컴퓨터그래픽스

01 Images

• Image의 정의 자체는 픽셀과 관련이 없다
• OLED와의 차이
• OLED는 항상 빛이 나감. 빛을 끄고 키냐에 따라 달라짐
• LCD는 빛을 꼬냐 안꼬냐에 따라 달라짐
• DLP는 빛의 이동방향에 따라
• Electronic paper: 장소의 밝기에 관계 없이 볼 수 있음. 반사광을 이용하기 때문에
• Inkjet printer: 온도로 점의 크기를 조절한다
• 20쪽: 24bpp(256)으로 가장 많이 표현한다. 왜냐하면 대부분의 모니터의 성능이 그럼. 72bpp가 사람이 인식할 수 있는 이미지에 가장 근접
• HDR은 밝기에 관한 정보가 추가적으로 있기 때문에 후보정의 폭이 넓다
• alpha는 투명도를 의미함
• 27쪽: floating-point HDR - 4byte * 3 = 12MB
• 그레이스케일로 바꿀 때 B와 G의 밝기가 다름. 따라서 변환할 때 초록에 가중치를 주면 더 그럴듯하다
• dithering을 consistent하게 하면 구림
• 38쪽: 레이저 프린터로 half toning한것이다
• 39쪽: 잉크젯 프린터는 잉크가 가까이 있으면 번져버리기 때문에 diffusion dither를 사용
• pixel value와 light intensity는 비례하고 non-linear함(exponential)
• 왜냐하면 사람 눈이 2%변화만큼만 인식할 수 있기 때문에 exponential하게 증가시켰을 때 사람이 인식하기에는 linear하게 느껴짐
• gamma correction
• 프로젝터는 껐을 때 흰색 -> k가 높음. LCD는 까만색 -> k가 낮음
• dynamic range를 키우기 위해서는 k가 아닌 Imax를 키워야한다

02 Perspective

• Perspective: 이미지에 depth를 표현하기 위한 방법
• one-point: vanishing point 1개 (뒤쪽)
• two-point: vanishing point 2개 (왼쪽 오른쪽)
• three-point: vanishing point 3개 (왼쪽, 오른쪽, 아래)
• multiview orthographic
• fov: field of view
• 망원렌즈를 사용하면 렌즈로부터의 거리가 압축되어 보인다
• 광각렌즈를 사용하면 건물같은 높이 있는 것들이 parallel하게 찍힘

03 Ray Tracing

• 렌더링 기법중 하나
• 원래는 빛에서부터 나가는 ray를 모두 계산. 하지만 너무 복잡하기 때문에 viewer에서 보이는 ray를 계산함 -> ray tracing
• 두가지 케이스
• ortho: point가 rect에 포함
• pers: point가 외부에 이씀
• r을 계산하기 위해 이미지 플레인이 어디이쓴지 정의해야함
• 9: w는 viewing direction의 반대방향
• u방향으로 u만큼 v방향으로 v만큼
• 이미지 플레인 좌표계는 2차원
• ray를 만들고 나면 intersection test를 해야한다
• 14쪽 t에 대한 이차식의 해 -> 근의공식으로 해를 구함
• 17: slab의 개념 - 평행한 두 플레인
• 18: ray가 두개의 슬랩 범위 내에 있으면 ray가 박스 안에 있다는 것을 알 수 있음
• 20: condition 2 벡터 ax가 n플레인에 수직

@shading

• BRDF: 물체를 놓고 카메라를 쭉 옮기면서 캡쳐하는것
• diffuse reflection: 완전한 난반사 surface를 가정
• 빛을 아무리 세게 비춰도 하이라이트 된 부분을 볼 수 없음. 모든 방향으로 빛이 고르게 나가기 때문
• 29쪽 밑의 그림: 기울수록 빛이 적게들어오니까 물체의 밝기가 줄어듬
• lambertian shading: 그것을 수식으로 나타낸것
• I는 빛의 강도
• 41쪽 k를 키우면 highlight의 양이 증가 p는 거울반사
• ambient shading: 전체적으로 깔려있ㅇ는 그림자. 모든 light source를 이용해 계산할 수 있지만 계산량이 쓸데없이 ㅏㄴㅎ아지기 때문에 그냥 상수를 더함

04 Vector Algebra Transformations

• 그래픽스에서는 포인트와 벡터의 컨셉에 차이가 있다
• linear transformation
• uniform scale
• nonuniform scale
• rotation
• reflection
• shear
• linear transformation을 compose할 때는 두 함수(행렬)를 곱하면 됨
• translation
• 선형변환이 아님. 행렬의 곱으로 표현할 수 없기 때문에
• 선형변환 말고 affine transformation 이라고 한다.
• affine transformation의 특징은 원점이 이동한다는점. 선형변환에서는 원점이 항상 0
• affine transformation을 compose할 때는 식이 따로 있는데 그거를 그냥 1차원 식으로 구하면 복잡하기 때문에 homogeneous coordination을 이용해서 행렬의 곱으로 나타냄
• homogeneous coordinates
• extra row & column이 있는 행렬에 벡터를 곱해서 linear transformation과 translation(affine transformation) 모두 표현 가능하다
• M11 M12  xt
  M21 M22 yt
  0     0       1
• 30쪽의 식이 원래 1차원 식보다 간단
• 31쪽의 4x4 행렬에서 column을 각각 하나의 벡터로 표현하면 마지막 column은 앞의 세 벡터(vx, vy, vz)의 원점을 나타낸다 -> affine basis
• homogeneous coordinates의 장점: 직관적이고 consistent하다
• frames에서 마지막 숫자가 1이면 벡터, 0이면 포인트이다
• 원점이 아닌 점으로 회전/y축이 아닌 축으로 회전할 때 composition을 할줄 안다면 이동, 회전등의 composition으로 간단하게 나타낼 수 있다
• composite affine transformations 는 순서가 중요함.

• rigid motions: affine transform의 subset 
• translation과 rotation만으로 이루어진 transform
• Q가 orthonormal(평면에 수직이고 길이 1)하면 
• euler angle: 축을 기준이로 세번 회ㅣ전시키는것
• roll-pitch-yaw
• 좌표계변환
• gimbal lock이라는 문제가 있다 
• pitch방향으로 90도 돌리면 yaw축이랑 같아지게 됨 => 하나의 축이 사라짐
• singularity: 축이 사라진 상황에서 1도를 돌리려면 실제 1도 보다 더 많이 회전해야하는 상황
• a축을 중심으로 30도 회전하는법
• a축을 z축으로 만들고 z축에 대해 30도 회전한 후 원래의 a축으로 돌려놓는다
• a를 단위벡터로 만들고 z축과 cross product한 결과 -> az plane에 수직한 벡터
• 그거를 기준으로 a랑 z의 각도를 구하고 회전
• 좀 비효율적임 -> 해결책: rodrigues’ rotation formula

• Rodrigues’ rotation formula
• 64쪽 공식있음

05 Pipeline and Rasterization

• pipeline의 종류
• forward approach(빛의 방향을 따라서), backward approach(빛의 반대방향)

• Modeling transformation
• 보통 원점 중심으로 만듬
• 배치 함수를 이용해서 배치함

• Camera transformation
• 사용자가 view direction, upvector(보통 y), eye position(zoom)을 입력
• x축은 BxA를 하면 나옴 = u
• y축은 UxB = v(새로운 업벡터)
• z축은 UxV = w
• 19쪽 행렬은 월드스페이스에서 카메라 좌표계로 바꾸는 행렬
• Window transformation
• 임의의 직사각형을 다른 직사각형으로 보내는 행렬
• Orthographic projection
• 윈도우 변환의 삼차원 버전
• Perspective projection
• z로 나누는 division을 표현하는 문제
• Official perspective matrix: 결론
• 외워
• midplane은 어디로 가는가 계산해보면 이해에 도움이된다
• Perspective transformation matrix는 두번째 세번째 곱한것

• Line Rasterization
• Point sampling: 정사각형의 중심이 line범위에 들어가면 색칠
• Midpoint algorithm: line과 가장 가까운 중심인 정사각형을 색칠 - column마다 하나의 정사각형을 칠하게 됨
• 45도 line은 얇게 그려짐
• 곱셈은 쓰면 안되기 때문에 optimization 진행
• Triangle Rasterization
• clipping 
• 어떤 vertex가 스크린 밖에 있을 때 밖에 있는 부분을 짤라줘야 한다

06 Rigging & Skinning

• kinematics
• 그래픽스의 모든 장면에서 kinematics가 쓰임. 스켈레톤 뿐만 아니라. 상대적인 위치를 정의하는 학문
• 팔꿈치: hinge. 어깨: ball and socket joint
• joint마다 joint각도가 잇음. 각도가 정해지면 위치랑 orientation을 알 수 잇다->forward kinematics
• 위치와 orientation을 알고 각도를 알아내는건 inverse kinematics
• forward kinematics가 더 쉬움
• 11쪽: 어떤 특정한 body part에 대해서 end effect 좌표를 얻어내는것이 목표
• 그림을 왼쪽부터 오른쪽으로 읽으면서 해당 하는것들의 매트릭스를 순서대로 곱함. local x,y축 이용 -> 좌표계변환
• 좀더 직관적인것은 global coordinate (정변환). global x,y축으로 변환
• 33: offset: parent에 대한 상대적 위치. channels: 회전
• skinning
• 좌표계 3개: rest 좌표계(rest pose의 모든 점의 정보), local 좌표계(bone마다 따로따로), global 좌표계(모든 bone의 configuration 정보)
• weight: 점마다 있는 정보로 해당 점이 어떤 bone에 붙어있는 부분인지 나타냄. weight의 총 합은 1이고 붙어있으면 양수, 안붙어있으면 0으로 나타냄

//시험범위 54쪽까지

07 Transformation, Input and Interaction

08 Pipeline Operations

• 3쪽: 왼쪽의 그림은 두가지로 해석될 수 있는 반면 오른쪽은 아님
• hidden surface elimination
• back-face culling
• 가장 쉬운 방법
• 정확한 방법은 아님. closed shape에 한해서 이용할 수 있음
• 카메라에 보이지 않는 backface를 버리는 방법
• 하나의 surface에 대해 보이는 부분이 있고 안보이는 부분이 있을 수 있는데, 그 부분은 안버리니까 그려지는 순서에 따라 이상하게 그려질수도 있음
• painter’s algorithm
• surface를 depth순으로 sorting해서 가장 뒤에 있는것부터 그림
• valid order를 구하기 쉬울 때 유용하다
• cycle이 있을 경우에는 적합하지 않다. 8쪽의 경우에 사각형을 두개로 자르면 해결할 수는 있지만 보통 그냥 안쓴다
• 9쪽과 같이 구름의 단층을 그리는 경우 등에 사용
• the z buffer(depth buffer)
• 화면의 해상도와 같은 크기를 가진 z buffer(pixel 개수와 같은 개수)
• 단점: 메모리, 계단현상 발생(aliasing)
• depth가 int로 저장됨. 
• 19쪽: 문제를 해결하려면 near-far plane간 space를 tight하게 설정
• Flat Shading
• 빠름, 하지만 하나의 삼각형에 대해 같은 색깔로 칠해지기 때문에 삼각형의 boundary가 그대로 보임. 삼각형을 작게 만들면 해결되지만 느려지니까…
• Gouraud Shading
• 최소한의 삼각형 개수로 그럴듯하게 shading 하는 방법
• 삼각형의 normal을 가지고 color를 구한 후 color을 interpolation한다
• 만약에 highlight가 삼각형 가운데에 있고 세 꼭짓점의 색깔이 같으면 제대로 작동하지 않음
• 결과적으로 smooth해지지만 boundary가 조금 보인다
• Phong Shading
• pixel마다 interpolation

09 Texture Mapping

• 수업안들음

10 Triangle Meshes

• topology: connectivity가 같은 도형
• geometry: 모양이 같은것
• topological validity: 어떤 도형이 sphere로 deform될 수 있는가
• geometric validity
• query가 빠른게 왜 중요한가
• 툴 성능의 차이: 트라이앵글이 많아지면 문제가 생기느냐 안생기느냐
• separate triangle은 깨지거나 이웃을 찾을수 없는 경우가 생기기때문에 잘 안쓴다
• indexed triangle set
• 12: 0,2,1 순서가 중요. 왜냐하면 순서에 따라 노말벡터가 정해지기 때문에. 노말벡터로 안/밖 구분을 해야하니까
• index는 삼각형을 따로띠로 저장한다면 triangle strip은 strip단위로 저장
• stroage requirement(메모리) 감소라는 장점이 있다. 왜냐하면 삼각형의 모든 점을 따로따로 저장하지 않으니까
• neighbor structure
• 삼각형의 이웃삼각형을 저장

11 Sampling and Reconstruction

• reconstruction: sampling의 반대(역함수)
• undersampling: 원래 함수의 freq보다 적게 샘플링을 하여 데이터 손실이 일어남
• 샘플링후 리컨스트럭션 하면 실제로는 원래의 함수가 나오지 않는다. undersampling때문에
• undersampling은 aliasing을 초래한다
• 막기 위한 방법: lowpass filters
• 가우시안을 쓰면 좋은 이유: 미분을 무한번 해도 smooth하게 나오기 때문
• discrete filtering in 2d: 밝기라는 축이 하나 더 추가됨

46쪽~

• Continuous-discrete convolution
• a:signal, f:filter
• sample은 discrete function(가로 i축)
• reconstructed signal은 continuous function(가로 x축)
• sample에 continuous한 필터를 적용해서 reconstruction을 진행하면 결과적으로 continuous한 signal이 나옴

• Resampling
• upsampling: increase sample rate to enlarge image size
• downsampling: decrease sample rate to reduce image size
• pseudocode

• Cont.-disc. convolution in 2D
• 축이 하나 추가됨 -> 4중루프
• 실제로는 separable filters를 사용
• 가로 축소와 세로 축소를 단계별로 따로따로(3중루프를 두차례 진행)
• 좌우상하대칭인 필터는 separable함
• 여러가지 필터
• box: discrete한 것과 continuous한것이 있음. 빠르지만 퀄리티가 구림
• tent: box보다는 느리지만 퀄리티가 좋음
• gaussian: 퀄리티는 좋음 smooth해서. 범위가 무한이기 때문에 아주 느려서 보통은 양쪽을 clip해서 사용함
• b-spline: 가우시안과 비슷하지만 finite해서 가우시안보다 빠름
• catmull-rom: 음수값이 있기 때문에 sharpness가 증가할 수 있음. interpolate함
• michell-netravali: b-spline은 smoothing feature가 있기 때문에 b와 catmull을 합쳐서 사용. overall best quality를 보장
• properties of filters
• 연속성
• impulse response
• interpolating
• ringing/overshoot
• overshoot: 필터의 음수값때문에 생김(catmull-rom)
• ripples: 대부분의 필터는 not ripple free하다. support region이 무한하지 않은 이상
• yucky details
• 가장자리에서 sampling을 할 경우 filter가 잘려서 적용됨
• 다양한 해결방법이 있음
• reducing and enlarging
• types of artifacts
• moire pattern: 샘플링을 너무 적게 해서 low freq. wave가 나오는 것
• dropping pixels하면 moire pattern(aliasing)이 나타남
• gaussian 하면 없음
• 어떤 필터가 좋은지는 푸리에변환 어쩌구저쩌구 근데 알 필요 없음

12 Antialiasing & Compositing

• antialiasing
• pixel의 freq < 실제 line의 freq라서 생기는 문제
• low pass filter를 적용하여 해결
• subpixel을 이용
• gaussian filtering
• subpixel에 weight를 줌
• 가까운 subpixel에 더 큰 weight 부여
• 하지만 느리기때문에 보통은 box filter를 씀
• ray tracing에서도 subpixel(ray)가 많을 수록 품질이 좋아짐. 하지만 그에 비례해서 시간은 오래걸림
• texture
• minification을 하면 moire pattern이 나타남
• magnification을 하면 texture의 pixel boundary가 크게 보임
• bilinear filtering
• 모서리 점들을 linear filtering + 위아래 linear filtering = bilinear filtering
• mipmap
• 여러 resolution 이미지를 미리 계산해놓음
• tri-linear interpolation(레벨 n에서 가져온것과 n+1에서 가져온것을 linear filtering)
• mipmap를 사용하면 blurring 된다는 문제가 있음
• 해결책: summed-area tables(anisotropic filtering)
• 얼마나 찌그러진것을 저장해놓느냐에 따라 4x, 16x anisotropic이라고함

13 2D Spline Curves

• 커브를 만드는게 어렵기때문에 조각조각 piece-wise로 구현한다
• 하나의 2d spline curve에 대해 7쪽처럼 두개의 그래프로 구현되어있음 (parametric cubic spline)
• 같은 입력으로 구현 방식에 따라 커브 모양이 달라짐(점을 지나냐 안지나나에 따라)
• hermite spline
• 시작 위치,속도 + 끝 위치, 속도를 파라미터로
• 모든 spline은 4x4 matrix로 표현됨
• continuity
• c0, c1, c2
• junction에서 기울기가 같으면 c1 continuous하다고 한다
• 32쪽 첫번째 그림에서 geometric하게는 continuous하지 않지만 parametric하게는 continuous함
• convex hull property
• 커브상의 어떤 점이 컨트롤 포인트의 가중평균으로 표현될 경우
• weight의 sum이 1이고 모든 weight가 0과 같거나 클경우
• bezier
• v를 p간의 차의 3배로 정의하면 convex hull property를 만족하게 됨.
• bernstein 어쩌구를 다 합하면 1이 되기 때문

• 56쪽: p1 p2 p3 p4로 커브를 만드나 l1 l2 l3 l4로 만드나 같은 커브를 만들 수 있다
• //과제 skeleton code 그대로 쓰면 된다 + dragging 해서 click하면 소의 control point를 지정하여 그거에 따라 움직일 수 있게. catmull-rom spline이용 - deadline 시험 다음날까지
• spline마다 다른점은 커널 function이 다르기 때문
• b-spline
• approximation spline
• 똑같이 cubic일 때 대부분의 spline은 c1 continuity만을 보장. b-spline은 c2 이상의 continuity를 보장
• 높은 continuity를 보장하는 high-order bezier curve를 사용하지 않는 이유
• bezier curve를 high order로 만들면 overly smooth하게 만들어지기 때문에 sharp corner를 만들 수 없다
• 높은 차수 때문에 computational burden이 생긴다
• chaining cubic bezier curve를 사용하지 않는 이유
• control하기가 어렵고 충분히 smooth해지지 않을 수 있다(c1 continuous하기 때문에)
• b-spline의 curve는 모든 control point와 만나지 않을 수도 있다 - approximation spline이기 때문
• joint point에서 더 smooth하다
• bezier spline과 마찬가지로 모든 차수에서 정의될 수 있다
• 두가지 종류
• non-periodic
• periodic
• 78쪽 matrix 안외워도됨
• 82쪽 공식 몰라두됨
• (0,1,2,3)/(0,0,0,1…) -> 시간 간격
• b-spline control point가 주어지고 그걸 hermit의 control로 conversion하고 싶을 때 85쪽의 공식을 이용 -> recursive subdivision 알고리즘을 돌려서 원래의 curve에 근사하도록 만듦 (충분히 작은 box에 들어갈 때 까지)
• 과제 - 30프레임으로 샘플링한다고 하면 1/30초마다 point를 정의ㅣ…..kinematics 강의자료 참고 math 코드에 있는 함수를 이용

14 Particle Systems

• 불꽃놀이, 눈, 먼지, 물거품, 비, 머리카락, 옷 등에 사용됨
• 파티클마다 연결되어있는 애들에서 오는 force를 모두 더하면 total force를 구할 수 있음
• 간단하기 때문에 병렬화해서 실시간 simulation이 가능하다
• force
• spring force
• l은 rest length(아무 힘도 작용하지 않을 때 길이)
• damping force(마찰력)
• damping을 안하면 무한히 진동하다가 터짐
• 파티클에 작용하는 힘을 모두 구하면 그걸 두번 적분해서 위치를 구할 수 있음
• 스프링 계수 k가 높을수록 잘 움직임
• collision detection은 너무 어려워서스킵

15 Compositing

• 두가지 장면 합치기
• 각 장면의 카메라 움직임 + 색 등 다양한 요소를 맞춰야함 
• 정교하게 합치기 위해 바이너리 이미지 마스크라는걸 씀
• 하지만 artifact가 생길 수 있기 때문에 alpha compositing(partially transparent mask)을 사용
• 두개의 이미지만 섞는 경우 식으로 간단하게 나타낼 수 있다
• 여러개의 이미지를 섞을 때 a@(b@c)이든 a@b)c이든 같은 결과가 나와야 하는데 앞의 식으로 하면 그렇지 않음. 따라서 식을 CC=αA CA+(1−αA)αBCB 이렇게 바꿔서 사용

16 Collision Detection 

• 파티클이 언제 어디를 뚫고 지나가는지 계산
• intersection test를 해서 만나는지 계산하고 inside/outside인지 외적을 통해 계산
• 시간이 오래 걸리기 때문에 barycentric coordinate를 이용하면 좋음
• mesh가 델타만큼 움직일 경우 mesh가 가만히 있다고 가정하고 particle을 델타만큼 옮겨서 계산
• 삼각형삼각형의 경우 
• 메쉬가 둘다 움직일 경우
• sphere tree는 길다란 물체에 대해서 fit이 안좋음