chore: Add GitHub Actions workflow for testing and building Docker image (#2778)

# Description

Please include a summary of the changes and the related issue. Please
also include relevant motivation and context.

## Checklist before requesting a review

Please delete options that are not relevant.

- [ ] My code follows the style guidelines of this project
- [ ] I have performed a self-review of my code
- [ ] I have commented hard-to-understand areas
- [ ] I have ideally added tests that prove my fix is effective or that
my feature works
- [ ] New and existing unit tests pass locally with my changes
- [ ] Any dependent changes have been merged

## Screenshots (if appropriate):

.github/workflows/test-build-image.yml

@@ -0,0 +1,46 @@
+name: Test Build Image
+
+on:
+  pull_request:
+    branches: ["main"]
+    paths:
+      - 'backend/**'
+
+jobs:
+  build:
+    name: Build Docker Image
+    runs-on: ubuntu-latest
+    strategy:
+      matrix:
+        platform: [linux/amd64, linux/arm64]
+
+    steps:
+      - name: Checkout
+        uses: actions/checkout@b4ffde65f46336ab88eb53be808477a3936bae11 # v4
+
+      - name: Login to GitHub Container Registry
+        uses: docker/login-action@465a07811f14bebb1938fbed4728c6a1ff8901fc # v2
+        with:
+          registry: ghcr.io
+          username: ${{ github.actor }}
+          password: ${{ secrets.GITHUB_TOKEN }}
+
+      - uses: docker/setup-qemu-action@68827325e0b33c7199eb31dd4e31fbe9023e06e3 # v3
+      - name: Set up Docker Buildx
+        uses: docker/setup-buildx-action@f95db51fddba0c2d1ec667646a06c2ce06100226 # v3
+
+      - name: Create Docker Cacha Storage Backend
+        run: |
+          docker buildx create --use --driver=docker-container
+
+      - name: Build image
+        id: build-image
+        uses: docker/build-push-action@4a13e500e55cf31b7a5d59a38ab2040ab0f42f56 # v5
+        env:
+          ECR_REGISTRY: ${{ steps.login-ecr.outputs.registry }}
+          IMAGE_TAG: ${{ github.sha }}
+        with:
+          context: ./backend/
+          platforms: ${{ matrix.platform }}
+          cache-from: type=gha
+          cache-to: type=gha,mode=max