T_(a, b) (O(f(x, y)))] + +[.question] +Formálně zapište vlastnost distributivity diskrétní konvoluce vzhledem ke sčítání. Tuto vlastnost dokažte. + +[.question] +Formálně nadefinujte 2D lineární filtr. Uveďte konkrétní příklad takového filtru a formálně dokažte, že vyhovuje Vaší +definici. diff --git a/pv131/03_fourier.ad b/pv131/03_fourier.ad new file mode 100644 index 0000000..8617097 --- /dev/null +++ b/pv131/03_fourier.ad @@ -0,0 +1,58 @@ += 3. 1D Fourier Transform + +[.question] +Mějme funkci stem:[f(x) = 0.4 * sin(2\pi(x-1))]. Určete její amplitudu, frekvenci a fázový posun. + +* Amplituda: 0.4, protože sinus +* Frekvence: 1 Hz, protože sinus má frekvenci stem:[2\pi], ale tady už stem:[2\pi] je +* Prostorová / časová frekvence: 1 Hz / stem:[m^-1] za předpokladu, že výsledek je v metrech +* Úhlová frekvence: stem:[2\pi] +* Posun: 1 + +[.question] +Mějme komplexní číslo z = 1 - i. Určete jeho amplitudu a fázi. + +* Amplituda: stem:[sqrt(1^2 + (-1)^2) = sqrt(2)] +* Fáze: stem:[\text{atan2}(y, x) = \text{atan2}(-1, 1) = -0.785] + +[.question] +Formálně nadefinujte 1D diskrétní Fourierovu transformaci a rozhodněte, zda se jedná o lineární operátor. Své tvrzení +podložte formálním důkazem. + +Lineární operátor stem:[f]:: +[stem] +==== +f(I * J) = f(I) * f(J ) +==== + +[.question] +Jaká operace ve frekvenční doméně odpovídá konvoluci v obrazové doméně? + +Násobení. + +[.question] +Co je obrazem Gaussiánu (umístěného do počátku souřadné soustavy) ve frekvenční doméně? + +Gaussian ale tak nějak menší nebo co. + +[source, matlab] +---- +G = fspecial('gaussian', 21, 5); +surf(G) +surf(abs(double(ft(G)))) +---- + +image::pv131/gaussiansd.jpg[] +image::pv131/gaussianfd.jpg[] + +[.question] +Kolik bázových funkcí je zapotřebí pro realizaci FT nad 1D diskrétním signálem délky N? Jak dlouhý bude výstupní +signál ve frekvenční doméně? + +1D DFT:: +[stem] +==== +Phi(omega_x) = sum_(x=0)^(N-1) f(x) * e^(-i(2pi)/N * omega_x * x) +==== + +Výstupní signál bude dlouhý stem:[N]. diff --git a/pv131/04_more_fourier.ad b/pv131/04_more_fourier.ad new file mode 100644 index 0000000..d33b2c9 --- /dev/null +++ b/pv131/04_more_fourier.ad @@ -0,0 +1,36 @@ += 4. 2D/3D Fourier Transform + Filtering in Frequency Domain + +[.question] +Odvoďte časovou složitost 2D DFT pro obraz velikosti M x N. + +2D DFT je 1D DFT aplikovaná dvakrát. Pokud složitost vnitřní DFT je stem:[O(N^2)] pak složitost vnější DFT bude +stem:[O(M^2*N^2)]. + +[.question] +Vysvětlete, proč se při zobrazení amplitud ve Fourierově doméně používá logaritmická škála. + +Protože je to pak líp vidět. ++ +Největší amplituda zkrátka bývá tak velká, že při lineárním zobrazení nejsou ostatní amplitudy vidět. Jenže ty ostatní +amplitudy nás zajímají taky, tak to celé dáme sežrat logaritmu, čímž se největší peak zmenší. + +[.question] +Dokažte, že 2D diskrétní Fourierova transformace je separabilní. + +[.question] +Znázorněte 1D frekvenční doménu digitálního signálu délky N. Popište osy a následně nakreslete 1D nízkofrekvenční box +filtr, který odstraní K nejvyšších frekvencí. + +Nakresli krabici s hranou stem:[2 * (N - K)] "položenou" na počátku. + +[.question] +Znázorněte 2D frekvenční doménu digitálního obrazu velikosti N x N. Popište osy a následně nakreslete 2D +nízkofrekvenční kruhový filtr, který odstraní frekvence vyšší než stem:[w_(threshold) = N/4]. + +Nakresli válec s poloměrem podstavy N/4 "položený" na počátku. + +[.question] +Znázorněte 2D frekvenční doménu digitálního obrazu velikosti 8 x 8. Určete, který Fourierův koeficient je komplexně +sdružený s Fourierovým koeficientem na pozici stem:[(w_x, w_y) = (5, 7)]. + +Ten středově souměrný podle počátku, stem:[(-5, -7)]. diff --git a/pv131/05_filtering.ad b/pv131/05_filtering.ad new file mode 100644 index 0000000..5aa0c54 --- /dev/null +++ b/pv131/05_filtering.ad @@ -0,0 +1,50 @@ += 5. Correspondence between Spatial and Frequency Filtering, Sampling Theorem + +[.question] +Zobrazte v Matlabu, jak vypadá Sobelův filtr převedený do frekvenční domény. Vysvětlete, které frekvence jsou +potlačovány a které naopak zvýrazňovány. + +Zelená čerchovaná čára. Potlačeny jsou velmi nízké a velmi vysoké frekvence. + +image::pv131/ftsobel.png[] + +[source, matlab] +---- +plot(abs(fft(padarray([1 0 -1]', 20)))) +---- + +[.question] +Vysvětlete původ vertikální čáry v levém dolním obrázku na slidu č. 3. + +image::pv131/stripesd.jpg[] +image::pv131/stripefd.jpg[] + +Kvůli prvnímu a poslednímu řádku vzniká periodicita. + +[.question] +Formálně nadefinujte 2D ideální nízkofrekvenční filtr s obdélníkovým profilem. Odvoďte i jemu odpovídající filtr v +prostorové doméně. + +Obdélník ve frekvenční doméně nad nízkými frekvencemi, které nás zajímají. V prostorové doméně vypadá jako +stem:[k * (\text{sinc}(ax)) / (ax)]. + +[.question] +Mějte libovolný 2D digitální obraz, který následně poškodíme gaussovským aditivním šumem. Vysvětlete, v čem se liší +frekvenční spektrum původního a zašuměného obrazu. + +Šum bude i ve frekvenční doméně. + +[.question] +Mějme 2D digitální šedotónový obraz velikosti 256x512 pixelů. Rozhodněte, zda je obraz frekvenčně omezený. Pokud ano, +určete nejvyšší frekvenci v něm obsaženou (v každé ose zvlášť) a uveďte min. vzorkovací frekvenci +(dle Nyquistova teorému), která navzorkuje obraz tak, aby byl plně rekonstruovatelný. Pokud ne, zdůvodněte proč. + +* Maximální frekvence podle osy x: 256 +* Maximální frekvence podly osy y: 512 + +Nyquistův teorém:: +Band-limited signal should be sampled at a rate of at least twice its highest-frequency component (i.e. the sampling +frequency should be equal at least to the Nyquist rate). If the band-limited signal is sampled at the Nyquist rate, +it is possible to reconstruct it exactly. + +Podle Nyquistova teorému chceme vzorkovací frekvenci 512 pro osu x a 1024 pro osu y. diff --git a/pv131/06_pyramids.ad b/pv131/06_pyramids.ad new file mode 100644 index 0000000..fc5ac79 --- /dev/null +++ b/pv131/06_pyramids.ad @@ -0,0 +1,67 @@ += 6. Multi-scale analysis, Pyramids, Wavelets + +[.question] +Vysvětlete princip konstrukce Laplaceovy pyramidy o 4 úrovních. + +Pyramida:: +Série postupně downsamplovaných verzí původního obrázku. + +Gaussovská pyramida:: +Na obrázky je při downsamplování aplikován Gaussián. Je low-pass filtering (LPF) pyramidou. + +Laplaceova pyramida:: +Pyramida map hran. Typicky vzniká aplikací Laplacián na Gaussovskou pyramidu, tedy operace při downsaplovaní je ve +finále Laplacian of Gaussian (LoG). Je high-pass filtering (HPF) pyramidou. + +Reprezentace obrázku pomocí pyramidy:: +HPF pyramida a LPF pyramida se vzájemně doplňují. LPF pyramida se však lépe komprimuje, čehož se využívá při uchovávání +obrázků. LPF úroveň k se dá zkombinovat s HPF úrovní k+1 a vznikne (skoro) LPF úroveň k+1. + +Konstrukce truncated HPF pyramidy o 4 úrovních:: +Originál je na úrovni j, považujme ho za stem:[LPF_j]. Nechť stem:[LPF_{j-1} = \text{downsample}(\text{blur}(LPF_j))] +a stem:[HPF_{j} = LPF_j - \text{blur}(\text{upsample}(LPF_{j-1}))]. ++ +Pyramida o 4 úrovních by se skládala z stem:[HPF_j, HPF_{j-1}, HPF_{j-2}, LPF_{j-3}]. + +[.question] +Mějme jednorozměrný diskrétní signál I délky 64. Pro výpočet 1D-DWT bylo použito stem:[j_0=4]. Určete, jaký dlouhý je +výsledný signál, tj. kolik je dohromady aproximačních a waveletových koeficientů. + +[stem] +++++ +J = 6 \\ +\#D_5 = 32 \\ +\#D_4 = 16 \\ +\#A_4 = 16 +++++ + +[.question] +Je dán dvourozměrný diskrétní obraz I o velikosti 256x256 pixelů a na něm provedená 2D-DWT s Haarovou bázovou funkcí +až do úrovně stem:[j_0=6]. Určete, kolik bude ve výsledném transformovaném obraze aproximačních koeficientů. + +[stem] +++++ +\#A_6 = 64 \times 64 +++++ + +[.question] +Je dán dvourozměrný diskrétní obraz I o velikosti 512x512 pixelůa na něm provedená 2D-DWT s Haarovou bázovou funkcí +až do úrovně, kdy je ve výsledném obraze přítomen jen 1 aproximační koeficient. Určete, kolik bylo třeba provést +rozkladů na nízkofrekvenčních a vysokofrekvenční složky. + +Jelikož nejnižší úroveň je stem:[j_0 = 0] a stem:[J = 9], je třeba 9 rozkladů. + +[.question] +Vynulování aproximačních koeficientů odpovídá eliminaci nejnižších frekvencí v původním obraze. Vytvořte matlabovskou +funkci, která tímto způsobem realizuje ve čtvercovém obraze detekci hran. + +[source, matlab] +---- +function [O] = detectEdges(I) + [C,S] = wavedec2(double(I), 8, 'haar'); + C(1:256) = 0; + O = dip_image(waverec2(C, S, 'haar')); +end +---- + +Pozn.: Nezdá se mi, že tohle je to, co chtějí. Navíc to funguje jen pro rozlišení 256x256. diff --git a/pv131/07_nonlinear.ad b/pv131/07_nonlinear.ad new file mode 100644 index 0000000..cc4ad2c --- /dev/null +++ b/pv131/07_nonlinear.ad @@ -0,0 +1,51 @@ += 7. Nonlinear Filtering + +[.question] +Mějme 2D diskrétní obrázek stem:[I] a dva 2D nelineární filtry stem:[A] a stem:[B]. Rozhodněte, zda platí +stem:[A(B(I))=B(A(I))]. + +Ne. Představte si za I šachovnici, za A max filtr a za B min filtr. (Viz slide č. 46.) + +[.question] +Mějme 2D diskrétní obrázek stem:[I], 2D nelineární filtr stem:[A] a 2D lineární filtr stem:[B[]. Rozhodněte, zda +platí stem:[A(B(I))=B(A(I))]. + +Ne. Představte si zašumelý obrázek, za A median a za B gaussian. (Viz slide č. 47.) + +[.question] +Vysvětlete princip efektivní implementace 1D mediánového filtru pomocí klouzajícího okna. + +Viz cvičení 7. + +Máme okno s velikostí N, které na začátku seřadíme. Pro každý pixel na pozici I nahradíme jeho intenzitu v okně +intenzitou pixelu I + N a do výstupního obrazu zapíšeme medián okna. + +[.question] +Zapište difúzní filtraci obrazu formou parciální diferenciální rovnice s počáteční podmínkou a vysvětlete jednotlivé +členy v této rovnici. + +Difúzní filtrování:: +Založeno na myšlence difúze z fyziky. ++ +[stem] +++++ +\partial_t u = \text{div}(g * \nabla u) = g * \text{div} (\nabla u) +++++ ++ +Pokud g je konstanta, pak se jedná o lineární difúzi. ++ +[stem] +++++ +\partial_t u = g * \Delta u +++++ ++ +[stem] +++++ +\frac{\partial u (x, y, t)}{\partial t} = u(x, y, t) \otimes L +++++ + +[.question] +Mějme obraz poškozený impulzním šumem. Rozhodněte, který z dvojice filtrů medián a gaussián je vhodný k potlačení +tohoto šumu. Své tvrzení zdůvodněte. + +Medián je supr na sůl a pepř, protože osamocenou změnu intenzity dopočítá z okolí, zatímco Gaussian ji rozmaže. diff --git a/pv131/08_segmentation.ad b/pv131/08_segmentation.ad new file mode 100644 index 0000000..870a9ff --- /dev/null +++ b/pv131/08_segmentation.ad @@ -0,0 +1,61 @@ += 8. Detection, Segmentation, Tracking + +[.question] +Vysvětlete rozdíl mezi detekcí a segmentací objektů. + +Detekce objektů:: +Snaha najít a klasifikovat v obrazu objekty. + +Segmentace objektů:: +Hledání *oblastí* obrazu, které odpovídají objektům. + +[.question] +Implementujte v Matlabu vlastní funkci realizující hysterézní prahování: +`[vystup] = function hystThreshold(vstup, nizkyPrah, vysokyPrah)` + +[source, matlab] +---- +% dipimage required +[vystup] = function hystThreshold(vstup, nizkyPrah, vysokyPrah) + low = vstup>nizkyPrah; + high = vstup>vysokyPrah; + vystup = reconstruction(nizkyPrah, vysokyPrah); +end +---- + +[.question] +Nadefinujte morfologickou operaci "black top-hat" a uveďte příklad jejího použití. + +Black Top-Hat:: +Rozdíl vstupního obrazu a jeho uzavření. Používá se k extrahování černých detailů, srovnání bílého pozadí, atd. + +Uzavření:: +Snaha o obnovu po dilataci. Takže eroze po dilataci. ++ +[stem] +++++ +\phi_B(x) = \epsilon_{\overline B}(\delta_B(x)) +++++ + +[.question] +Uveďte příklad alespoň 3 texturových deskriptorů, které lze použít při předzpracování obrazu tak, aby následná +segmentace byla realizovatelná jen s pomocí standardního globálního prahování. + +Barva:: +To dává smysl, ne? + +Rosenfeld et al.:: +Textura je "změřena" v okolí daného pixelu a převedena na amplitudu. + +Thompson:: +Detekce přechodů mezi texturami. Převádí na problém segmentace podle hran. + +[.question] +Vysvětlete pojem značka (marker) a význam značek například v algoritmu watershed. Uveďte příklad, jak lze značky najít +pro dotýkající se kávová zrnka. + +Marker:: +Pixel nebo malá skupina pixelů, která spadá do oblasti, jejíž segment chceme. V případě watershedu jsou markery místa +odkud voda stoupá. + +U kávových zrnek se markery nejspíš dají najít pomocí prahovaní a eroze. diff --git a/pv131/09_hough.ad b/pv131/09_hough.ad new file mode 100644 index 0000000..f0d6ef1 --- /dev/null +++ b/pv131/09_hough.ad @@ -0,0 +1,86 @@ += 9. Hough and Radon transform + +[.question] +Mějme 2D diskrétní obraz obsahující pouze 4 body představující rohy čtverce s hranou délky 2a umístěného svým středem +do bodu stem:[[2a, 2a\]] a osově orientovaného. Popište, jak vypadá obsah akumulátoru po aplikaci diskrétní přímkové +Houghovy transformace. Určete zejména počet a souřadnice význačných bodů. + +Houghova transformace:: +Detekuje tvary zadané rovnicí stem:[f(x, a) = 0], kde stem:[x] je pozice a stem:[a] jsou parametry v +obrázku. ++ +U přímek to funguje zhruba tak, že máme 2D pole, kterému říkáme akumulátor, kde jedna dimenze je stem:[\rho], vzdálenost +přímky od počátku, a druhá je stem:[\theta], úhel, který svírá osa x s kolmicí od počátku k přímce. Velikost tohohle +pole určuje počet možných přímek (úhlů a vzdáleností od počátku), které je možné detekovat. Algoritmus samotný probíhá +takto: ++ +1. Zkvantizuj prostor parametrů (zvol počet stem:[\rho] a stem:[\theta] -- řádků a sloupců akumulátoru). +2. Inicializuj akumulátor na samé 0. +3. Pro každý pixel s intenzitou, která nám přijde zajímavá, inkrementuj každou buňku akumulátoru, která odpovídá čáře, + tímto bodem prochází. +4. Najdi lokální maxima v akumulátoru, odpovídají totiž těm přímkám, co hledáš. + +*** + +image::pv131/2a.png[] + +Obsah akumulátoru závisí na počtu přímek, které připadají v úvahu. Nicméně, pokud má rozumnou velikost, pak v jeho +grafu bude 4 oblouků, které odpovídají rohům čtverce a které se protínají v šesti bodech. Souřadnice těchto bodů +budou stem:[(3a, \pi/2), (a, \pi/2), (3a, 0), (a, 0), (sqrt(2) * 2a, \pi/4), (0, 3\pi/4)]. Odpovídají postupně 4 stranám +a 2 úhlopříčkám čtverce. + +[.question] +Mějme 2D diskrétní obrázek velikost stem:[M \times N], na který aplikujeme diskrétní přímkovou HT danou rovnicí +stem:[\rho = x cos \theta + y sin \theta] s následujícím vzorkováním akumulátoru: 1 pixel podle osy stem:[\rho] a +1 úhlový stupeň ve směru osy stem:[\theta]. Určete, jak závisí velikost akumulátoru na velikosti vstupního obrázku. + +Úhlových stupnů je 360, z čehoh 180 jich nepotřebujeme, a proto 180 je tedy i velikost akumulátoru v dimenzi +stem:[\theta]. + +Nejzažší bod od počátku, kde lze se přímku detekovat je stem:[[M-1, N-1\]] tedy velikost akumulátoru v dimenzi +stem:[\rho] je stem:[\sqrt{(M-1)^2 * (N-1)^2}]. + +[.question] +Uvažujme, že máme zavedenu diskrétní eliptickou HT. Určete dimenzi akumulátoru. + +Pět. Pozice středu (x a y), orientace a dva poloměry. + +Existuje link:https://en.wikipedia.org/wiki/Hough_transform#Efficient_ellipse_detection_algorithm[algoritmus], +kterému stačí jeden. + +[.question] +Uvažujme, že máme zavedenu diskrétní parabolickou HT. Určete dimenzi akumulátoru. + +Čtyři. Pozice vrcholu (x a y), orientace a koeficient stem:[a] z stem:[ax^2]. + +[.question] +Uveďte základní kroky potřebné při rekonstrukci signálu pomocí filtrované zpětné projekce (filtered backprojection). +Popište, například pomocí obrázku, co se při jednotlivých krocích děje. + +Radonova tranformace:: +Podívá se na obraz pod nějakým úhlem -- skrze nějakou přímku danou směrnicí stem:[a] a odsazením stem:[b] a integruje. ++ +[stem] +++++ +R_I(a, b) = \int_{-\infty}^{\infty} I(x, ax + b)dx +++++ ++ +Svým způsobem je to zase Fourierka. + +Filtered Backprojection:: +Jeden ze způsobů, jak provést inverzní Radonovu transformaci. Dalším je Fourier Slice Theorem. + +.Postup +1. High-pass filtrování ++ +[stem] +++++ +R_I^\text{filtered}(\rho, \theta) = \text{IFT}_{\upsilon \to \rho}\{|\upsilon| \text{FT}_{\rho' \to \upsilon} \{ R_I(\rho', \theta)\} \} +++++ ++ +2. Zpětná projekce ++ +[stem] +++++ +I(x, y) = \int_0^\pi R_I^\text{filtered}(x cos \theta + y * sin \theta, \theta) d\theta +++++ diff --git a/pv131/10_description.ad b/pv131/10_description.ad new file mode 100644 index 0000000..e65cb22 --- /dev/null +++ b/pv131/10_description.ad @@ -0,0 +1,136 @@ += 10. Description, Classification, Understanding, Evaluation +:stem: asciimath + +[.question] +Mějme 3-bitový 2D obrázek M=[1 2 1; 2 4 2; 1 2 1]. Spočítejte pro něho odpovídající matici současného výskytu +(co-occurrence matrix). + +Co-occurence matrix:: +Obecně je to matice popisující, kolikrát se prvek odpovídající řádku setkal s prvek odpovídajícím sloupci. V tomhle +kontextu je to Haralickova fíčura a udává, jak často se v obrazu vedle sebe vyskytují intenzity dané sloupcem a řádkem. ++ +Počítá se nejprve zvlášť pro separační vektory: (1, 0), (0, 1), (1, -1), a (1, 1), kde např. (1, 1) říká, že pixely, +které nás zajímají budou na budou sousedět buď levým horním nebo pravým dolním (tedy uvažujeme i záporný směr). ++ +Tyhle 4 matice se znormalizují (vydělí počtem hodnot, které jsou v dané matici započítány) a sečtou a to je výsledek. + +*** + +[stem] +++++ +A = [[1,2,1], + [2,4,2], + [1,2,1]] +++++ + +Představte si, že následující matice jsou 8x8, jelikož 3-bitová čísla odpovídají intervalu stem:[[0, 8)]. Řádky a +sloupce odpovídající intezitám 0, 5, 6, a 7 jsou vynechány. + +_Příklad: stem:[C_{(1, 0)}(1, 2)] říká kolik pixelů s intenzitou 1 sousedí s pixelem s intenzitou 2 na ose x, ve +kterémkoli směru._ + +[stem] +++++ +C_{(1, 0)} = [[0,4,0,0], + [4,0,0,2], + [0,0,0,0], + [0,2,0,0]], +C_{(0, 1)} = [[0,4,0,0], + [4,0,0,2], + [0,0,0,0], + [0,2,0,0]], +++++ + +[stem] +++++ +C_{(1, 1)} = [[0,0,0,2], + [0,2,0,0], + [0,0,0,0], + [2,0,0,0]], +C_{(1, -1)} = [[0,0,0,2], + [0,2,0,0], + [0,0,0,0], + [2,0,0,0]] +++++ + +Počet "zajímavých dvojic" z stem:[A]: + +[stem] +++++ +\#R_{(1, 0)} = \#R_{(0, 1)} = 4 * 3 = 12 +++++ + +[stem] +++++ +\#R_{(1, 1)} = \#R_{(1, -1)} = (2 + 1 + 1) * 2 = 8 +++++ + +[stem] +++++ +C = \frac{C_{(1, 0)}}{\#R_{(1, 0)}} + \frac{C_{(0, 1)}}{\#R_{(0, 1)}} + + \frac{C_{(1, 1)}}{\#R_{(1, 1)}} + \frac{C_{(1, -1)}}{\#R_{(1, -1)}} = ... + = [[0,2/3,0,1/2], + [2/3,1/2,0,1/3], + [0,0,0,0], + [1/2,1/3,0,0]] +++++ + +[.question] +Vysvětlete rozdíl mezi pojmy "image-oriented bounding box" a "object-oriented bounding box". + +Bounding box:: +Nejmenší obdélník, který pojme objekt. Šířka a výška bounding boxu se dá použít jako minimální a maximální průměr +objektu. + +Image-oriented bounding box:: +Bounding box orientovaný podél os obrázku. + +Object-oriented bounding box:: +Bounding box orientovaný podél os objektu (které je třeba nejdřív spočítat). + +[.question] +Uveďte algoritmus pro výpočet deskriptoru "object-oriented bounding box". + +Pomocí Feretova průměru. + +1. Spočítat minimální Feretův průměr. +2. Spočítat Feretův průměr, jenž jen kolmý k 1. + +V Matlabu lze použít `measure(...'Feret')`. + +Podle slidu 17 prý existuje v matlabu funkce, která vrací trojici (maxFeret, minFeret, feret kolmý k minFeret), ale já +ji nenašel. + +[.question] +Nadefinujte pojem "kulatost binárního objektu". + +[stem] +++++ +C = \frac{4\pi * A}{P^2} +++++ + +.Vysvětlivka +* C = circularity (kulatost) +* A = plocha objektu +* P = poloměr objektu + +Výsledkem je hodnota mezi 0 a 1, popisující, jak moc je objekt kulatý. + +[.question] +Nadefinujte Jaccardův index podobnosti dvou binárních objektů a vyhodnoťte ho nad čtvercem se stranou +"a" a kruhem jemu vepsaným. + +[stem] +++++ +J = \frac{TP}{FP + TN + FN} +++++ + +.Vysvětlivka +* GT = ground truth +* AR = výsledek algoritmu +* TP = true positive = stem:[AR \cap GT] +* FP = false positive = stem:[AR \cap \bar{GT}] +* TN = true negative = stem:[\bar{AR} \cap \bar{GT}] +* FN = false negative = stem:[\bar{AR} \cap GT] + +Pro čtverec a vepsanou kružnici stem:[J = \frac{\pi * (\frac{a}{2})^2}{a^2} = \frac{\pi}{4}]. diff --git a/pv131/11_techniques.ad b/pv131/11_techniques.ad new file mode 100644 index 0000000..e035c02 --- /dev/null +++ b/pv131/11_techniques.ad @@ -0,0 +1,94 @@ += 11. Selected Image Analysis Techniques + +[.question] +Zaveďte vhodnou míru, která bude vyhodnocovat míru podobnosti dvou obrázků stejné velikosti bez vzájemného posunu, +a to na základě jejich intenzit. Míru formálně nadefinujte a nastavte tak, aby pro naprosto shodné obrázky vracela +hodnotu "1", naopak pro zcela odlišné obrázky hodnotu "0". + +Předpokládejme BÚNO, že intenzita spadá do intervalu stem:[\[0, 1\]], pak pro obrázky A a B: + +[stem] +++++ +R = \frac{\sum_i A[i] * B[i]}{\text{pixelCount}} +++++ + +[.question] +Vysvětlete rozdíl mezi operacemi "konvoluce" a "korelace". Uveďte příklad 2D matice, se kterou operace korelace i +konvoluce nad libovolným 2D diskrétním obrázkem vrátí totožný výsledek. + +Konvoluce:: +Lineární operátor na dvou maticích. (`conv2` v matlabu) ++ +stem:[g(x) = \sum_a f(x - a) * h(a)], (f = obraz, h = konvoluční jádro) + +Korelace (Cross-correlation):: +Míra podobnosti mezi signály. (`xcorr2` v matlabu) ++ +stem:[g(x) = \sum_a f(x + a) * h(a)] + +Pokud je matice shodná po převrácení podél osy x i y ("středově souměrná"), pak se výsledek neliší. Jinými slovy filtr +(resp. jeho kernel) musí být symetrický. + +____ +A symmetric kernel is one that is unchanged when the two variables are permuted; it is a kernel function K such that +K(t, u) = K(u, t).[1] +____ + +Nicméně je nutné pamatovat, že kernely se neindexují jako klasické matice, ale mají (0, 0) uprostřed. Nestačí tedy +matici transponovat. + +[1]: https://en.wikipedia.org/wiki/Integral_transform + +[.question] +Vysvětlete princip segmentace obrazu s pomocí metody grafových řezů. Jak se nad 2D diskrétním obrázkem s rozměry 5x5 +pixelů sestrojí počáteční graf (z kolika vrcholů je složen a jak jsou jednotlivé vrcholy pospojovány hranami)? + +Grafové řezy jsou přístup k segmentaci, jenž se snaží najít minimum rovnice +[stem] +++++ +E_\text{total}(\text{model}) = \alpha * E_\text{internal}(\text{model}) + \beta * E_\text{external}(\text{model, image}) +++++ +pomocí grafů. Další metodou jsou Deformable Models, jenž jsou popsány buď parametricky, a nebo implicitně. + +Takže grafové řezy jsou vlastně taková redukce z problému segmentace na problém nejkratší cesty: + +* pixely -> uzly +* 4/8-sousedé -> hrany +* váhy hran jsou pravděpodobnosti, že dva sousední pixely jsou součástí jednoho objektu +* nízká váha hrany -> mezi pixely nejspíš prochází hrana objektu +* váhové funkce bývají založeny na inverzi gradientu, např. G-aussian |\Delta I| nebo MaxIntensity - |\Delta I| + +Tenhle převod je dále rozšířen o přidání source a sink uzlu. Vysledkem pak není nejkratší cesta, ale dvě plochy, +pozadí a popředí. Váhy mezi obrazem a novými uzly jsou dány pravděpodobností, že pixel je součástí pozadí/popředí. + +[.question] +Nakreslete hrubé schéma konvoluční neuronové sítě a popište stručně, k čemu slouží jednotlivé komponenty, resp. jaké +operace jsou v nich vykonávány. + +image::pv131/cnn.png[] + +Neuronky mají, podobně jako cibule, zlobři a `.psd` soubory, vrstvy. Pro učení neuronek se používá backpropagation +alogoritmus s cross-entropy loss, whatever that is. + +Fully connected layer:: +* každý neuron ve skryté vrstvě je napojen na každý neuron ve vstupní vrstvě +* žádná prostorová informace, velké množství parametrů +* klasická aktivační operace modelovaná podle reálného neuronu +* používá se k zploštění konvoluční vrstvy a ke klasifikaci + +Convolution layer:: +* každý neuron ve skryté vrstvě je napojen pouze na oblast neuronů ve vstupní vrstvě +* každá podvrstva reprezentuje jeden filtr, jedno konvoluční jádro, které se aplikuje na vstupní obraz +* pamět je potřeba jednak na váhy jádra a bias pro každý filtr, druhak na vlastní výpočet z aktuálního obrazu +* stride je délka kroku -- posunu jádra + +Max pooling layer:: +* Downsampling pomocí maxima z regionu +* nelineární operace (filtry lineární jsou, proto nás to zajímá) + +ReLU (Rectified Linear Unit) layer:: +* nuluje záporné hodnoty +* nelinární operace (filtry lineární jsou, proto nás 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      +++++ + +[.question] +==== +Mějme fotografii, kterou chápeme jako spojitý dvourozměrný obraz stem:[F (x, y)] o rozměrech 15x10 cm. +Zadefinujte takovou vzorkovací funkci stem:[S(x, y)] pro její naskenování, aby výsledný digitální obraz +stem:[I(x, y) = F(x, y) * S(x, y)] byl naskenován v kvalitě 300 dpi (dots per inch). Pro jednoduchost +použijte převod 1 palec (inch) = 2.5 cm. +==== + +[.record-content] +==== +[stem] +++++ +S(x, y) = sum_(j=-oo)^oo sum_(k=-oo)^oo delta(x - j * Delta X), " kde " Delta x = 6/(6*300) = 4/(4*300) = Delta y = 1/300 +++++ + +Pozn. 15 cm = 6 in, 10cm = 4 in +==== + +[.question] +==== +Je dán 2D diskrétní obraz o velikosti 512x512 pixelů a na něm provedená 2D diskrétní vlnková +transformace s Haarovou bázovou funkcí až do úrovně, kdy je ve výsledném obraze přítomen jen +jeden aproximační koeficient. Určete, kolik bylo třeba provést úrovní rozkladů na nízkofrekvenční +a vysokofrekvenční složky, a celkový počet detailních koeficientů pro diagonální směr přes všechny +úrovně rozkladů dohromady. Řešení bez odvození nebo vysvětlení bude hodnoceno 0 body. +==== + +[.record-content] +==== +stem:[J_0 = 0], stem:[J = log(512) = 9]. Je tedy třeba provést 9 úrovní rozkladu a výsledek bude mít 8 HPF variant, tedy +počet diagonálních detailních koeficientů je: + +[stem] +++++ +2^8 * 2^8 + 2^7 * 2^7 + 2^6 * 2^6 + 2^5 * 2^5 + 2^4 * 2^4 + 2^3 * 2^3 + 2^2 * 2^2 + 2^1 * 2^1 = 87358 +++++ +==== + +[.question] +==== +Mějme 1D diskrétní signál délky stem:[N = 15] s konstantní úrovní stem:[a \in \mathbb{R}]. Znázorněte jemu odpovídající +frekvenční spektrum s důrazem na amplitudy jednotlivých Fourierových koeficientů a také na popis souřadného systému v +rámci jedné zvolené periody. +==== + +[.record-content] +==== +stem:[[15a, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0\]] +==== + +[.question] +==== +Dvourozměrné konvoluční jádro zadané maticí stem:[X] je separabilní, právě tehdy když hodnost této +matice je rovna 1. Pro níže uvedené konvoluční jádro uveďte jednu konkrétní konfiguraci hodnot +neznámých proměnných a, b, c, d, e tak, aby toto konvoluční jádro bylo separabilní: + +[stem] +++++ +X = [[a,2,b],[1,c,d],[3,e,1]] +++++ +==== + +[.record-content] +==== +[stem] +++++ +[[3, 2, 1], + [1, 2/3, 1/3], + [3, 2, 1]] +++++ +==== + +[.question] +==== +Mějme 2D diskrétní obraz stem:[I], dva 2D lineární filtry stem:[A] a stem:[B] a dva 2D nelineární filtry stem:[C] a +stem:[D]. Uveďte příklady filtrů, kdy neplatí: + +[loweralpha] +. stem:[A(B(I)) = B(A(I))], +. stem:[C(D(I)) = D(C(I))], +. stem:[A(C(I)) = C(A(I))]. +==== + +[.record-content] +==== +[loweralpha] +a. Nejsou příklady. Lineární filtry lze vždy vyjářit konvolucí a ta je asociativní. +b. C = maximový filtr, D = minimový filtr +c. A = gaussian, D = median +==== + +[.question] +==== +Mějme dva libovolné 2D diskrétní obrazy stem:[A] a stem:[B] o velikosti M x N pixelů, na které aplikujeme +2D diskrétní Fourierovu transformaci. Uveďte: + +[loweralpha] +. počet Fourierových koeficientů, které tato transformace spočítá pro každý z obrazů stem:[A] a stem:[B]; +. výčtem prvků nejmenší (ve smyslu množinové inkluze) možnou podmnožinu Fourierových + koeficientů, která umožní jednoznačně rozhodnout, zda obrazy stem:[A] a stem:[B] mají stejnou průměrnou + intenzitu. +==== + +[.record-content] +==== +[loweralpha] +a. M x N +b. Jeden. Koeficient v počátku indikuje intenzitu. +==== + +[.question] +==== +Do přiloženého dvouúrovňového obrazu stem:[A] zakreslete přibližné pozice značek, které povedou algoritmus záplavy, +aplikovaný na velikost gradientu obrazu stem:[A], k očekávanému výsledku v obraze stem:[B]. + +image::pv131zk/07.png[] +==== + +[.record-content] +==== +image::pv131zk/07solution.png[] +==== + +[.question] +==== +Uveďte hodnoty minimálního a maximálního Feretova průměru pro základní geometrické útvary: + +[loweralpha] +. čtverec o straně délky a, +. obdélník se stranami délky a a b, +. kruh o poloměru r, +. kružnice o poloměru r, +. rovnoramenný pravoúhlý trojúhelník s přeponou délky a. +==== + +[.record-content] +==== +[loweralpha] +a. min = a, max = stem:[sqrt(2)a] +b. min = min(a, b), max = stem:[sqrt(a^2 + b^2)] +c. min = max = 2r +d. min = max = 2r +e. max = a, min = stem:[a/2] +==== + +[.question] +==== +Mějme dvě diferenciální rovnice: + +[stem] +++++ +(\partial I(x, y, t))/(\partial t) = x^2 + y^2 + +(\partial J(x, y, t))/(\partial t) = I(x, y, t) - 5 + x +++++ + +s počátečními podmínkami stem:[I(x, y, 0) = 0] a stem:[J(x, y, 0) = 2], které definují změny dvou 2D diskrétních +obrazů stem:[I] a stem:[J] o rozměrech 5x5 pixelů v čase t. Odvod’te, jak budou vypadat obrazy stem:[I(x, y, 3)] a +stem:[J(x, y, 3)] při použití časového kroku stem:[Delta t = 1] pro obraz stem:[I] a stem:[Delta t = 3] pro obraz +stem:[J]. +==== + +[.record-content] +==== +[stem] +++++ +I(x, y, t) = I(x, y, t - 1) + x^2 + y^2 + +I(x, y, 3) = [ + [00,03,12,27,48], + [03,06,15,30,51], + [12,15,24,39,60], + [27,30,39,54,75], + [48,51,60,75,96] +] + +J(x, y, t) = J(x, y, t - 3) + 3 * (I(x, y, t) - 5 + x) + +J(x, y, 3) = [ + [-13,-01,029,077,143], + [-04,008,038,086,152], + [023,035,065,113,179], + [068,080,110,158,224], + [131,143,173,221,287] +] +++++ +==== + +[.question] +==== +Mějme libovolný dvourozměrný digitální šedotónový obraz velikosti 65x33 pixelů. Určete nejvyšší +možné prostorové frekvence v něm obsažené (pro každou osu zvlášť) a uveďte, jak vypadají jim +odpovídající bázové funkce. +==== + +[.record-content] +==== +Nejvyšší frekvence je taková, která se "dotkne" ka6dého pixelu v dané ose: + +. pro x: 65 Hz / m (za předpokladu 1 pixel = 1 metr) +. pro y: 33 Hz / m + +Bázové funkce: +[stem] +++++ +omega_x in [0, 65) + +b_x(x) = e^(-i(2pi/65 * omega_x * x)) + +omega_y in [0, 33) + +b_y(y) = e^(-i(2pi/33 * omega_y * y)) +++++ +==== + +[.question] +==== +Odvoďte předpis jednorozměrného spojitého signálu, který vznikne jednorozměrnou spojitou konvolucí +stem:[f(x) \otimes f(x)]. Funkce stem:[f : \mathbb{R} \to \mathbb{R}] je definována následovně: + +[stem] +++++ +f(x) = {(1\text{,},-1/2 <= x <= 1/2\text{,}),(0\text{,},\text{jinak.}):} +++++ +==== + +[.record-content] +==== +Fourierova transformace: +[stem] +++++ +Phi(omega_x) = int_-oo^oo f(x) * e^(-i omega_x x) dx +++++ + +[stem] +++++ +Phi(omega_x) = int_-1/2^1/2 e^(-i omega_x x) dx overset("integral calculator")(=) + [(i * e^(-i omega_x x))/omega_x]_(-1/2)^(1/2) = + (2 * "sin"(omega_x/2))/omega_x + +++++ +==== + +[.question] +==== +Mějme dvourozměrný binární obraz, jehož popředí obsahuje právě čtyři pixely na pozicích +stem:[(a, 0), (2a, 0), (a, a)] a stem:[(2a, a)], přičemž předpokládejme, že souřadný systém má počátek (tj. bod stem:[(0, 0)]) +vlevo dole, osa x je orientována vpravo, osa y svisle nahoru a souřadnice obrazových bodů jsou uváděny jako +stem:[(x, y)]. + +[loweralpha] +. Popište, jak vypadá obsah akumulátoru po aplikaci diskrétní přímkové Houghovy transformace. Určete zejména počet a + souřadnice význačných bodů. +. Určete pozici jednoho pixelu v původním binárním obraze, po jehož přidání do popředí tohoto + obrazu by příslušný akumulátor obsahoval právě dvě globální maxima, která by odpovídala + úhlopříčkám čtverce s vrcholy v původních čtyřech pixelech popředí tohoto obrazu. +==== + +[.record-content] +==== +[loweralpha] +a. Pokud má akumulátor rozumnou velikost, pak v něm budou 4 "oblouky" a 6 význačných bodů v místech, kde se 2 oblouky + protínají. Souřadnice: stem:[(a, 0), (2a, 0), (a, pi/2), (0, pi/2), (sqrt(2)a, pi/4), (sqrt(2)a, -pi/4)] +b. stem:[(3/2a, a/2)] + +==== + +[.question] +==== +Pomocí konvolučního teorému dokažte vlastnost distributivity jednorozměrné diskrétní konvoluce +vzhledem k operaci sčítání. Nezapomeňte jednotlivé kroky důkazu patřičně zdůvodnit, jinak budou hodnoceny 0 body. +Stejně bude hodnocen i důkaz nevyužívající konvoluční teorém. +==== + +[.record-content] +==== +Chci dokázat stem:[f otimes (g + h) = f otimes g + f otimes h]. + +Konvoluční teorém: stem:["DFT"(f otimes g) = "DFT"(f) * "DFT"(g)] + +Nechť stem:[i = g + h], pak +[stem] +++++ +"DFT"(f otimes i) = "DFT"(f) * "DFT"(i) = "DFT"(f) * "DFT(g + h)" \ + overset("lin. op.")(=) "DFT"(f) * ("DFT"(g) + "DFT"(h)) \ + overset("asoc. nás.")(=) "DFT"(f) * "DFT"(g) + "DFT"(f) * "DFT"(h) \ + overset("konv. teorém")(=) f otimes g + f otimes h. +++++ +==== + +[.question] +==== +Dokažte, že aplikací Fourierovy transformace na libovolný jednorozměrný reálný diskrétní signál +sudé délky stem:[N > 0] získáme alespoň dva čistě reálné Fourierovy koeficienty. +Tip: Uvědomte si, že je rozdíl mezi pojmy sudý signál a signál sudé délky. +==== + +[.record-content] +==== +1D DFT: +[stem] +++++ +sum_(x=0)^(N-1) f(x) * e^(-u((2pi)/N omega_x x)) + = sum_(x=0)(N-1) f(x) * ("cos"((2pi)/N omega_x x) - i * "sin"((2pi)/N omega_x x)) +++++ + +f(x) má sudou délku > 0, tedy stem:[N = 2 * k, k > 0]. +[stem] +++++ +sum_(x=0)^(2k-1) f(x) * ("cos"(pi/k omega_x x) -i * "sin"(pi/k omega_x x)) +++++ + +Pro x = 0 platí stem:[-i * "sin"(0) = 0], tedy stem:[F(0)] je reálné číslo. + +Pro x = k platí stem:[-i * "sin"(pi/k * k * omega_x) = -i * "sin"(pi omega_x)], což je vždy 0, tedy i stem:[F(k)] je +reálné číslo. + +Tedy platí, že signál nenulové sudé délky má alespoň 2 reálné Fourierovy koeficienty. +==== + +++++ +
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